text:
\begin{document}

\begin{abstract} We classify circle actions with the Rokhlin property on separable, nuclear, unital $C^*$-algebras that absorb the Cuntz
algebra $\mathcal{O}_2$ tensorially. The invariant we use is the induced action on the primitive
ideal space. In particular, any two such actions on $\mathcal{O}_2$ are
conjugate. The range of the invariant is discussed and computed for a class of $\mathcal{O}_2$-absorbing
$C^*$-algebras. Additionally, we show that circle actions with the Rokhlin property on separable $\mathcal{O}_2$-absorbing unital $C^*$-algebras
are generic, in a suitable sense.\\
\indent A crucial result in our work, which is of independent interest, is as follows: if $\alpha$ is a
circle action with the Rokhlin property on a unital, separable $C^*$-algebra $A$ that absorbs the UHF-algebra $M_{n^\infty}$ tensorially, then
the restriction of $\alpha$ to $\mathbb{Z}_n$ has the Rokhlin property. If $A$ moreover absorbs $\mathcal{O}_2$, then a circle
action has the Rokhlin property if and only if all of its restrictions to finite subgroups have the Rokhlin property. We give several
examples to show that our results cannot be extended to all unital Kirchberg algebras in the UCT class. \end{abstract}

\maketitle

\tableofcontents

\section{Introduction}

The interplay between \ca s and dynamics has a long and rich history. Crossed products have provided some of the most interesting examples of
\ca s. Some algebraic properties are preserved under formation of crossed products in great generality. For example, crossed products of type
I $C^*$-algebras by compact groups are type I, and crossed products of nuclear $C^*$-algebras by amenable groups are nuclear, regardless of the
action. On the other hand, for preservation of other (usually stronger) properties, one must assume some kind of freeness condition on the action.
This is best seen in the commutative setting, where Theorem 1.1.1 in \cite{phillips equivariant k-theory book} shows how free actions on compact
spaces enjoy a number of nice analytic and algebraic properties.\\
\indent In the noncommutative setting, there are several different notions of freeness for actions, and many of them are surveyed in
\cite{phillips freeness of actions}. For finite groups, and in roughly decreasing
order of strength, there are: the \Rp\ (see \cite{izumi finite group actions I}), finite Rokhlin dimension (see \cite{finite rokhlin dim}), the tracial
 Rokhlin property (see \cite{phillips tracial rp first paper}), pointwise outerness, and hereditary saturation
(see \cite{phillips equivariant k-theory book}), just to mention a few. Among these, the \Rp\ is the strongest one, and is therefore less common that
the other notions of freeness. For example, it implies that the algebra has non-trivial projections, ruling out the existence of such actions on many
$C^*$-algebras of interest, such as the Jiang-Su algebra $\mathcal{Z}$. There are also less obvious $K$-theoretic obstructions
to the \Rp. See Theorem 3.13 in \cite{izumi finite group actions I}. On the other hand, the \Rp\ implies very strong structure preservation results
for crossed products (see Theorem 2.3 in \cite{phillips freeness of actions} for a list of properties that are preserved by Rokhlin actions, and
see \cite{hirshberg winter}, \cite{osaka phillips} and \cite{phillips tracial rp first paper} for the proofs of most of them), and it is the hypothesis
in most theorems on classification of group actions (see Theorems 3.4 and 3.5 in \cite{izumi finite group actions II}, and see Theorem 4.7 in
\cite{gardella Kirchberg 1}). These have been the main uses
of the \Rp: obtaining structure results of the \cp, and classification of actions.\\
\indent Besides finite groups, the \Rp\ has been extensively studied for automorphisms (see \cite{herman ocneanu automorphisms with rp} and
\cite{kishimoto automorphisms with rp}) and flows (see \cite{kishimoto flows}). In \cite{hirshberg winter}, Hirshberg and Winter introduced the \Rp\
for an action of a second countable compact group on a unital \ca, and proved that absorption of a strongly self-absorbing \ca\ and approximate
divisibility pass to crossed products by such actions. \\
\indent It is natural to try to generalize the results on the structure of the \cp\ in \cite{osaka phillips} to arbitrary compact groups. This will
be done in \cite{gardella rokhlin property}. Another natural direction is to explore the classification of Rokhlin actions of compact groups on
certain classes of classifiable \cas, generalizing or at least complementing Izumi's work for finite group actions with the \Rp. This work focuses
on Rokhlin actions of the circle on $\Ot$-absobring \ca s, where $\Ot$ is the Cuntz algebra on two generators (homomorphisms into such algebras were
classified in \cite{kirchberg}), with the goal of classifying them up to
conjugacy by an approximately inner automorphism. See Theorem \ref{classification on Ot} below. Circle actions with the \Rp\ on unital Kirchberg
algebras are studied in \cite{gardella Kirchberg 1} and \cite{gardella Kirchberg 2}. \\
\indent Similarly to what happens with finite groups, Rokhlin actions of compact groups are rare, and there are \ca s that do not have any
action of any compact group with the \Rp, such as the Cuntz algebra $\OI$ or the Jiang-Su algebra $\mathcal{Z}$. In this sense, $C^*$-algebras that
absorb $\Ot$ form a distinguished class since they have many actions
with the \Rp. In fact, the set of all circle actions with the \Rp\ on a separable $\Ot$-absorbing \ca\ is a dense $G_\delta$ set in the space of all
circle actions. See Theorem \ref{rokhlin action are a Gdelta set}. \\
\ \\
\indent This paper is organized as follows. In Section 2, we establish the notation that will be used throughout, as well as some standard results and
definitions that will be relevant thereafter. In Section 3, we introduce the definition of the Rokhlin property for circle actions on \uca, and derive
some of its basic properties which will be frequently used in the later sections. We also provide a number of examples of circle actions on
\ca s with the \Rp, mostly on simple \ca s. In contrast, we show in Theorem \ref{no direct limit actions with the Rp on AF-algebras} that
no direct limit action of the circle on a UHF-algebra can have the \Rp. We specialize to circle actions on $C^*$-algebras
that absorb the Cuntz algebra $\Ot$ in Section 4.
This class of \ca s is special from the point of view of the Rokhlin property, for at least two reasons. First, circle actions with the \Rp\ are
generic on separable, unital, $\Ot$-absorbing \ca s (Theorem \ref{rokhlin action are a Gdelta set}), a fact that
should be contrasted with Theorem \ref{no direct limit actions with the Rp on AF-algebras}. Second, if $A$ is a \ca\ as above and $\alpha
\colon \T\to\Aut(A)$ is an action with the \Rp, then $\alpha|_{\Z_n}\colon \Z_n\to\Aut(A)$ has the \Rp\ for every $n$ in $\N$ (Theorem
\ref{restriction of Rp has Rp}). Again, this is special
to algebras that absorb $\Ot$, as Example \ref{eg restriction on S1} and Example \ref{eg restriction on pi ca} show. \\
\indent Finally, in Section 5 we use the results of Section 4 and a Theorem in \cite{kirchberg} to classify circle actions on
nuclear $\Ot$-absorbing \ca s with the Rokhlin property up to conjugacy by an approximately inner automorphism. See Theorem \ref{classification on Ot}. As
an application, we show that a circle action on an $\Ot$-absorbing \ca\ has the \Rp\ \ifo infinitely many restrictions to finite subgroups have the
\Rp. See Theorem \ref{equivalences restrictions}. We also exhibit several examples that show how these results fail for general Kirchberg algebras.
We also discuss the range of the invariant used in Theorem \ref{classification on Ot}. A complete description of the
range of the invariant, despite being open in the general case, can be obtained if one assumes that the primitive ideal space of
the \ca\ is Hausdorff and finite dimensional. See Proposition \ref{range of the invariant}. \\
\ \\
\noindent \textbf{Acknowledgements.} Some of this work was carried out while the author was participating in the Thematic Program
on Abstract Harmonic Analysis, Banach and Operator Algebras, at the Fields Institute for Research in Mathematical Sciences
at the University of Toronto, in January-June 2014. He wishes to thank the Fields Institute for its hospitality and for providing a
stimulating research environment.\\
\indent The author is grateful to Chris Phillips for several helpful conversations and for carefully reading a first draft of the present
work, as well as Eberhard Kirchberg for valuable feedback and electronic correspondence. Finally, he would
also like express his gratitude to Luis Santiago for calling his attention to the results in the paper \cite{kirchberg}, and for his
encouragement, especially in the early stages of this project.

\section{Notation and preliminaries}
We adopt the convention that $\{0\}$ is not a \uca, this is, we require that $1\neq 0$ in a \uca. We take $\N=\{1,2,\ldots\}$. For a separable \ca\ $A$, we denote by $\Aut(A)$ the automorphism group of $A$, which is equipped with the topology of pointwise norm convergence. In this topology, a sequence $(\varphi_n)_{n\in\N}$ converges to $\varphi\in\Aut(A)$ if and only if for every $\varepsilon>0$ and every compact set $F\subseteq A$, there exists $m\in\N$ such that $\|\varphi_m(a)-\varphi(a)\|<\varepsilon$ for all $a$ in $F$.\\
\indent If $A$ is moreover unital, then $\U(A)$ denotes the unitary group of $A$, and two automorphisms $\varphi$ and $\psi$ of $A$ are said to be approximately unitarily equivalent if $\varphi\circ\psi^{-1}$ is approximately inner.\\
\indent For a locally compact group $G$, an action of $G$ on $A$ is always assumed to be a \emph{continuous} group homomorphism from $G$ into $\Aut(A)$, unless otherwise stated. If $\alpha\colon G\to\Aut(A)$ is an action of $G$ on $A$, then we will denote by $A^\alpha$ the fixed-point subalgebra of $A$ under $\alpha$. \\
\indent For a \ca\ $A$, we set
\begin{align*} \ell^\I(\N,A)&=\left\{(a_n)_{n\in\N}\in A^{\N}\colon \sup_{n\in\N}\|a_n\|<\I\right\};\\
c_0(\N,A)&=\left\{(a_n)_{n\in\N}\in\ell^\I(\N,A)\colon \lim\limits_{n\to\I}\|a_n\|= 0\right\};\\
A_\I&= \ell^\I(\N,A)/c_0(\N,A).\end{align*}
We identify $A$ with the constant sequences in $\ell^\I(\N,A)$ and with their image in $A_\I$. We write $A_\I\cap A'$ for the central sequence algebra of $A$, that is, the relative commutant of $A$ in $A_\I$. For a sequence $(a_n)_{n\in\N}$ in $A$, we denote by $\overline{(a_n)}_{n\in\N}$ its image in $A_\I$. We have
$$A_\I\cap A'=\left\{\overline{(a_n)}_{n\in\N}\in A_\I \colon \lim_{n\to\I}\|a_na-aa_n\|\to 0 \mbox{ for all } a\in A\right\}.$$
\indent If $\alpha\colon G\to\Aut(A)$ is an action of $G$ on $A$, then there are actions of $G$ on $A_\I$ and on $A_\I\cap A'$, both denoted by $\alpha_\I$. Note that unless the group $G$ is discrete, these actions will in general not be continuous.\\
\indent Given $n\in\N\cup\{\I\}$, we denote by $\mathcal{O}_n$ the Cuntz algebra with canonical generators $\{s_j\}_{j=1}^n$ satisfying the usual relations (see for example Section 4.2 in \cite{Rordam classification}).\\
\indent The circle will usually be denoted by $\T$ whenever it is regarded as a group, and by $\T $ whenever it is regarded as a topological space. The finite cyclic group of order $n$ will be denote by $\Z_n$, and we will usually identify $\Z_n$ with the $n$-th roots of unity in $\T$, and in this fashion we will regard $\Z_n$ as sitting inside the circle.\\
\ \\
\indent We recall the definition of the \Rp\ for a finite group action on a \uca.

\begin{df} Let $A$ be a \uca, let $G$ be a finite group, and let $\alpha\colon G\to\Aut(A)$ be an action. We say that $\alpha$ has the \emph{\Rp} if for every $\varepsilon>0$ and for every finite set $F\subseteq A$ there exist orthogonal projections $e_g$ in $A$ for $g$ in $G$ such that
\be\item $\|\alpha_g(e_h)-e_{gh}\|<\varepsilon$ for all $g$ and $h$ in $G$
\item $\|e_ga-ae_g\|<\varepsilon$ for all $g$ in $G$ and all $a$ in $F$
\item $\sum\limits_{g\in G}e_g=1$.\ee\end{df}

The definition of the \Rp\ for finite groups was originally introduced by Izumi in \cite{izumi finite group actions I}, although a similar
notion has been studied by Herman and Jones in \cite{herman jones} for $\Z_2$ actions on UHF-algebras, and by Herman and Ocneanu in
\cite{herman ocneanu automorphisms with rp} for integer actions. The Rokhlin property also played a crucial role in the classification
of finite group actions on von Neumann algebras.\\
\indent Izumi's definition is in terms of central
sequences and therefore one gets positive elements $b_g$ for $g$ in $G$ such that the following quantities are abitrarily small for all $g$ and $h$ in $G$ and all $a$ in $F$:
\bi\item $\|\alpha_g(b_h)-b_{gh}\|$
\item $\|b_ga-ab_g\|$
\item $\left\|1-\sum\limits_{g\in G}b_g\right\|$
\item $\|b_gb_h-\delta_{g,h}b_g\|$\ei
However, using that $C(G)$ is semiprojective (see Definition 14.1.1 in \cite{Loring}), it is easy to see that the two definitions are equivalent
whenever the algebra is separable. \\
\indent The following is part of Proposition 2.14 in \cite{phillips freeness of actions}, and we include the proof for the convenience of the reader.
This result should be compared with Example \ref{eg restriction on S1} and Example \ref{eg restriction on pi ca}.

\begin{prop}\label{finite gp Rp properties} Let $A$ be a \uca, let $G$ be a finite group, and let $\alpha\colon G\to\Aut(A)$ be an action with the \Rp.
If $H\subseteq G$ is a subgroup, then $\alpha|_H$ has the \Rp.\end{prop}
\begin{proof} Set $n=\card(G/H)$. Given $\varepsilon>0$ and a finite subset $F\subseteq A$, choose projections $e_g$ for $g$ in $G$ as in the
definition of the \Rp\ for $F$ and $\frac{\ep}{n}$. We claim that the projections $f_h=\sum\limits_{\overline{x}\in G/H} e_{hx}$ for $h$ in $H$, form a
family of Rokhlin projections for the action $\alpha|_H$, the finite set $F$ and tolerance $\varepsilon$. \\
\indent Given $h$ and $k$ in $H$, we have
\begin{align*} \|\alpha_k(f_h)-f_{kh}\|&=\left\|\sum\limits_{\overline{x}\in G/H} \alpha_k(e_{hx})-e_{khx}\right\|\\
 &\leq \sum\limits_{\overline{x}\in G/H}\left\|  \alpha_k(e_{hx})-e_{khx}\right\| \leq \mbox{card}(G/H)\frac{\ep}{n}=\ep.
\end{align*}
Finally, for $a$ in $F$ and $h$ in $H$, we have
\[\|af_h-f_ha\|\leq \sum\limits_{\overline{x}\in G/H}\left\| ae_{hx}-e_{hx}a\right\|< \ep.\]
\end{proof}

\section{The \Rp\ for circle actions}

Hirshberg and Winter defined the \Rp\ for an arbitrary action of a compact, second countable group in \cite{hirshberg winter}. In the case of
the circle, and using semiprojectivity of $C(S^1)$, one can show that their definition is equivalent to the following.

\begin{df}\label{def RP} Let $A$ be a unital \ca\ and let $\alpha\colon\T\to\Aut(A)$ be a continuous action. Then $\alpha$ is said to have the \emph{Rokhlin property} if for every finite subset $F\subseteq A$ and every $\varepsilon>0$, there exists a unitary $u\in\U(A)$ such that
\be\item $\|\alpha_\zeta(u)-\zeta u\|<\varepsilon$ for all $\zeta$ in $\T$ and
\item $\|ua-au\|<\varepsilon$ for all $a$ in $F$.\ee\end{df}

\begin{rem} In order to check condition (2) in Definition \ref{def RP}, it is enough to consider finite subsets of any set of
 generators of $A$. It is also immediate to show that if in
Definition \ref{def RP} one allows the set $F$ to be compact instead of finite, one obtains an equivalent definition.
These easy observations will be used repeatedly and without reference.\end{rem}

We present some basic properties of Rokhlin circle actions, some of which resemble those of free actions on spaces. For instance, the
proposition below is the analog of the fact that a diagonal action on a product space is free if one of the factors is free.

\begin{prop}\label{Properties RP}
Let $A$ be a \uca, and let $\af \colon \T \to \Aut (A)$ have the \Rp. If $\bt \colon \T \to \Aut (B)$ is any action of $\T$ on a \uca,
then the tensor product action $\zeta \mapsto \af_\zeta \otimes \bt_\zeta$ of $\T$ on $\Aut (A \otimes B),$ for any $C^*$-tensor product
on which it is defined, has the \Rp.\end{prop}
\begin{proof} Let $\varepsilon>0$ and let $F'\subseteq A$ and $F''\subseteq B$ be finite subsets of the respective unit balls of $A$ and $B$,
and set
$$F=\{a\otimes b\colon a\in F',b\in F''\},$$
which is a finite subset of $A\otimes B$. Using the \Rp\ for $\alpha$, choose a unitary $u$ in $A$ such that the conditions in Definition \ref{def RP}
are satisfied for $\ep$ and $F'$.
Set $v=u\otimes 1\in \U(A\otimes B)$. For $x=a\otimes b$ in $F$, we have
$$\|vx-xv\|=\|(ua-au)\otimes b\|\leq \|ua-au\|\|b\|<\ep.$$
On the other hand,
\begin{align*}\|(\alpha\otimes\beta)_\zeta(v)-\zeta v\|&=\|(\alpha_\zeta(u)\otimes \beta_\zeta(1))-\zeta (u\otimes 1)\|=\|\alpha_\zeta(u)-\zeta u\|<\varepsilon,\end{align*}
for all $\zeta\in\T$, which finishes the proof. \end{proof}

We point out that a tensor product action may have the \Rp\ without any of the tensor factors having the \Rp, even if one of the actions
is the trivial action. See Example \ref{nice example on Ot without proof} and Proposition \ref{nice example on Ot with proof}. This is analogous to
the fact that a diagonal action on a product
space may be free without any of the factors being free, except that examples with the trivial action do not exist.\\
\indent The proposition below is the analog of the fact that an equivariant inverse limit of free actions is free.

\begin{prop} \label{direct limit and Rp} If $A = \varinjlim (A_n,\iota_n)$ is a direct limit of \ca s with unital maps, and $\af \colon \T \to \Aut (A)$ is an action obtained as the direct limit of actions $\af^{(n)} \colon \T \to \Aut (A_n),$ such that $\af^{(n)}$ has the \Rp\  for all $n,$ then $\af$ has the \Rp. \end{prop}
\begin{proof} Let $F\subseteq A$ be a finite set, and let $\varepsilon>0$. Write $F=\{a_1,\ldots,a_N\}$. Since $\bigcup\limits_{n\in\N} \iota_{\I,n}(A_n)$ is dense in $A$, there exist $n\in\N$ and $F'=\{b_1,\ldots,b_n\}\subseteq A_n$ such that $\|a_j-\iota_{\I,n}(b_j)\|<\frac{\ep}{3}$ for $j=1,\ldots,N$. Since $\af^{(n)}$ has the \Rp, there exists a unitary $u$ in $A_n$ such that $\|\af^{(n)}_\zeta(u)-\zeta u\|<\frac{\ep}{3}$ for all $\zeta$ in $\T$ and $\|b_ju-ub_j\|<\frac{\ep}{2}$ for all $j=1,\ldots,N$. Notice that $\iota_{\I,n}(u)$ is a unitary in $A$, since the connecting maps are unital. Moreover, if $\zeta\in\T$, then
$$\|\af_\zeta(\iota_{\I,n}(u))-\zeta \iota_{\I,n}(u)\|=\|\iota_{\I,n}(\alpha^{(n)}_\zt(u))-\iota_{\I,n}(\zeta u)\| <\frac{\ep}{3}<\varepsilon.$$
Finally,
\begin{align*} &\|\iota_{\I,n}(u)a_j-a_j\iota_{\I,n}(u)\|\\
 &\ \ \ \ \ \ \leq \|\iota_{\I,n}(u)a_j-\iota_{\I,n}(u)\iota_{n,\I}(b_j)\|+\|\iota_{\I,n}(u)\iota_{n,\I}(b_j)-\iota_{n,\I}(b_j)\iota_{\I,n}(u)\|\\
&\ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\|\iota_{n,\I}(b_j)\iota_{\I,n}(u)-a_j\iota_{\I,n}(u)\|\\
&\ \ \ \ \ \ \leq\|a_j-\iota_{n,\I}(b_j)\|+\|ub_j-b_ju\|+\|\iota_{n,\I}(b_j)-a_j\|\\
&\ \ \ \ \ \ <\frac{\ep}{3}+\frac{\ep}{3}+\frac{\ep}{3}=\varepsilon.\end{align*}
Hence $\iota_{\I,n}(u)$ is the desired unitary for $F$ and $\varepsilon$, and thus $\alpha$ has the \Rp. \end{proof}

We have the following convenient result, which turns out to be crucial in some proofs, in particular in the classification of
Rokhlin actions of the circle on Kirchberg algebras; see \cite{gardella Kirchberg 1}.

\begin{prop}\label{can replace estimate by equality} Let $A$ be a \uca, let $\alpha\colon \T\to\Aut(A)$ be an action with the Rokhlin property,
let $\ep>0$ and let $F\subseteq A$ be a finite subset. Then there exists a unitary $u$ in $A$ such that
\be\item $\alpha_\zeta(u)=\zeta u$ for all $\zeta\in\T$.
\item $\|ua-au\|<\varepsilon$ for all $a$ in $F$.\ee\end{prop}
The definition of the Rokhlin property differs from the conclusion of this proposition in that in condition (1), one only requires
$\|\alpha_\zeta(u)-\zeta u\|<\varepsilon$ for all $\zeta\in\T$.
\begin{proof} One can normalize $F$ so that $\|a\|\leq 1$ for all $a$ in $F$.
Set $\ep_0=\min\left\{\frac{1}{3},\frac{\ep}{7+2\ep}\right\}$. Choose a unitary $v$ in $A$ such that conditions (1) and (2) in
Definition \ref{def RP} are satisfied for the finite set $F$ with
$\varepsilon_0$ in place of $\varepsilon$. Denote by $\mu$ the normalized Haar measure on $\T$, and set
$$x =\int_{\T}\overline{\zeta}\alpha_\zeta(v)\ d\mu(\zeta).$$
Then $\|x\|\leq 1$ and $\|x-v\|\leq \varepsilon_0$. One checks that $\|x^*x-1\|\leq 2\varepsilon_0< 1$ and that $\alpha_\zt(x)=\zt x$
for all $\zt$ in $\T$.\\
\indent We have
$$\left\|(x^*x)^{-1}\right\|\leq \frac{1}{1-\|1-x^*x\|}\leq\frac{1}{1-2\ep_0},$$
and thus $\left\|(x^*x)^{-\frac{1}{2}}\right\|\leq\frac{1}{\sqrt{1-2\ep_0}}$.\\
\indent Set $u = x(x^*x)^{-\frac{1}{2}}$, which is a unitary in $A$. Using $\|x\|\leq 1$ at the first step,
and $0\leq 1-(x^*x)^{\frac{1}{2}}\leq 1-x^*x$ at the third step, we get
\begin{align*} \|u-x\| & \leq \left\|(x^*x)^{-\frac{1}{2}}-1\right\|\\
& \leq \left\|(x^*x)^{-\frac{1}{2}}\right\|\left\|1-(x^*x)^{\frac{1}{2}}\right\|\\
& \leq \frac{1}{\sqrt{1-2\ep_0}}\|1-x^*x\|\leq \frac{2\ep_0}{\sqrt{1-2\ep_0}}.
\end{align*}
We deduce that
$$ \|u-v\|\leq \frac{2\ep_0}{\sqrt{1-2\ep_0}}+\ep_0.$$
For $\zt$ in $\T$, we have $\alpha_\zt(x^*x)=x^*x$ and hence $\alpha_\zeta(u)=\zeta u,$
so $u$ satisfies condition (1) of the statement. Finally, for $a$ in $F$, we have
\begin{align*} \|ua-au\| & \leq \|ua-va\|+\|va-av\|+\|av-au\|\\
&< \|u-v\|\|a\|+\ep_0+\|a\|\|v-u\|\\
&\leq  \frac{4\ep_0}{\sqrt{1-2\ep_0}}+3 \ep_0< \frac{4\ep_0}{1-2\ep_0}+3\ep_0\\
&< \frac{7\ep_0}{1-2\ep_0}<\ep,
\end{align*}
as desired.
\end{proof}

We now turn to examples of circle actions with the Rokhlin property. As in the finite group case, the \Rp\ is rare, and it is challenging to
construct many examples on simple \ca s. We will give an explicit construction of a family of circle actions with the \Rp\ on simple \ATa s,
and also on the Cuntz algebra $\Ot$. For more examples on purely inifinite \ca s, see \cite{gardella Kirchberg 2} (the construction of the
examples there is not explicit).

\begin{eg}\label{eg of Rp on simple AT-algebra} This is an example of a circle action on a simple, unital \ATa\ with the Rokhlin property. For
$n\in\N$, let $A_n=C(\T )\otimes M_{n!}$. Consider the action $\alpha^{(n)}\colon\T\to\Aut(A_n)$ given by $\alpha^{(n)}_\zeta(f)(w)=f(\zeta^{-1}w)$
for $\zeta$ and $w$ in $\T$ and for $f$ in $A_n\cong C(\T ,M_{n!})$. In other words, $\alpha^{(n)}$ is the tensor product of the regular representation of
$\T $ with the trivial action on $M_{n!}$. Then $\alpha^{(n)}$ has the Rokhlin property by Proposition \ref{Properties RP}.\\
\indent We construct a direct limit algebra $A=\dirlim (A_n,\iota_n)$ as follows. Fix a countable dense subset $X=\{x_1,x_2,x_3,\ldots\}\subseteq
\T $, and assume that $x_1=1$. With $f_{x}(\zeta)=f(x^{-1}\zeta)$ for $f$ in $A_n$, for $x$ in $X$ and for $\zeta$ in $\T $, define maps $\iota_n\colon A_n\to
A_{n+1}$ for $n\in\N$, by
$$\iota_n(f)=\left(
               \begin{array}{cccc}
                 f_{1} & 0 & \cdots & 0 \\
                 0 & f_{x_2} & \cdots & 0 \\
                 \vdots & \vdots& \ddots & \vdots \\
                 0 & 0 & \cdots & f_{x_n} \\
               \end{array}
             \right) \ \ \mbox{ for } f \in A_n.$$
Then $\iota_n$ is unital and injective, for all $n\in\N$. The limit algebra $A=\dirlim(A_n,\iota_n)$ is a unital \ATa.\\
\indent It is easy to check that
$$\iota_n\circ\alpha^{(n)}_\zeta=\alpha^{(n+1)}_\zeta\circ\iota_n$$
for all $n\in\N$ and all $\zeta\in\T$, so
that $\left(\alpha^{(n)}\right)_{n\in\N}$ induces a direct limit action $\alpha=\varinjlim \alpha^{(n)}$ of $\T$ on $A$. Then $\alpha$ has the
Rokhlin property by Proposition \ref{direct limit and Rp}. Simplicity of $A$ follows from Proposition 2.1 in \cite{dadarlat nagy nemethi pasnicu},
since $X$ is assumed to be dense in $\T$.\end{eg}

Using the absorption properties of $\Ot$, we can construct an action of the circle on $\Ot$ with the \Rp.

\begin{eg}\label{eg of Rp on Ot} Let $A$ and $\alpha$ be as in the example above. Then $A$ is a separable, unital, nuclear, simple \ca. Use Theorem 7.1.12 in \cite{Rordam classification} to choose an isomorphism $\varphi\colon A\otimes \Ot\to \Ot$, and define an action $\gamma\colon\T\to\Aut(\Ot)$ by $\gamma_\zeta=\varphi\circ (\alpha_\zeta\otimes \id_{\Ot})\circ \varphi^{-1}$ for $\zeta\in\T$. Since $\alpha$ has the \Rp, it follows from Proposition \ref{Properties RP} that $\gamma$ has the \Rp\ as well.\end{eg}

\begin{eg} If $A$ is any unital \ca\ such that $A\otimes \Ot\cong A$, then one can construct a circle action on $A$ with the Rokhlin property by tensoring the trivial action on $A$ with any action on $\Ot$ with the \Rp, such as the one constructed in Example \ref{eg of Rp on Ot}.\end{eg}

Our next goal is to prove some non-existence results for circle actions with the \Rp. We begin with an easy lemma which already rules out
such actions on matrix algebras.

\begin{lemma} \label{lma: pointwise outer}
Let $A$ be a \uca\ and let $\alpha\colon \T\to\Aut(A)$ be an action with the \Rp. Then $\alpha_\zt$ is not inner for all $\zt$ in $\T$
 with $\zt\neq 1$.\end{lemma}
\begin{proof}
Let $\zt$ in $\T\setminus\{1\}$, and assume that there exists a unitary $v$ in $A$ such that $\alpha_\zt=\Ad(v)$. Let $\ep>0$ satisfy
$\ep<\frac{|1-\zt|}{2}$ and using the \Rp\ for $\alpha$, find a unitary $u$ in $A$ such that, in particular, $\|\alpha_\zt(u)-\zt u\|<\ep$
and $\|uv-vu\|<\ep$. Then
\begin{align*}
\ep &>\|\alpha_\zt(u)-\zt(u)\|=\|vuv^*-\zt u\|\geq \left| \|u-\zt u\| - \|vuv^*-u\|\right|\\
&=|1-\zt|-\|vu-uv\|> \frac{|1-\zt|}{2}>\ep,
\end{align*}
which is a contradiction. This shows that $\alpha_\zt$ is not inner.
\end{proof}

\begin{cor} Let $n$ in $\N$. Then there are no actions of the circle on $M_n$ with the \Rp.\end{cor}
\begin{proof} This is an immediate consequence of Lemma \ref{lma: pointwise outer}, since every automorphism of $M_n$ is inner.\end{proof}

We will generalize the corollary above in Theorem \ref{no direct limit actions with the Rp on AF-algebras} below, where we show that there are
no direct limit actions of the circle with the \Rp\ on UHF-algebras. We need a series of preliminary lemmas.

\begin{nota} Let $n\in\N$. We denote by $\U_n(\C)$ the unitary group of $M_n$, and by $P\U_n(\C)$ the quotient group $P\U_n(\C)=\U_n(\C)/\T $,
where $\T $ sits inside of $\U_n(\C)$ via $\zeta\mapsto \mbox{diag}(\zeta,\ldots,\zeta)$. If $\mathcal{Z}(\U_n(\C))$ denotes the center of
the unitary group $\U_n(\C)$, we have $\mathcal{Z}(\U_n(\C))=\T \cdot 1_n$.\end{nota}

\begin{prop}\label{Lemma actions on Mn} Let $n\in\N$ and let $\gamma\colon\T\to\Aut(M_n)$ be a continuous action. Then there exists a
 continuous map $v\colon \T\to\U_n(\C)$ such that $\gamma_\zeta=\Ad(v(\zeta))$ for all $\zeta$ in $\T$.
\end{prop}

\begin{proof} Recall that every automorphism of $M_n$ is inner, so that for every $\zeta\in\T$ there exists a unitary $u(\zeta)\in\U_n(\C)$ such that $\alpha_\zeta=\Ad(u(\zeta))$. Moreover, $u(\zeta)$ is uniquely determined up to multiplication by elements of $\T =\mathcal{Z}(\U_n(\C))$ and hence $\gamma_\zeta$ determines a continuous group homomorphism $u\colon \T\to P\U_n(\C)$. Denote by $\rho\colon \U_n(\C)\to P\U_n(\C)$ the canonical projection. We want to solve the following lifting problem:
\beqa\xymatrix{ & \U_n(\C)\ar[d]^-{\rho}\\
\T\ar@{-->}[ur]^-{v}\ar[r]_-{u}&P\U_n(\C).}\eeqa
The map $u$ determines an element $[u]\in \pi_1(P\U_n(\C))$ and $\rho$ induces a group homomorphism $\pi_1(\rho)\colon \pi_1(\U_n(\C))\to\pi_1(P\U_n(\C))$. The quotient map $\rho\colon \U_n(\C)\to P\U_n(\C)$ is actually a fiber bundle, since $\U_n(\C)$ is a manifold and the action of $\T $ on $\U_n(\C)$ is free. See the Theorem in Section 4.1 of \cite{palais}. The long exact sequence in homotopy for this fiber bundle is
\beqa\xymatrix{ \cdots \ar[r] &\pi_1(\T )\ar[r] & \pi_1(\U_n(\C))\ar[r]^-{\pi_1(\rho)} &\pi_1(P\U_n(\C)) \ar[r]& \pi_0(\T ).}\eeqa
Recall that $\pi_1(\U_n(\C))\cong\Z$, and that $\pi_0(\T )\cong 0$. The map $\pi_1(\T )\to \pi_1(\U_n(\C))$ is induced by $\zeta\mapsto \diag(\zeta,\ldots,\zeta)$, which on $\pi_1$ corresponds to multiplication by $n$. In other words, the above exact sequence is
\beqa\xymatrix{ \cdots \ar[r] &\Z\ar[r]^-{\cdot n} & \Z\ar[r]^-{\pi_1(\rho)} &\pi_1(P\U_n(\C)) \ar[r]& 0,}\eeqa
which implies that $\pi_1(P\U_n(\C))\cong \Z_n$ and that the map $\pi_1(\rho)$ is surjective. It follows that $u$ is homotopic to a map $\widehat{u}\colon \T\to P\U_n(\C)$ that is liftable. The homotopy lifting property for fiber bundles implies that $u$ itself is liftable, this is, there exists a continuous map $v\colon \T\to\U_n(\C)$ such that $u(\zeta)=\rho(v(\zeta))$ for all $\zeta\in\T$. (See the paragraph below Theorem 4.41 in \cite{hatcher} for the definition of the homotopy lifting property. Proposition 4.48 in \cite{hatcher} shows that every fiber bundle has this property.) \end{proof}

\begin{lemma}\label{unitaries in Mn at least 2 away} Let $n\in\N$ and let $v\colon \T\to \U_n(\C)$ be any continuous map. Then for every $u$ in $\U_n(\C)$, there exists $\zeta$ in $\T$ such that
$$\|v(\zeta)uv(\zeta)^*-\zeta u\|\geq 2.$$\end{lemma}
\begin{proof} Assume that there exists $u\in\U_n(\C)$ such that $\|v(\zeta)uv(\zeta)^*-\zeta u\|< 2$ for all $\zeta\in\T$. Set
$w(\zeta)=\overline{\zeta}v(\zeta)uv(\zeta)^*u^*$. Then $\|w(\zeta)-1\|<2$ for all $\zeta\in\T$, and thus $w$, as an element of
$C(\T,\U(M_n))=\U(C(\T,M_n))$, is homotopic to 1. Hence, the map $\T \to \C\setminus\{0\}$ given by $\zeta\mapsto \det(v(\zeta))$
is homotopic to the constant path, and therefore the winding number of $\zeta\mapsto \det(w(\zeta))$ is zero. On the other hand,
$$\det(w(\zeta))=\det(\overline{\zeta}v(\zeta)uv(\zeta)^*u^*)=\overline{\zeta}^n,$$
so the winding number is actually $-n$. This is a contradiction.\end{proof}

\begin{thm}\label{no direct limit actions with the Rp on AF-algebras} Assume that $A=\varinjlim(M_{k_n},\iota_n)$ is an unital UHF-algebra with
unital connecting maps. If $\alpha=\varinjlim \alpha^{(n)}$ is a direct limit action of the circle on $A$, then $\alpha$ does not have the
Rokhlin property.\end{thm}
\begin{proof} Assume that $\alpha$ has the Rokhlin property. Let $F\subseteq A$ be a finite set, and let $\ep =2$. A standard approximation argument
shows that there exist $n\in\N$ and $u\in\U_{k_n}(\C)$ such that
$$\|ua-au\|<2 \ \ \mbox{ and } \ \ \left\|\alpha^{(n)}_\zeta(u)-\zeta u\right\|<2$$
for all $a$ in $F$ and for all $\zeta$ in $\T$. By Proposition \ref{Lemma actions on Mn}, there is a continuous map $v\colon \T\to\U_{k_n}(\C)$ such that
$\alpha^{(n)}_\zeta=\Ad(v(\zeta))$ for all $\zeta\in\T$. Now, Lemma \ref{unitaries in Mn at least 2 away} implies that
$\|v(\zeta)uv(\zeta)^*-\zeta u\|\geq 2$ for all $\zeta\in\T$. Therefore, $2> \left\|\alpha^{(n)}_\zeta(u)-\zeta u\right\|\geq 2$, which is a contradiction.
Thus, $\alpha$ does not have the Rokhlin property. \end{proof}

\section{Circle actions on $\Ot$-absorbing \ca s}

In this section, we specialize to circle actions with the \Rp\ on $\Ot$-absorbing \ca s. This class of \ca s is special in many ways, at least
in our context. First, circle actions with the \Rp\ are generic on separable, unital, $\Ot$-absorbing \ca s; see Theorem \ref{rokhlin action are a Gdelta set}.
This fact should be contrasted with Theorem \ref{no direct limit actions with the Rp on AF-algebras}. Second, if $A$ is a \ca\ as above and $\alpha
\colon \T\to\Aut(A)$ is an action with the \Rp, then $\alpha|_{\Z_n}\colon \Z_n\to\Aut(A)$ has the \Rp\ for every $n$ in $\N$. Again, this is special
to algebras that absorb $\Ot$, as Example \ref{eg restriction on S1} and Example \ref{eg restriction on pi ca} show. What we prove is actually
stronger, and for the conclusion to hold it is enough that $A$ absorb the universal UHF-algebra.

\subsection{Circle actions with the Rokhlin property are generic.} Throughout this subsection, $A$ will be a separable, unital \ca.

\begin{df} Given an enumeration $S=\{a_1,a_2,\ldots\}\subseteq A$ of a countable dense subset of $A$, define metrics on $\Aut(A)$ by
$$\rho_S^{(0)}(\alpha,\beta)=\sum\limits_{k=1}^\I \frac{\|\alpha(a_k)-\beta(a_k)\|}{2^k} \ \ \ \mbox{ and } \ \ \ \rho_S(\alpha,\beta)=\rho_S^{(0)}(\alpha,\beta)+\rho_S^{(0)}(\alpha^{-1},\beta^{-1}).$$\end{df}

Denote by $\mbox{Act}_\T(A)$ the set of all circle actions on $A$. Following \cite{phillips tracial rp is generic}, for any enumeration $S=\{a_1,a_2,\ldots\}\subseteq A$ as above, define a metric on $\mbox{Act}_\T(A)$ by
$$\rho_{\T,S}(\alpha,\beta)=\max_{\zeta\in\T} \rho_S(\alpha_\zeta,\beta_\zeta).$$

\begin{lemma}\label{space of T-actions is complete} For any $S$ as above, the function $\rho_{\T,S}$ is a complete metric on
$\mbox{Act}_\T(A)$.\end{lemma}
\begin{proof} Let $\left(\alpha^{(n)}\right)_{n\in \N}$ be a Cauchy sequence in $\mbox{Act}_\T(A)$, this is, for every $\varepsilon>0$
there is $n_0$ in $\N$ such that for every $n,m\geq n_0$, we have $\rho_{\T,S}\left(\alpha^{(n)},\alpha^{(m)}\right)<\varepsilon$. We
want to show that there is $\alpha\in\mbox{Act}_\T(A)$ such that $\lim\limits_{n\to\I}\rho_{\T,S}\left(\alpha,\alpha^{(n)}\right)= 0$.\\
\indent Given $\zeta\in\T$, we have $\rho_S\left(\alpha_\zeta^{(n)},\alpha_\zeta^{(m)}\right)\leq \rho_{\T,S}\left(\alpha^{(n)},\alpha^{(m)}\right)$,
and hence $\left(\alpha_\zeta^{(n)}\right)_{n\in \N}$ is Cauchy in $\Aut(A)$. By Lemma 3.2 in \cite{phillips tracial rp is generic},
the pointwise norm limit of the sequence $\left(\alpha^{(n)}_\zeta\right)_{n\in \N}$ exists, and we denote it by $\alpha_\zeta$. It
also follows from Lemma 3.2 in \cite{phillips tracial rp is generic} that $\alpha_\zeta$ is an automorphism of $A$, with inverse
$\alpha_{\zeta^{-1}}$. Moreover, the map $\alpha\colon\T\to\Aut(A)$ given by $\zeta\mapsto \alpha_\zeta$ is a group homomorphism, since
it is the pointwise norm limit of group homomorphisms. It remains to check that it is continuous, and this follows from an $\frac{\ep}{3}$
argument from $\lim\limits_{n\to\I}\left\|\alpha_\zeta^{(n)}(a_k)-\alpha_\zeta(a_k)\right\|= 0$ for all $k$ in $\N$, and the fact that
$\alpha^{(n)}\colon\T\to\Aut(A)$ is continuous for all $n\in\N$.\end{proof}

\begin{nota} Given a finite subset $F\subseteq A$ and $\varepsilon>0$, let $W_\T(F,\varepsilon)$ be the set of all actions $\alpha\in\mbox{Act}_\T(A)$ such that there exists $u\in\U(A)$ with $\|ua-au\|<\varepsilon$ for all $a$ in $F$ and $\|\alpha_\zeta(u)-\zeta u\|<\varepsilon$ for all $\zeta\in\T$.\end{nota}

It is easy to check that an action $\alpha\in \mbox{Act}_\T(A)$ has the \Rp\ \ifo $\alpha\in W_\T(F,\varepsilon)$ for all finite subsets
$F\subseteq A$ and all positive numbers $\varepsilon>0$.

\begin{lemma}\label{lma} Let $S$ be a countable dense subset of $A$, and let $\mathcal{F}$ be the set of all finite subsets of $S$. Then $\alpha\in\mbox{Act}_\T(A)$ has the \Rp\ \ifo
$$\alpha\in \bigcap_{F\in \mathcal{F}}\bigcap_{n=1}^\I W_\T(F,\frac{1}{n}).$$\end{lemma}
\begin{proof} One just needs to approximate any finite set by a finite subset of $S$. We omit the details.\end{proof}

Using the notation of the lemma above, observe that the family $\mathcal{F}$ is countable.

\begin{lemma}\label{W(F,epsilon) is dense} Let $A$ and $\D$ be unital, separable \ca s, such that there is an action
$\gamma\colon\T\to\Aut(\D)$ with the \Rp. Suppose that there exists an isomorphism $\varphi\colon A\otimes \D\to A$
such that $a\mapsto \varphi(a\otimes 1_\D)$ is approximately unitarily equivalent to $\id_A$. Then for every finite subset
$F\subseteq A$ and every $\varepsilon>0$, the set $W_\T(F,\varepsilon)$ is open and dense.\end{lemma}
\begin{proof} We first check that $W_\T(F,\varepsilon)$ is open. Choose an enumeration $S=\{a_1,a_2,\ldots\}$ of a countable
dense subset of $A$. Let $\alpha$ in $W_\T(F,\varepsilon)$, and choose $u\in\U(A)$ such that $\|ua-au\|<\varepsilon$ for all
$a$ in $F$ and $\|\alpha_\zeta(u)-\zeta u\|<\varepsilon$ for all $\zeta\in\T$. Set
$$\varepsilon_0=\max_{\zeta\in\T}\|\alpha_\zeta(u)-\zeta u\|,$$
so that $\varepsilon_1=\varepsilon-\varepsilon_0>0$. Choose $k$ in $\N$ such that $\|a_k-u\|<\frac{\varepsilon_1}{3}$. Now, we claim that if
$\alpha'\in \mbox{Act}_\T(A)$ satisfies $\rho_{\T,S}(\alpha',\alpha)<\frac{\varepsilon_1}{2^{k}3}$, then $\alpha'\in W_\T(F,\varepsilon)$. Indeed,
\begin{align*}\|\alpha'_\zeta(u)-\zeta u\|&\leq \|\alpha'_\zeta(u)-\alpha_\zeta(u)\|+\|\alpha'_\zeta(u)-\zeta u\|\\
&\leq \frac{2\ep_1}{3} +\|\alpha_\zeta'(a_k)-\alpha_\zeta(a_k)\|+ \varepsilon_0\\
&\leq \frac{2\ep_1}{3} + 2^k\rho_{\T,S}(\alpha,\alpha')+\varepsilon_0\\
&=\varepsilon_1+\varepsilon_0=\varepsilon.\end{align*}
This proves that $W_\T(F,\varepsilon)$ is open.\\
\indent We will now show that $W_\T(F,\varepsilon)$ is dense in $\mbox{Act}_\T(A)$. Let $\alpha$ be an arbitrary action in $\mbox{Act}_\T(A)$,
let $T\subseteq A$ be a finite set, and let $\delta>0$. We want to find $\beta\in\mbox{Act}_\T(A)$ such that $\beta\in W_\T(F,\varepsilon)$
and $\rho_{\T,S}(\alpha,\beta)<\delta$. \\
\indent Choose $\delta'>0$ such that $\delta'<\min\{\delta,\varepsilon\}$. Since $\alpha$ is continuous, there is $\delta_0>0$ such that
whenever $\zeta,\zeta'\in \T$ and $|\zeta-\zeta'|<\delta_0$, then $\|\alpha_{\zeta}(a)-\alpha_{\zeta'}(a)\|<\frac{\dt'}{4}$ for all $a\in T$.
Choose $m\in\N$ and $\zeta_1,\ldots,\zeta_m\in\T $ such that for every $\zeta\in\T$ there is $j\in\N$ with $1\leq j\leq m$ and such that
$|\zeta-\zeta_j|<\delta_0$. Choose $w\in\U(A)$ such that $\|w\varphi(1\otimes a)w^*-a\|<\frac{\dt'}{2}$ for all
$a\in T\cup \bigcup\limits_{j=1}^m\alpha_{\zeta_j}(T)$. Set $\psi=\Ad(w)\circ\varphi$ and for $\zeta\in\T$, define an action
$\beta\in\mbox{Act}_\T(A)$ by
$$\beta_\zeta=\psi\circ(\gamma_\zeta\otimes\alpha_\zeta)\circ\psi^{-1}.$$
We claim that $\beta\in W_\T(F,\varepsilon)$. Choose $w'\in A\otimes\D$ of the form $w'=\sum\limits_{\ell=1}^rx_\ell\otimes d_\ell$ for
some $d_1,\ldots,d_r\in \D$ and some $x_1,\ldots,x_r\in A$, such that $\|w-w'\|<\frac{\dt}{3}$. Since $\gamma$ has the \Rp, use Proposition
\ref{can replace estimate by equality} to choose $u\in\U(\D)$ such that $\gamma_\zeta(u)=\zeta u$ for all $\zeta\in\T$ and $\|ud_\ell-d_\ell u\|<\frac{\ep}{4}$ for all $\ell=1,\ldots,r$. Then
$$\|(1_A\otimes u)w'-w'(1_A\otimes u)\|<\frac{\dt}{3}$$
and hence $\|(1_A\otimes u)w-w(1_A\otimes u)\|<\delta$. Set $v=\varphi(1_A\otimes u)$. Then
\begin{align*}\|\beta_\zeta(v)-\zt v\| &=\|w\varphi\left((\alpha_\zeta\otimes\gamma_\zeta)(\varphi^{-1}(w^*\varphi(1_A\otimes u)w))\right)w^*-\zt\varphi(1_A\otimes u)\|\\
& \leq \|w\varphi\left((\alpha_\zeta\otimes\gamma_\zeta)(\varphi^{-1}(w^*\varphi(1_A\otimes u)w))\right)w^*-w\varphi\left((\alpha_\zeta\otimes\gamma_\zeta)(1_A\otimes u)\right)w^*\|\\
& \ \ \ +\|w\varphi\left((\alpha_\zeta\otimes\gamma_\zeta)(1_A\otimes u)\right)w^*-\zt\varphi(1_A\otimes u)\|\\
&<\frac{\dt'}{2} + \|w\varphi\left(\zeta 1_A\otimes u\right)w^*-\zt\varphi(1_A\otimes u)\|\\
&<\frac{\dt'}{2}+\frac{\dt'}{2}=\delta'<\varepsilon\end{align*}
for all $\zeta\in\T$, and thus $\|\beta_\zeta(u)-\zeta v\|<\varepsilon$ for all $\zeta\in\T$. On the other hand, given $a\in F$, we have
\begin{align*}\|va-av\|& = \|\varphi(1_A\otimes u)a-a\varphi(1_A\otimes u)\| \\
&\leq \|\varphi(1_A\otimes u)a-\varphi(1_A\otimes u)w\varphi(a\otimes 1_\D)w^*\|\\
& \ \ \ +\|\varphi(1_A\otimes u)w\varphi(a\otimes 1_\D)w^*-w\varphi(a\otimes 1_\D)w^*\varphi(1_A\otimes u)\|\\
& \ \ \ +\|w\varphi(a\otimes 1_\D)w^*\varphi(1_A\otimes u)-a\varphi(1_A\otimes u)\|\\
&<\frac{\dt'}{2}+0+\frac{\dt'}{2}=\delta'<\varepsilon\end{align*}
because $a\otimes 1_\D$ and $1_A\otimes u$ commute. This proves the claim.\\
\indent It remains to prove that $\|\beta_\zeta(a)-\alpha_\zeta(a)\|<\delta$ for all $a$ in $T$ and all $\zt$ in $\T$. For fixed $\zeta\in\T$ and $a\in T$, we have
\begin{align*}\|\beta_\zeta(a)-\alpha_\zeta(a)\|&=\|w\varphi\left((\alpha_\zeta\otimes\gamma_\zeta)(\varphi^{-1}(w^*aw))\right)-\alpha_\zeta(a)\| \\
&\leq \|w\varphi\left((\alpha_\zeta\otimes\gamma_\zeta)(\varphi^{-1}(w^*aw))\right)-w\varphi\left((\alpha_\zeta\otimes\gamma_\zeta)(a\otimes 1_\D)\right)\|\\
& \ \ \ + \|w\varphi\left((\alpha_\zeta\otimes\gamma_\zeta)(a\otimes 1_\D)\right)-\alpha_\zeta(a)\|\\
&< \frac{\dt'}{2}+\|w\varphi(\alpha_\zeta(a)\otimes 1_\D)w^*-\alpha_\zeta(a)\|\\
&\leq \frac{\dt'}{2}+ \|w\varphi(\alpha_\zeta(a)\otimes 1_\D)w^*-w\varphi(\alpha_{\zeta_j}(a)\otimes 1_\D)w^*\|\\
&\ \ \ +\|w\varphi(\alpha_{\zeta_j}(a)\otimes 1_\D)w^*-\alpha_{\zeta_j}(a)\|+\|\alpha_{\zeta_j}(a)-\alpha_{\zeta}(a)\|\\
&<\frac{\dt'}{2}+\frac{\dt'}{4}+\frac{\dt'}{4}=\delta'<\delta.\end{align*}
This finishes the proof.\end{proof}

\begin{thm}\label{Rokhlin are generic on D-absorbing algs} Let $A$ and $\D$ be unital, separable \ca s, such that there is an action
$\gamma\colon\T\to\Aut(\D)$ with the \Rp. Suppose that there exists an isomorphism $\varphi\colon A\otimes \D\to A$ such that
$a\mapsto \varphi(a\otimes 1_\D)$ is approximately unitarily equivalent to $\id_A$. Then the set of all circle actions with the \Rp\ on $A$ is a dense
$G_\delta$-set in $\mbox{Act}_\T(A)$. \end{thm}
\begin{proof} By Lemma \ref{lma}, the set of all circle actions on $A$ that have the \Rp\ is precisely the countable intersection
$$\bigcap_{F\in \mathcal{F}}\bigcap_{n\in\N} W_\T(F.\frac{1}{n}).$$
By Lemma \ref{W(F,epsilon) is dense}, each $W_\T(F,\frac{1}{n})$ is open and dense in $\mbox{Act}_\T(A)$, which is a complete metric space
by Lemma \ref{space of T-actions is complete}, so the result follows from the Baire Category Theorem.\end{proof}

Recall that a unital, separable \ca\ $\D$ is said to be strongly self-absorbing if it is infinite-dimensional and the map $\D\to \D\otimes \D$ given by $d\mapsto d\otimes 1$ is approximately unitarily equivalent to an isomorphism. (Strongly self-absorbing \ca s are always nuclear, so there is no ambiguity when talking about tensor products.) The only known examples are the Jiang-Su algebra $\mathcal{Z}$, the Cuntz algebras $\Ot$ and $\mathcal{O}_\I$, UHF-algebras of infinite type, and tensor products of $\OI$ by such UHF-algebras. See \cite{toms winter ssa} for more details and results on strongly self-absorbing \ca s.\\

\begin{rem}\label{remark} In the context of the above theorem, suppose additionally that $\D$ is a unital, separable strongly self-absorbing \ca. Then, according to Theorem 7.2.2 in \cite{Rordam classification}, \tfae:
\be\item There exists an isomorphism $\varphi\colon A\otimes\D\to A$ such that $a\mapsto \varphi(a\otimes 1_\D)$ is approximately unitarily equivalent to $\id_A$;
\item There exists some isomorphism $\psi\colon A\otimes \D\to A$.\ee\end{rem}

\begin{thm}\label{rokhlin action are a Gdelta set} Let $A$ be a separable \uca\ such that $A\otimes\Ot\cong A$. Then the set of all circle actions on $A$ with the \Rp\ is a dense $G_\delta$-set in $\mbox{Act}_\T(A)$.\end{thm}
\begin{proof} By Example \ref{eg of Rp on Ot}, there is an action $\gamma\colon\T\to\Aut(\Ot)$ with the \Rp. Since $A$ absorbs $\Ot$ tensorially,
the hypotheses of Theorem \ref{Rokhlin are generic on D-absorbing algs} are met by Remark \ref{remark}, and the result follows. \end{proof}

It is a consequence of the theorem above that the \Rp\ is generic for circle actions on $\Ot$. Nevertheless, we do not know of any such action for which it is possible to describe what the images of the canonical generators of $\Ot$ are. In particular, we do not have a model action on $\Ot$. \\
\indent The example below does not admit an explicit description since we are not able to write down a formula for the isomorphism $\Ot\otimes\Ot\cong\Ot$, but it is more concrete than Example \ref{eg of Rp on Ot}.

\begin{eg}\label{nice example on Ot without proof} Let $\gamma\colon \T\to\Aut(\Ot)$ be the gauge action, this is, the action determined by $\gamma_\zt(s_j)=\zt s_j$ for all $\zt\in\T$ and for $j=1,2$. Choose an isomorphism $\varphi\colon \Ot\otimes\Ot\to\Ot$ and let $\alpha\colon\T\to\Aut(\Ot)$ be given by $\alpha_\zt=\varphi\circ(\gamma_\zt\otimes\id_\Ot)\circ\varphi^{-1}$ for all $\zt\in\T$. Then $\alpha$ has the \Rp. \end{eg}

We do not yet have enough tools to prove that the action described above indeed has the \Rp. We will come back to this example in Section 5.
See Corollary \ref{pointwise outer tensor with trivial} and Proposition \ref{nice example on Ot with proof}.

\subsection{Restrictions of Rokhlin actions are again Rokhlin actions}

This subsection is devoted to proving that that the restriction of a circle action with the \Rp\ on an $\Ot$-absorbing \ca\ to a finite cyclic
group again has the \Rp. See Theorem \ref{restriction of Rp has Rp}. This phenomenon cannot be expected to hold in full generality since the
\Rp\ for a circle action does not guarantee the existence of any non-trivial projections, and moreover there are serious $K$-theoretical
obstructions to the \Rp\ for finite groups. See Examples \ref{eg restriction on S1} and \ref{eg restriction on pi ca} below.\\
\indent On the other hand, this result will allow us to classify, up to conjugacy by approximately inner automorphisms, all circle actions on
unital $\Ot$-absorbing \ca s with the \Rp. See Section 5, and in particular Theorem \ref{classification on Ot}.\\
\indent We give a very rough idea of what our strategy will be. We will first focus on cyclic group actions which are restrictions of circle
actions with the \Rp. These have what we call the ``unitary Rokhlin property'', which is a weakening of the \Rp\ that asks for a unitary instead
of projections; see Definition \ref{df: uRp}. The dual actions of actions with the unitary Rokhlin property can be characterized, and we do so in
Proposition \ref{strongly approx inner and circle Rp}. The relevant notion is that of ``strong approximate innerness''; see Definition \ref{df: sai}.
We will later show in (the proof of) Theorem \ref{restriction of Rp has Rp} that under a number of assumptions, every strongly approximately inner
action of $\Z_n$ is approximately representable, which is the notion dual to the \Rp, as was shown by Izumi in \cite{izumi finite group actions I}.
The conclusion is then that the original restriction, which a priori had the unitary Rokhlin property, actually has the \Rp.\\
\ \\
\indent The following is Definition 3.6 in \cite{izumi finite group actions I}.

\begin{df}\label{df: sai} Let $B$ be a \uca, and $\beta$ be an action of a finite abelian group $G$ on $A$.
\be\item We say that $\beta$ is \emph{strongly approximately inner} if there exist unitaries $u(g)$ in $(B^\beta)^\I$, for  $g$ in $G$, such that
$$\beta_g(b)=u(g)bu(g)^*$$
for $b$ in $B$ and $g$ in $G$.
\item We say that $\beta$ is \emph{approximately representable} if $\beta$ is strongly approximately inner and the unitaries $\{u(g)\}_{g\in G}$ above can be taken so that they form a representation of $G$ in $(B^\beta)^\I$.\ee\end{df}

\begin{nota}\label{generating automorphism} Let $B$ be a \ca, let $G$ be a cyclic group (this is, either $\Z$ or $\Z_n$ for some $n$ in $\N$ with additive notation), and let $\beta\colon G\to\Aut(B)$ be action of $G$ on $B$. We will usually make a slight abuse of notation and also denote by $\beta$ the generating automorphism $\beta_1$.\end{nota}

If $G$ is a finite cyclic group, we have the following characterization of strong approximate innerness in terms of elements in $B$, rather than in $(B^\beta)^\I$.

\begin{lemma} Let $B$ be a \uca, let $n\in\N$, and let $\beta$ be an action of $\Z_n$ on $B$. Then $\beta$ is strongly approximately inner \ifo for every finite subset $F\subseteq B$ and every $\varepsilon>0$, there is a unitary $w$ in $\U(B)$ such that $\|\beta(w)-w\|<\varepsilon$ and $\|\beta(b)-wbw^*\|<\varepsilon$ for all $b$ in $F$. Moreover, $\beta$ is approximately representable \ifo the unitary $w$ above can be chosen so that $w^n=1$.\end{lemma}
\begin{proof} Assume that $\beta$ is strongly approximately inner. Use a standard perturbation argument to choose a sequence
$(u_m)_{m\in\N}$ of unitaries in $B^\beta$ that represents $u(1)$ in $(B^\beta)^\I$. Then $\lim\limits_{m\to\I}\|\beta(u_m)-u_m\|= 0$,
and for $b$ in $B$, we have $\lim_{m\to\I}\|\beta(b)-u_mbu_m^*\|= 0$. \\
\indent Given a finite set $F\subseteq B$ and $\varepsilon>0$, choose
$M\in\N$ such that $\|\beta(u_M)-u_M\|<\varepsilon$ and $\|\beta(b)-u_Mbu_M^*\|<\varepsilon$ for all $b$ in $F$, and set $w=u_M$. \\
\indent Conversely, set $\varepsilon=\frac{1}{m}$ and let $w_m$ be as in the statement.
Then
$$u=\overline{(w_m)}_{m\in\N} \in (B^\beta)^\I$$
satisfies $\beta(b)=ubu^*$ for all $b$ in $F$, and hence $\beta$ is strongly approximately inner.\\
\indent For the second statement, observe that a unitary of order $n$ in $(B^\beta)^\I$ can be lifted to a sequence unitaries of order
$n$ in $B^\beta$. Indeed, a standard functional calculus argument shows that if $v$ is a unitary in $B^\beta$ such that $\|v^n-1\|$ is small,
then $v$ is close to a unitary $\widetilde{v}$ in $B^\beta$ such that $\widetilde{v}^n=1$. \end{proof}

The following is Lemma 3.8 in \cite{izumi finite group actions I}.

\begin{lemma} \label{approx repres and Rp} Let $B$ be a separable \uca, and let $\beta$ be an action of a finite abelian group $G$ on $B$.
\be\item The action $\beta$ has the \Rp\ \ifo the dual action $\widehat{\beta}$ is approximately representable.
\item The action $\beta$ is approximately representable \ifo the dual action $\widehat{\beta}$ has the \Rp.\ee\end{lemma}

The lemma above should be regarded as the assertion that for finite abelian group actions, the \Rp\ and approximate representability are dual notions.
It is therefore natural to ask what condition on $\beta$ is equivalent to its dual action being strongly approximately inner, rather than
approximately representable. Such a condition will necessarily be weaker than the \Rp. We define the relevant condition below.

\begin{df}\label{df: uRp}
Let $B$ be a \uca, let $n\in\N$ and let $\beta\colon\Z_n\to\Aut(B)$ be an action. We say that $\beta$ has the \emph{unitary \Rp} if for every
$\varepsilon>0$ and for every finite subset $F\subseteq B$, there exists $u\in\U(B)$ such that $\|ub-bu\|<\varepsilon$ for all $b$ in $F$ and
$\left\|\beta_k(u)-e^{2\pi i k/n}u\right\|<\varepsilon$ for all $k\in\Z_n$.\end{df}

Let $A$ be a \uca. Given a continuous action $\alpha\colon\T\to\Aut(A)$, and $n\in\N$, we denote by $\alpha|_n$ the restriction $\alpha|_{\Z_n}\colon\Z_n\to\Aut(A)$ of $\alpha$ to
$$\{1,e^{2\pi i/n},\ldots,e^{2\pi i (n-1)/n}\}\cong \Z_n.$$
Recall that if $v$ is the canonical unitary in $A\rtimes_{\alpha|_n}\Z_n$ implementing $\alpha|_n$, then the dual action
$$\widehat{\alpha|_n}\colon\Z_n\cong\widehat{\Z_n}\to \Aut(A\rtimes_{\alpha|_n}\Z_n)$$
of $\alpha|_n$ is given by $\left(\widehat{\alpha|_n}\right)_k(a)=a$ for all $a$ in $A$ and $\left(\widehat{\alpha|_n}\right)_k(v)=e^{2\pi i k/n}v$ for all $k\in\Z_n$.

The following easy lemmas provide us with many examples of cyclic group actions with the unitary \Rp.

\begin{lemma}\label{restriction has uRp} If $\alpha\colon\T\to\Aut(A)$ has the \Rp, then $\alpha|_n$ has the unitary \Rp\ for all $n\in\N$.\end{lemma}
\begin{proof} Given $\ep >0$ and a finite subset $F\subseteq A$, choose a unitary $u$ in $\U(A)$ such that $\|ua-au\|<\ep $ for all
$a$ in $F$ and $\|\alpha_\zeta(u)-\zeta u\|<\varepsilon$ for all $\zeta\in\T$. If $n\in\N$, then
$$\left\|(\alpha|_n)_k(u)-e^{2\pi i k/n}u\right\|=\left\|\alpha_{e^{2\pi i k/n}}(u)-e^{2\pi i k/n}u\right\|<\varepsilon$$
for all $k\in\Z_n$, as desired.\end{proof}

\begin{lemma}\label{Rp implies uRp} If $\beta\colon\Z_n\to\Aut(B)$ has the \Rp, then it has the unitary \Rp.\end{lemma}
\begin{proof} Given $\varepsilon>0$ and a finite subset $F\subseteq B$, choose projections $e_0,\ldots,e_{n-1}$ as in the definition
of the Rokhlin property for the tolerance
$\frac{\varepsilon}{n}$ and the finite set $F$, and set $u=\sum\limits_{j=0}^{n-1}e^{-2\pi i j/n}e_j$. Then $u$ is a unitary in $B$. Moreover, $\|ub-bu\|<\varepsilon$ for all $b$ in $F$ and
$$\left\|\beta_k(u)-e^{2\pi i k/n}u\right\|=\left\|\sum\limits_{j=0}^{n-1}e^{2\pi ij/n}\beta_k(e_j)-e^{2\pi i k/n}\sum\limits_{j=0}^{n-1}e^{-2\pi i j/n}e_j\right\|<\varepsilon$$
since $\|\beta_k(e_j)-e_{j+k}\|<\frac{\varepsilon}{n}$ for all $j,k\in\Z_n$, and the projections $e_0,\ldots,e_{n-1}$ are pairwise orthogonal.\end{proof}

The converse of the preceding lemma is not in general true, since the unitary Rokhlin property does not ensure the existence of any non-trivial projections on the algebra. We present two examples of how this can fail.

\begin{eg}\label{eg restriction on S1} Consider the action of left translation of $\T$ on $C(\T )$. It has the Rokhlin property, so its restriction to any $\Z_n\subseteq \T$ has the unitary \Rp. However, no finite group action on $C(\T )$ can have the \Rp\ since $C(\T )$ has no non-trivial projections. \end{eg}

Besides merely the lack of projections, there are less obvious $K$-theoretic obstructions for the restrictions of an action with the \Rp\ to have the \Rp.\\
\indent We need a lemma first.

\begin{prop}\label{trivial action on K-thy} Let $G$ be a connected metric group, let $A$ be a \uca, and let $\alpha\colon G\to\Aut(A)$ be a
continuous action. Then $K_\ast(\alpha_g)=\id_{K_\ast(A)}$ for all $g$ in $G$.\end{prop}
\begin{proof} We just prove it for $K_0$; the proof for $K_1$ is similar, or follows by replacing $(A,\alpha)$ with $(A\otimes B,\alpha\otimes\id_B)$,
where $B$ is any \ca\ satisfying the UCT such that $K_0(B)=0$ and $K_1(B)=\Z$, and using the K\"unneth formula. (For example, $B=C_0(\R)$ will do.)\\
\indent Denote the metric on $G$ by $d$. Let $n\in\N$ and let $p$ be a projection in $M_n(A)$. Set $\alpha^{(n)}=\alpha\otimes\id_{M_n}$, the
augmentation of $\alpha$ to $M_n(A)$. Since $\alpha^{(n)}$ is continuous, there exists $\delta>0$ such that
$\left\|\alpha^{(n)}_g(p)-\alpha^{(n)}_h(p)\right\|<1$ whenever $g$ and $h$ in $G$ satisfy $d(g,h)<\delta$. Since $\alpha^{(n)}_g(p)$ and $\alpha^{(n)}_h(p)$
are projections in $M_n(A)$, it follows that $\alpha^{(n)}_g(p)$ and $\alpha^{(n)}_h(p)$ are homotopic, and hence their classes in $K_0(A)$ agree,
that is, $K_0(\alpha_g)([p]_0)=K_0(\alpha_h)([p]_0)$. Denote by $e$ the unit of $G$. Since $g$ and $h$ satisfying $d(g,h)<\delta$ are arbitrary,
and since $G$ is connected, it follows that
$$K_0(\alpha_g)([p]_0)=K_0(\alpha_e)([p]_0)=[p]_0$$
for any $g$ in $G$. Since $p$ is an arbitrary projection in $A\otimes\K$, it follows that $K_0(\alpha_g)=\id_{K_0(A)}$ for all $g$ in $G$,
as desired.\end{proof}

\begin{eg}\label{eg restriction on pi ca} This is an example of a purely infinite simple separable nuclear \uca\ (in particular, with a great deal of projections), and an action of the circle on it satisfying the \Rp, such that no restriction to a finite subgroup of $\T$ has the \Rp. \\
\indent Let $\{p_n\}_{n\in\N}$ be an enumeration of the prime numbers, and for every $n$ in $\N$, set $q_n=p_1\cdots p_n$. Fix a countable dense subset $X=\{x_1,x_2,x_3,\ldots\}$ of $\T$ with $x_1=1$. For $x$ in $X$ and $f$ in $C(\T)$, denote by $f_x$ in $C(\T)$ the function given by $f_x(\zeta)=f(x^{-1}\zeta)$ for $\zeta\in\T$. For $n$ in $\N$, define a unital injective map
$$\iota_n\colon M_{q_n}(C(\T))\to M_{q_{n+1}}(C(\T))$$
by
$$\iota_n(f)=\left(
               \begin{array}{cccc}
                 f_{1} & 0 & \cdots & 0 \\
                 0 & f_{x_2} & \cdots & 0 \\
                 \vdots & \vdots& \ddots & \vdots \\
                 0 & 0 & \cdots & f_{x_{p_n}} \\
               \end{array}
             \right)$$
for $f$ in $M_{q_n}(C(\T))$. The direct limit $A=\varinjlim (M_{q_n}(C(\T)),\iota_n)$ is a unital A$\T$-algebra, and an argument similar to the one exhibited in Example \ref{eg of Rp on simple AT-algebra} shows that $A$ is simple. For $n\in\N$, let $\alpha^{(n)}\colon \T\to\Aut(M_{q_n}(C(\T)))$ be the tensor product of the trivial action on $M_{q_n}$ with the action coming from left translation on $C(\T)$. Then $\alpha^{(n)}$ has the \Rp\ by Proposition \ref{Properties RP}. Since $\iota_n\circ\alpha^{(n)}_\zeta=\alpha^{(n+1)}_\zeta\circ\iota_n$ for all $n\in\N$ and all $\zeta\in\T$, the sequence $\left(\alpha^{(n)}\right)_{n\in\N}$ induces a direct limit action $\alpha=\varinjlim\alpha^{(n)}$ of $\T$ on $A$, which has the \Rp\ by Proposition \ref{direct limit and Rp}.\\
\indent Now set $B=A\otimes\OI$ and define $\beta\colon\T\to\Aut(B)$ by $\beta=\alpha\otimes\id_{\OI}$. Then $B$ is a purely infinite, simple, separable, nuclear \uca, and $\beta$ has the \Rp, again by Proposition \ref{Properties RP}. We claim that for every $m\in\N$, the restriction $\beta|_m\colon \Z_m\to\Aut(B)$ does not have the \Rp.\\
\indent Fix $m\in\N$, and assume that $\beta|_m$ has the \Rp. By Proposition \ref{trivial action on K-thy}, we have $K_\ast(\beta_\zeta)=\id_{K_\ast(B)}$ for all $\zeta\in\T$. By Theorem 3.4 in \cite{izumi finite group actions II}, it follows that every element of $K_0(B)$ is divisible by $m$. On the other hand,
$$ (K_0(B),[1_B])\cong (K_0(A),[1_A])$$
\beqa\cong \left(\left\{\frac{a}{b}\colon a\in\Z, b=p_{k_1}\cdots p_{k_n}\colon n, k_1,\ldots,k_n\in\N, k_j\neq k_\ell \mbox{ for } j\neq \ell\right\},1\right),\eeqa
where not every element is divisible by $m$. This is a contradiction. \end{eg}

\indent We will nevertheless show that the restrictions of an action of the circle with the \Rp\ to any finite cyclic subgroup again have the
\Rp\ if the algebra is separable and absorbg the universal UHF-algebra $\mathcal{Q}$. See Theorem \ref{restriction of Rp has Rp} below. \\
\ \\
\indent The following result is analogous to Proposition \ref{can replace estimate by equality}, and so is its proof.

\begin{prop}\label{can replace estimate by equality in def of uRp} Let $B$ be a separable, unital \ca, let $n\in\N$ and let $\beta\colon \Z_n\to\Aut(B)$ be an action on $B$. Then $\beta$ has the unitary Rokhlin property if and only if for every finite set $F\subseteq B$ and every $\varepsilon > 0$, there is a unitary $u\in\U(B)$ such that
\be \item $\beta_k(u)=e^{2\pi i k/n} u$ for all $k\in\Z_n$;
\item $\|ub-bu\|<\varepsilon$ for all $b$ in $F$.\ee\end{prop}
Similarly to what was pointed out after the statement of Proposition \ref{can replace estimate by equality}, the definition of the unitary Rokhlin property differs in that in condition (1), one only requires $\|\beta_k(u)-e^{2\pi ik/n} u\|<\varepsilon$ for all $k\in\Z_n$.
\begin{proof} Recall that $(C(\T ), \T ,\verb"lt")$ is equivariantly semiprojective. Since the quotient $\T /\Z_n$ is compact, it follows from Theorem 3.11 in \cite{phillips thiel sorensen} that the restriction $(C(\T ),\Z_n,\verb"lt")$ is equivariantly semiprojective as well. The result now follows using an argument similar to the one used in the proof of Proposition \ref{can replace estimate by equality}. The details are left to the reader. \end{proof}

\begin{lemma}\label{unitary Rp and central sequence} Let $A$ be a separable \uca, let $n\in\N$ and let $\alpha\colon \Z_n\to\Aut(A)$ be an
action of $\Z_n$ on $A$. Regard $\Z_n\subseteq \T $ as the $n$-th roots of unitry, and let $\gamma\colon\Z_n\to \Aut(C(\T ))$ be the
restriction of the action by left translation of $\T $ on $C(\T )$. Let $\alpha_\I\colon \Z_n\to\Aut(A_\I\cap A')$ be the action on
$A_\I\cap A'$ induced by $\alpha$. Then $\alpha$ has the unitary \Rp\ \ifo there exists a unital equivariant homomorphism
\[\varphi\colon (C(\T ),\gamma) \to (A_\I\cap A',\alpha_\I).\]\end{lemma}
\begin{proof} Choose an increasing sequence $(F_m)_{m\in\N}$ of finite subsets of $A$ such that $\overline{\bigcup\limits_{m\in\N}F_m}=A$. For each $m\in\N$, there exists a unitary $u_m$ in $A$ such that
$$\|u_ma-au_m\|<\frac{1}{m} \ \ \mbox{ and } \ \ \left\|\alpha_{j}(u_m)-e^{2\pi i j/n}u_m\right\|<\frac{1}{m}$$
for every $a$ in $F_m$ and for every $j$ in $\Z_n$. Denote by $u=\overline{(u_m)}_{m\in\N}$ the image of the sequence of unitaries $(u_m)_{m\in\N}$
in $A_\I$. Then $u$ belongs to the relative commutant of $A$ in $A_\I$. Consider the unital map $\varphi\colon C(\T )\to A_\I\cap A'$ given by
$\varphi(f)=f(u)$ for $f$ in $C(\T )$. One checks that
$$\alpha_{j}(\varphi(f))=\varphi(\gamma_{e^{2\pi ij/n}}(f))$$
for all $j\in\Z_n$ and all $f$ in $C(\T )$, so $\varphi$ is equivariant.\\
\indent Conversely, assume that there is an equivariant unital homomorphism
$$\varphi\colon C(\T ) \to A_\I\cap A'.$$
Let $z\in C(\T )$ be the unitary given by $z(\zt)=\zt$ for all $\zt$ in $\T $, and let $v=\varphi(z)$. By semiprojectivity of $C(\T )$, we
can choose a representing sequence $(v_m)_{m\in\N}$ in $\ell^\I(\N,A)$ consisting of unitaries. It follows that
$$\lim_{m\to\I}\left\|\alpha_j(v_m)-e^{2\pi i j/n}v_m\right\|= 0=\lim_{m\to\I}\|v_ma-av_m\|$$
for every $a$ in $A$, and this is clearly equivalent to $\alpha$ having the unitary \Rp.\end{proof}

\begin{prop}\label{strongly approx inner and circle Rp} Let $n$ in $\N$ and let $\beta\colon \Z_n\to\Aut(B)$ be an action of $\Z_n$ on
a unital separable \ca\ $B$.
\be\item The action $\beta$ has the unitary \Rp\ \ifo its dual action $\widehat{\beta}$ is strongly approximately inner.
\item The action $\beta$ is strongly approximately inner \ifo its dual action $\widehat{\beta}$ has the unitary \Rp.\ee\end{prop}
\begin{proof} Part (a). Assume that $\beta$ has the unitary \Rp. Use Lemma \ref{unitary Rp and central sequence} to choose a unital equivariant \hm\ $\varphi\colon C(\T )\to B_\I\cap B'$. Denote by $u\in B_{\I}\cap B'$ the image under this \hm\ of the unitary $z\in C(\T)$ given by $z(\zeta)=\zt $ for $\zt\in\T$, and denote by $\lambda$ the implementing unitary representation of $\Z_n$ in $B\rtimes_\beta \Z_n$ for $\beta$. In $(B\rtimes_\beta\Z_n)_\I$, we have $u^*\lambda_j u=e^{2\pi i j/n}\lambda_j$ for all $j\in\Z_n$, and $ub=bu$ (this is, $ubu^*=b$) for all $b$ in $B$. Therefore, if $\beta$ has the unitary \Rp, then $\widehat{\beta}$ is implemented by $u^*$, and thus it is approximately representable. The converse follows from the same computation, as we have $(B\rtimes_\beta \Z_n)^{\widehat{\beta}}=B$.\\
\indent Part (b). Denote by $v$ the canonical unitary in the crossed product, and assume that $\beta$ is strongly approximately inner. Let $F\subseteq B\rtimes_{\beta}\Z_n$ be a finite subset, and let $\varepsilon>0$. Since $B$ and $v$ generate $B\rtimes_{\beta}\Z_n$, we can assume that there is a finite subset $F'\subseteq B$ such that $F=F'\cup\{v\}$. Choose $w\in\U(B)$ such that $\|\beta(w)-w\|<\varepsilon$ and $\|\beta(b)-wbw^*\|<\varepsilon$ for all $b$ in $F'$. Since $\beta(b)=vbv^*$ for every $b$ in $B$, if we let $u=w^*v$, the first of these conditions is equivalent to $\|vu-uv\|<\varepsilon$, while the second one is equivalent to $\|ub-bu\|<\varepsilon$ for all $b$ in $F'$. On the other hand, $\widehat{\beta}_k(u)=\widehat{\beta}_k(w^*v)=w^*(e^{2\pi i k/n}v)=e^{2\pi i k/n}u$ for $k\in\Z_n$. Thus, $u$ is the desired unitary, and $\widehat{\beta}$ has the unitary Rokhlin property.\\
\indent Conversely, assume that $\widehat{\beta}$ has the unitary Rokhlin property. Let $F'\subseteq B$ be a finite subset, and let $\varepsilon>0$. Set $F=F'\cup\{v\}$. Use Proposition \ref{can replace estimate by equality in def of uRp} to choose $u$ in the unitary group of $A\rtimes_\beta\Z_n$ such that $\|ub-bu\|<\varepsilon$ for all $b$ in $F$, and $\widehat{\beta}_k(u)=e^{2\pi i k/n}u$ for all $k\in\Z_n$. Set $w=vu^*$. Then $w\in B$ since
$$\widehat{\beta}_k(w)=e^{2\pi i k/n}v\overline{e^{2\pi i k/n}}u^*=vu^*=w$$
for all $k\in\Z_n$ and $(B\rtimes_{\beta}\Z_n)^{\Z_n}=B$. On the other hand,
$$\|\beta(b)-wbw^*\|=\|vbv^*-vu^*buv^*\|=\|b-u^*bu\|=\|ub-bu\|<\varepsilon,$$
for all $b$ in $F$. Hence $w$ is an implementing unitary for $F'$ and $\varepsilon$, and $\beta$ is strongly approximately inner.\end{proof}

\begin{cor}\label{restriction of Rokhlin, dual is str approx inner} Let $\alpha\colon\T\to\Aut(A)$ be an action with the \Rp, and let $n\in\N$. Then $\widehat{\alpha|_n}\colon\Z_n\to\Aut(A\rtimes_{\alpha|_n}\Z_n)$ is strongly approximately inner.\end{cor}
\begin{proof} The restriction $\alpha|_n$ has the unitary \Rp\ by Lemma \ref{restriction has uRp}, and by part (a) of Proposition
\ref{strongly approx inner and circle Rp}, its dual $\widehat{\alpha|_n}$ is strongly approximately inner.\end{proof}

The next ingredient needed is showing that crossed products by restrictions of Rokhlin actions of compact groups preserve absorption of
strongly self-absorbing \ca s. For Rokhlin actions, this was shown by Hirshberg and Winter in \cite{hirshberg winter}. The more general
statement is proved using similar ideas.

\begin{prop}\label{sequence unital embeddings fixed point restriction} Let $A$ be a separable unital \ca, let $G$ be a compact Hausdorff second countable group, and let $\af\colon G \to \Aut(A)$ be an action satisfying the Rokhlin property. Let $H$ be a closed subgroup of $G$. If $B$ is a unital separable \ca\ which admits a central sequence of unital homomorphisms into $A$, then $B$ admits a unital homomorphism into the fixed point subalgebra of $\alpha|_H$ in $A_\I\cap A'$.\end{prop}
\begin{proof} Notice that $\left(A_\I\cap A'\right)^\alpha$ is a subalgebra of $\left(A_\I\cap A'\right)^{\alpha|_H}$. The result now follows from Theorem 3.3 in \cite{hirshberg winter}.\end{proof}

\begin{rem} In the proposition above, if $B$ is moreover assumed to be simple, for example if it is strongly self-absorbing, it follows that the unital homomorphism obtained is actually an embedding, since it is not zero.\end{rem}

Recall the following result by Hirshberg and Winter.

\begin{lemma}\label{embedding fixed by the action} (Lemma 2.3 of \cite{hirshberg winter}.) Let $A$ and $\D$ be unital separable \ca s. Let $G$ be a
compact group and let $\alpha\colon G\to\Aut(A)$ be a continuous action. If there is a unital homomorphism $\D\to (A_\I\cap A')^G$, then
there is a unital homomorphism
\[\D \to \left(M(A\rtimes_\alpha G)\right)_\I\cap (A\rtimes_\alpha G)'.\] \end{lemma}

\begin{thm}\label{crossed product of restriction absorbs D} Let $\D$ be a strongly self-absorbing \ca, let $A$ be a $\D$-absorbing, separable \uca,
and let $\alpha\colon\T\to\Aut(A)$ be an action with the \Rp. Then, for every $n\in\N$, the crossed product $A\rtimes_{\alpha|_n}\Z_n$ is a unital
separable $\D$-absorbing \ca.\end{thm}
\begin{proof} By Theorem 7.2.2 in \cite{Rordam classification}, there exists a unital embedding of $\D$ into $A_\I\cap A'$, which is equivalent to
the existence of a central sequence of unital embeddings of $\D$ into $A$. Use Proposition \ref{sequence unital embeddings fixed point restriction}
to obtain a unital homomorphism of $\D$ into the fixed point subalgebra of $\alpha|_{\Z_n}$ in $A_\I\cap A'$. It follows that this homomorphism is
actually an embedding, since it is not zero and $\D$ is simple, by Theorem 1.6 in \cite{toms winter ssa}. Lemma \ref{embedding fixed by the action}
provides us with a unital embedding of $\D$ into $(A\rtimes_\alpha \Z_n)_\I\cap (A\rtimes_\alpha \Z_n)'$, which again by Theorem 7.2.2 in
\cite{Rordam classification} is equivalent to $A\rtimes_\alpha \Z_n$ being $\D$-absorbing, since $\D$ is strongly self-absorbing. \end{proof}

The following is the main theorem of this subsection.

\begin{thm}\label{characterize restriction has Rp} Let $A$ be a separable \uca, let $n\in\N$ and let $\alpha\colon\T\to\Aut(A)$ be an action with the \Rp. Suppose that $A$ absorbs $M_{n^\I}$. Then ${\alpha|_n}$ has the \Rp.\end{thm}
\begin{proof} By Lemma \ref{approx repres and Rp}, it is enough to show that $\widehat{\alpha|_n}\colon\Z_n\to\Aut(A\rtimes_{\alpha|_n}\Z_n)$ is approximately representable. Recall that by Lemma \ref{restriction of Rokhlin, dual is str approx inner}, the action $\widehat{\alpha|_n}$ is strongly approximately inner. In view of Lemma 3.10 in \cite{izumi finite group actions I}, to show that it is in fact approximately representable, it is enough to show that there is a unital map
$$M_n\to \left((A\rtimes_{\alpha|_n}\Z_n)^{\widehat{{\alpha|_n}}}\right)_\I\cap (A\rtimes_{\alpha|_n}\Z_n)',$$
where the relative commutant is taken in $(A\rtimes_{\alpha|_n}\Z_n)_\I$. \\
\ \\
\indent \textbf{\underline{Claim:}} $\left((A\rtimes_{\alpha|_n}\Z_n)^{\widehat{{\alpha|_n}}}\right)_\I\cap (A\rtimes_{\alpha|_n}\Z_n)'=(A_\I \cap A')^{(\alpha|_n)_\I}$.\\
\indent Since $(A\rtimes_{\alpha|_n}\Z_n)^{\widehat{\alpha|_n}}=A$, we have
\begin{align*}\left((A\rtimes_{\alpha|_n}\Z_n)^{\widehat{{\alpha|_n}}}\right)_\I\cap (A\rtimes_{\alpha|_n}\Z_n)'=A_\I\cap (A\rtimes_{\alpha|_n}\Z_n)'\\
=\left\{\overline{(a_m)}_{m\in\N}\in A_\I\colon \lim_{m\to\I}\|a_mx-xa_m\|= 0 \mbox{ for all } x\in A\rtimes_{\alpha|_n}\Z_n\right\}. \end{align*}
\indent Let $v$ be the canonical unitary in $A\rtimes_{\alpha|_n}\Z_n$ that implements $\alpha|_n$ in the crossed product. Then for a bounded sequence $(a_m)_{m\in\N}$ in $A$, the condition $\|a_mx-xa_m\|\to 0$ as $m\to\I$ for all $x$ in $A\rtimes_{\alpha|_n}\Z_n$ is equivalent to $\|a_ma-aa_m\|\to 0$ for all $a$ in $A$ and $\|a_mv-va_m\|\to 0$, both as $m\to\I$. Hence, the above set is equal to
$$\left\{\overline{(a_m)}_{m\in\N}\in A_\I\colon \lim_{m\to\I}\|a_ma-aa_m\|= 0 \mbox{ for all } a\in A \mbox{ and } \lim_{m\to\I}\|(\alpha|_n)(a_m)-a_m\|= 0\right\},$$
which is precisely the same as the subset of $A_\I\cap A'$ that is fixed under the action on $A_\I\cap A'$ induced by $\alpha|_n$. This proves the claim.\\
\ \\

\indent Since $A$ absorbs the UHF-algebra $M_{n^\I}$, there is a unital embedding $\iota\colon M_n\to A_\I\cap A'$. By Proposition \ref{sequence unital embeddings fixed point restriction}, there is a unital homomorphism $M_n \to (A_\I\cap A')^{(\alpha|_n)_\I}$.
Using the claim above, we conclude that there is a unital homomorphism
$$M_n \to \left((A\rtimes_{\alpha|_n}\Z_n)^{\widehat{{\alpha|_n}}}\right)_\I\cap (A\rtimes_{\alpha|_n}\Z_n)'.$$
This homomorphism is necessarily an embedding, since it is not zero. Apply Lemma 3.10 in \cite{izumi finite group actions I} to the action $\widehat{\alpha|_n}\colon\Z_n\to \Aut(A\rtimes_{\alpha|_n}\Z_n)$ to conclude that $\widehat{\alpha|_n}$ is approximately representable, and hence that $\alpha|_n$ has the \Rp, thanks to Lemma \ref{approx repres and Rp}. \end{proof}

The following partial converse to Theorem \ref{characterize restriction has Rp} holds. The same result is likely to be true for a larger class of \ca s.

\begin{prop}\label{converse to restriction has Rp if algebra absorbs UHF}
Let $A$ belong to one of the following classes of \ca s:
\be\item Unital Kirchberg algebras satisfying the UCT;
\item Simple, nuclear, separable \uca s with tracial rank zero satisfying the UCT;
\item Unital direct limits of one-dimensional noncommutatice CW-complexes with trivial $K_1$-group (this includes all AF- and AI-algebras).\ee
Let $\alpha\colon\T\to\Aut(A)$ be a continuous action and let $n\in\N$. If $\alpha|_n$ has the \Rp, then $A$ absorbs $M_{n^\I}$. \end{prop}
\begin{proof} Assume that $\alpha|_n$ has the \Rp. Since $\alpha$ induces the trivial action on $K$-theory by Proposition \ref{trivial action on K-thy},
so does $\alpha|_n$, and hence Theorem 3.4 in \cite{izumi finite group actions II}, Theorem 3.5 in \cite{izumi finite group actions II}, or
Theorem 4.2 in \cite{gardella santiago}, depending on which one is applicable, implies that $A$ is isomorphic to $A\otimes M_{n^\I}$.\end{proof}

Denote by $\mathcal{Q}$ the universal UHF-algebra, that is, the unique, up to isomorphism, UHF-algebra with $K$-theory
$$\left(K_0(\mathcal{Q}),\left[1_{\mathcal{Q}}\right]\right)\cong (\Q,,1).$$
It is well-known that $\mathcal{Q}\otimes M_{n^\I}\cong \mathcal{Q}$ for all $n$ in $\N$, and that $\mathcal{Q}\otimes \Ot\cong \Ot$.

\begin{thm}\label{restriction of Rp has Rp} Let $A$ be a separable, $\mathcal{Q}$-absorbing \uca, let $\alpha\colon\T\to\Aut(A)$ be an action
with the \Rp\ and let $n\in\N$. Then ${\alpha|_n}$ has the \Rp. In particular, restrictions of circle actions with the \Rp\ on separable, unital
$\Ot$-absorbing \ca s, again have the \Rp. \end{thm}

\section{Classification and range of the invariant}

The goal of this section is to classify, up to conjugacy by approximately inner automorphisms, all circle actions on unital, separable, nuclear, $\Ot$-absorbing \ca s with the \Rp. See Theorem \ref{classification on Ot}.\\
\indent The invariant is the induced action on the lattice of ideals (see Definition \ref{map on lattice of ideals}), and this reflects the fact that homomorphisms between $\Ot$-absorbing \ca s are classified up to approximate unitary equivalence by this invariant; see \cite{kirchberg}.
A considerable amount of work is needed to go from approximate unitary equivalence of the automorphisms given by a single group element, to conjugacy of the actions themselves. The \Rp\ for the restrictions to finite cyclic groups (see Theorem \ref{restriction of Rp has Rp} above) is crucial in this step. \\
\ \\
\indent We present some general results first, and then use them in our particular setting to obtain classification.

\begin{lemma}\label{limit actions exist} Let $A$ be a separable \ca, let $G$ be a compact group, and let $\alpha\colon G\to\Aut(A)$ be a continuous
action. Let $(\varepsilon_n)_{n\in\N}$ be a sequence such that $\sum\limits_{n\in\N}\varepsilon_n<\I$, and choose an increasing family of compact
subsets $F_n\subseteq A$ for $n\in\N$ whose union is dense in $A$. Assume that we can inductively choose a unitary $u_n$ in $A$ for $n\in\N$ such
that, with $\alpha^{(1)}=\alpha$ and $F'_1=\bigcup\limits_{g\in G}\alpha_g(F_1)$, if we let
$$\alpha^{(n-1)}_g=\Ad(u_{n-1})\circ\alpha^{(n-2)}_g\circ \Ad(u_{n-1}^*)$$
for $n\geq 2$, then $\|u_na-au_n\|<\varepsilon_n$ for all $a$ in a compact set $F_n'$ that contains
$$\bigcup\limits_{g\in G}\alpha_g^{(n-1)}\left(F'_{n-1}\cup F_n\cup \Ad(u_{n-1}\cdots u_1)(F_{n-1}') \right).$$
Then $\lim\limits_{n\to\I}\Ad(u_n\cdots u_1)$ exists in the topology of pointwise norm convergence in $\Aut(A)$ and defines an approximately inner automorphism $\mu$ of $A$. Moreover, for every $g$ in $G$ and for every $a$ in $A$, the sequence $\left(\alpha^{(n)}_g(a)\right)_{n\in\N}$ converges and
$$\lim\limits_{n\to\I} \alpha^{(n)}_g(a)=\mu\circ\alpha_g(a)\circ\mu^{-1}.$$
In particular, $g\mapsto \lim\limits_{n\to\I} \alpha^{(n)}_g$ is a continuous action of $G$ on $A$.\end{lemma}
\begin{proof} We will first show that $\lim\limits_{n\to\I}\Ad(u_n\cdots u_1)$ exists and defines an automorphism of $A$. Such an automorphism will clearly be approximately inner.\\
\indent  For $n\geq 1$, set $v_n=u_n\cdots u_1$. Let
$$S=\{a\in A\colon (v_nav_n^*)_{n\in\N}\mbox{ converges in }A\}.$$
We claim that $S$ is dense in $A$. Indeed, $S$ contains the set $F_n$ for all $n$, and since $\bigcup\limits_{n\in\N}F_n$ is dense in $A$, the claim follows.\\
\indent In particular, $S$ is a dense $\ast$-subalgebra of $A$. For each $a$ in $S$, denote by $\mu_0(a)$ the limit of the sequence $(v_nav_n^*)_{n\in\N}$. The map $\mu_0\colon S\to A$ is linear, multiplicative, preserves the adjoint, and is isometric, and therefore extends by continuity to a \hm\ $\mu \colon A\to A$ with $\|\mu(a)\|=\|a\|$ for all $a$ in $A$. Given $a$ in $A$ and given $\varepsilon>0$, choose $b$ in $S$ such that $\|a-b\|<\frac{\ep}{3}$, using density of $S$ in $A$. Choose $N\in\N$ such that $\|v_Nbv_N^*-\mu(b)\|<\frac{\ep}{3}$. Then
\begin{align*} \|\mu(a)-v_Nav_N^*\|&\leq \|\mu(a-b)\|+\|\mu(b)-v_Nbv_N^*\|+\|v_Nav_N^*-v_Nbv_N^*\|\\
&< \frac{\ep}{3} + \frac{\ep}{3}+\frac{\ep}{3}=\varepsilon,\end{align*}
It follows that $\mu=\lim\limits_{n\to\I} \Ad(u_n\cdots u_1)$ is an endomorphism of $A$, and it is isometric since it is a norm-pointwise limit of isometric maps. In particular, it is injective. We still need show that it is an automorphism. For this, we construct its inverse. By a similar reasoning, the norm-pointwise limit $\lim\limits_{n\to\I} \Ad(u_1^*\cdots u_n^*)$ defines another injective endomorphism of $A$, since $\|u_n^*au_n-a\|<\varepsilon_n$ for all $n\in\N$ and for all $a$ in $F_n'$. We denote this endomorphism by $\nu$. We claim that $\nu$ is a right inverse for $\mu$, which will imply that $\mu$ is an automorphism since it is injective. For $a$ in $F_k$ and $k\leq n\leq m$, and using that $\Ad(u_j\cdots u_n)(a)\in F_j'$ and that $u_{j+1}$ commutes with the elements of $F_j'$ up to $\varepsilon_j$ in the last step, we have
\begin{align*} \|a-\left(\mu_m \circ\mu_n^{-1}\right)(a)\|&= \|u_{m}u_{m-1}\cdots u_{n+1}au_{n+1}^*\cdots u_m^*-a\|\\
&\leq \sum\limits_{j=n}^{m-1}\left\|\Ad(u_{j+1}\cdots u_n)(a)-\Ad(u_{j}\cdots u_n)(a)\right\|\\
& < \sum\limits_{j=n}^{m-1}\varepsilon_j.\end{align*}
The estimate above implies that $\mu(\nu(a))=a$ since $\sum\limits_{j=1}^\I \varepsilon_j$ converges. Now, the union of the sets
$F_k$ with $k\in\N$ is dense in $A$, so this proves the first part of the statement.\\
\indent We now show that given $g$ in $G$ and $a$ in $A$, the sequence $\left(\alpha^{(n)}_g(a)\right)_{n\in\N}$ converges by showing that it is Cauchy. Since $\bigcup\limits_{n=1}^\I F_n$ is dense in $A$, it suffices to consider elements in this union. Let $n\in\N$ and choose $a$ in $F_n\subseteq F_n'$. If $g\in G$ and $m\geq k\geq n$, then
\begin{align*} \ \ \ &\| \alpha^{(k)}_g(a)-\alpha^{(m)}_g(a)\| \\
&= \| \alpha^{(k)}_g(a) - \Ad(u_m)\circ\cdots\circ\Ad(u_{k+1})\circ \alpha^{(k)}_g\circ \Ad(u_{k+1}^{*})\circ\cdots\circ \Ad(u_m^*)(a)\|\\
&\leq \sum\limits_{j=k}^{m-1}\big( \|\Ad(u_{j+1}\cdots u_k)\circ\alpha^{(k)}_g\circ \Ad(u_{k}^*\cdots u_{j+1}^*)(a)\\
& \ \ \ \ \ \ \ \ \ \ \ -\Ad(u_{j}\cdots u_k)\circ\alpha^{(k)}_g\circ \Ad(u_{k}^*\cdots u_{j}^*)(a)\|\big)\\
&<\sum\limits_{j=k}^{m-1} \varepsilon_j,\end{align*}
where in the last step we use that
$$\left(\Ad(u_{j}\cdots u_k)\circ\alpha^{(k)}_g\circ \Ad(u_{k}^*\cdots u_{j}^*)\right)(a)\in F_j'$$
and that $u_{j+1}$ commutes with the elements of $F_j'$ up to $\varepsilon_j$. Since $\sum\limits_{j\in\N} \varepsilon_j<\I$, the claim follows. \\
\indent Finally, we claim that for every $g\in G$ and every $a\in A$,
$$\lim\limits_{n\to\I}(\Ad(u_n\cdots u_1)\circ\alpha_g\circ\Ad(u_1^*\cdots u_n^*))(a)=(\mu\circ\alpha_g\circ\mu^{-1})(a).$$
Fix $g\in G$ and $a\in A$, and let $\varepsilon>0$. Choose $n\in\N$ such that
$$\|\Ad(u_n\cdots u_1)(x)-\mu(x)\|<\frac{\ep}{2}$$
for all $x\in \{a\}\cup \bigcup\limits_{g\in G}\alpha_g(\mu^{-1}(a))$. Then
\begin{align*} & \|(\Ad(u_n\cdots u_1)\circ\alpha_g\circ\Ad(u_1^*\cdots u_n^*))(a)-(\mu\circ\alpha_g\circ\mu^{-1})(a)\|\\
& \ \ \leq \|(\Ad(u_n\cdots u_1)\circ\alpha_g\circ\Ad(u_1^*\cdots u_n^*))(a)-(\Ad(u_n\cdots u_1)\circ\alpha_g\circ\mu^{-1})(a)\|\\
& \ \ \ \ \ +\|(\Ad(u_n\cdots u_1)\circ\alpha_g\circ\mu^{-1})(a)-(\mu\circ\alpha_g\circ\mu^{-1})(a)\|\\
& \ \ = \|\Ad(u_1^*\cdots u_n^*)(a)-\mu^{-1}(a)\|+\|(\Ad(u_n\cdots u_1)\circ\alpha_g\circ\mu^{-1})(a)-(\mu\circ\alpha_g\circ\mu^{-1})(a)\|\\
& \ \ \leq \frac{\ep}{2}+\frac{\ep}{2}=\varepsilon.\end{align*}
This finishes the proof. \end{proof}

The following is a variant of an argument used by Evans and Kishimoto in \cite{evans kishimoto}, which has become standard by now. Some of the hypotheses of the theorem can be relaxed, but this version is good enough for our purposes. In particular, the connecting maps need not be injective, and conditions (5) and (6) can assumed to hold only approximately on the sets $X_n$ and $Y_n$ respectively. On the other hand, the actions $\alpha$ and $\beta$ are not assumed to be direct limit actions, but only limit actions.

\begin{thm} \label{equivariant approximate intertwining} (Approximately equivariant intertwining.) Let $G$ be a locally compact group. Suppose there are direct systems $(A_n,i_n)$ and $(B_n,j_n)$, in which $i_n\colon A_n\to A_{n+1}$ and $j_n\colon B_n\to B_{n+1}$ are inclusions for all $n\in\N$. Suppose there are actions $\af^{(n)}\colon G\to\Aut(A_n)$ and $\beta^{(n)}\colon G\to \Aut(B_n)$ that induce norm-pointwise limit actions $\af=\lim\limits_{n\to\I} \af^{(n)}$ and $\beta=\lim\limits_{n\to\I} \beta^{(n)}$ of $G$ on $A=\varinjlim A_n$ and $B=\varinjlim B_n$ respectively. This is, $\alpha_g(a)=\lim\limits_{n\to\I}\alpha_g^{(n)}(a)$ exists for every $g$ in $G$ and every $a$ in $A$, and $g\mapsto \alpha_g$ defines a continuous action of $G$ on $\varinjlim A_n$, and similarly with $\beta$.\\
\indent Suppose there are a sequence $(\varepsilon_n)_{n\in\N}$ of positive numbers, maps $\varphi_n\colon A_n \to B_n$ and $\psi_n\colon B_n\to A_{n+1}$, increasing compact subsets $X_n\subseteq A_n$ and $Y_n\subseteq B_n$, and a family of subsets $G_n\subseteq  G$ such that
\be\item $\sum\limits_{n\in\N}\varepsilon_n <\I$;
\item $\left(\bigcup\limits_{m=n}^\I X_m\right)\cap A_n$ is dense in $A_n$ and $\left(\bigcup\limits_{m=n}^\I Y_m\right)\cap B_n$ is dense in $B_n$ for all $n\in\N$;
\item $\varphi_n(X_n)\subseteq Y_n$ and $\psi_n(Y_n)\subseteq X_{n+1}$ for all $n\in\N$;
\item $\alpha_g^{(n)}(X_n)\subseteq X_n$ and $\beta_g^{(n)}(Y_n)\subseteq Y_n$ for all $g$ in $G$ and all $n\in\N$;
\item $(\psi_n\circ\varphi_n)(a)=i_n(a)$ for all $a$ in $A_n$ and all $n\in\N$;
\item $(\varphi_{n+1}\circ\psi_n)(b)=j_n(b)$ for all $b$ in $B_n$ and all $n\in\N$;
\item $\left\|\left(\varphi_n\circ\alpha_g^{(n)}\right)(a)-\left(\beta_g^{(n)}\circ\varphi_n\right) (a)\right\|<\varepsilon_n$ for all $a$ in $X_n$ and $g$ in $G_n$;
\item $\left\|\left(\psi_n\circ\beta_g^{(n)}\right)(b)-\left(\alpha_g^{(n+1)}\circ\psi_n \right)(b)\right\|<\varepsilon_n$ for all $b$ in $Y_n$ and $g$ in $G_n$;
\item $\left\|\left(i_n\circ\alpha^{(n)}_g\right)(a)-\left(\alpha^{(n+1)}_g\circ i_n\right)(a)\right\|<\ep _n$ for all $a$ in $X_n$ and $g$ in $G_n$;
\item $\left\|\left(j_n\circ\beta^{(n)}_g\right)(b)-\left(\beta^{(n+1)}_g\circ j_n\right)(b)\right\|<\ep _n$ for all $b$ in $Y_n$ and $g$ in $G_n$;
\item $\overline{\bigcup\limits_{n\geq N}G_n}=G$ for all $N\in\N$.\ee
Then the map $\varphi_0\colon\bigcup\limits_{n\in\N}A_n\to \bigcup\limits_{n\in\N}B_n$, given by
$$\varphi_0(a)=\varphi_n(a)$$
for $a$ in $A_n$, is well defined and extends by continuity to an isomorphism $\varphi\colon A\to B$ such that
$$\varphi\circ\alpha_g=\beta_g\circ\varphi$$
for all $g$ in $G$.  \end{thm}
Note that the subsets $G_n\subseteq G$ are not assumed to be subgroups or even closed in $G$, and the family $(G_n)_{n\in\N}$ is not assumed to be increasing.
\begin{proof} Consider the map $\varphi_0\colon\bigcup\limits_{n\in\N}A_n\to \bigcup\limits_{n\in\N}B_n$ defined above. It is straightforward to show that $\varphi_0$ is well-defined and norm-decreasing, and hence it extends by continuity to a \hm\ $\varphi\colon A\to B$. Also, the proof of Proposition 2.3.2 in \cite{Rordam classification} shows that $\varphi$ is an isomorphism. It remains to check that it intertwines the actions $\alpha$ and $\beta$. \\
\indent Fix $n\in\N$ and choose $g$ in $G_n$ and $a$ in $X_n$. We claim that  $$(\varphi\circ\alpha_g)(a)=(\beta_g\circ\varphi)(a).$$
For $m,k\in\N$ with $m\geq k$, denote by $i_{m-1,k}\colon A_k\to A_m$ and $j_{m-1,k}\colon B_k\to B_m$ the compositions $i_{m-1,k}=i_{m-1}\circ\cdots\circ i_k$ and $j_{m-1,k}=j_{m-1}\circ\cdots\circ j_k$, respectively. We have
\begin{align*} (\varphi\circ\alpha_g)(a) =  \lim\limits_{m\to\I} (\varphi\circ\alpha^{(m)}_g\circ i_{m-1,n})(a)=\lim\limits_{m\to\I} (\varphi_m\circ\alpha^{(m)}_g\circ i_{m-1,n})(a)\end{align*}
and likewise,
\begin{align*} (\beta_g\circ \varphi)(a) &= \lim\limits_{m\to\I} (\beta_g\circ \varphi_m\circ i_{m-1,n})(a)=\lim\limits_{m\to\I}(\beta^{(m)}_g\circ \varphi_m\circ i_{m-1,n})(a).\end{align*}
Moreover, if $r\geq m$, then
\begin{align*} \ \ \ &\|(\varphi_r\circ\alpha^{(r)}_g\circ i_{r-1,n})(a)-(j_{r-1,m}\circ\beta^{(m)}_g\circ \varphi_m\circ i_{m-1,n})(a)\|\\
&\ \ \  \leq \|(\varphi_r\circ\alpha^{(r)}_g\circ i_{r-1,n})(a)-(\beta^{(r)}_g\circ \varphi_r\circ i_{r-1,n})(a)\| \\
&\ \ \ \ \ \ \  \ \ \ +\|(\beta^{(r)}_g\circ \varphi_r\circ i_{r-1,n})(a)-(j_{r-1,m}\circ \beta^{(m)}_g\circ \varphi_m\circ i_{m-1,n})(a)\|\\
&\ \ \ <\varepsilon_r+\sum\limits_{k=m}^{r-1}\|(\beta^{(k+1)}_g\circ \varphi_{k+1}\circ i_{k,n})(a)-(j_k\circ \beta^{(k)}_g\circ \varphi_{k}\circ i_{k-1,n})(a)\|\\
&\ \ \ =\varepsilon_r+\sum\limits_{k=m}^{r-1}\|(\beta^{(k+1)}_g\circ \varphi_{k+1}\circ i_k(i_{k-1,n})(a))-(j_k\circ \beta^{(k)}_g\circ \varphi_{k}(i_{k-1,n})(a))\|\\
&\ \ \ <\varepsilon_r+\sum\limits_{k=m}^{r-1}\|(\beta^{(k+1)}_g\circ \varphi_{k+1}\circ i_k(i_{k-1,n})(a))-(j_k\circ\beta^{(k)}_g\circ \varphi_{k+1}\circ i_k(i_{k-1,n})(a))\| \\
&\ \ \ <\sum\limits_{k=m}^{r}\varepsilon_k. \end{align*}
Since $\sum\limits_{j\in\N}\varepsilon_j<\I$, we conclude that
$$\lim\limits_{m\to\I}(\varphi_m\circ\alpha^{(m)}_g\circ i_{m-1,n})(a)=\lim\limits_{m\to\I}(\beta^{(m)}_g\circ \varphi_m\circ i_{m-1,n})(a)$$
and the claim follows.\\
\indent If $a\in X_n$ is fixed, the conclusion is that
$$(\varphi\circ\alpha_g)(a)=(\beta_g\circ\varphi)(a)$$
holds for all $g$ in $G_n$. Since the family $(X_n)_{n\in\N}$ is increasing, the identity above holds for all $g$ in $\bigcup\limits_{m\geq n} G_m$. Now $g\mapsto (\varphi^{-1}\circ\beta_g\circ\varphi)(a)$ and $g\mapsto \af_g(a)$ are two continuous functions from $G$ to $A$ that agree on the dense subset $\bigcup\limits_{m\geq n} G_m$ of $G$, hence they must agree everywhere. Hence $(\varphi\circ\alpha_g)(a)=(\beta_g\circ\varphi)(a)$ hold for all $g$ in $G$, and for all $a$ in $X_n$. Thus it holds for all $a$ in $\bigcup\limits_{n\in\N}X_n$ and by continuity, it holds for all $a$ in $A$, and the result follows.\end{proof}

\begin{df}\label{map on lattice of ideals} Let $A$ be a \ca, let $G$ be a locally compact group, and let $\alpha$ and $\beta$ be continuous actions of $G$ on $A$. We say that $\alpha$ and $\beta$ \emph{induce the same action on the lattice of ideals} of $A$ if for every $g$ in $G$ we have $\alpha_g(I)=\beta_g(I)$ for every closed two-sided ideal $I$ of $A$. \end{df}

\begin{rem} In the context of the above definition, the actions $\alpha$ and $\beta$ define the same action on the lattice of ideals \ifo the ideals of $A$ generated by $\alpha_g(a)$ and $\beta_g(a)$ coinciding for every $a$ in $A$ and for every $g$ in $G$. \end{rem}

The following technical result will allow us to verify the hypotheses of Lemma \ref{limit actions exist} in the proof of Theorem \ref{classification if many restrictions have the Rp}.

\begin{lemma}\label{conjugacy with unitaries that almost commute}
Let $A$ be a \uca, and let $\alpha$ and $\beta$ be circle actions on $A$ such that there exists an increasing sequence $(k_n)_{n\in\N}$
such that $\alpha|_{k_n}$ and $\beta|_{k_n}$ have the \Rp\ for all $n$ in $\N$. Suppose that for each $\zt$ in $\T$, the automorphisms
$\alpha_\zt$ and $\beta_\zt$ are unitarily approximately equivalent.\\
\indent Given $\varepsilon>0$ and a compact subset $F\subseteq A$, there exist $n$ in $\N$ and a unitary $u$ in $A$ such that
$$\| \Ad(u)\circ\alpha_\zt\circ\Ad(u^*)(a)-\beta_\zt(a)\|<\varepsilon \ \ \mbox{ for all } a\in F \mbox{ and all } \zt\in \T$$
and
$$\|ua-au\|<\varepsilon+ \max_{\zt\in \Z_{k_n}} \|\alpha_\zt(a)-\beta_\zt(a)\| \ \ \mbox{ for all } a\in F.$$\end{lemma}
\begin{proof} The proof is a careful analysis of the construction of the unitaries in the proof of Lemma 3.3 in Izumi's paper
\cite{izumi finite group actions I}. First, we must choose, for each $\zt$ in $\T$, a unitary $u_\zt$ in $A$ such that
$$\|u_\zt^*\alpha_\zt(a)u_\zt-\beta_\zt(a)\|<\ep$$
for all $a$ in $F$. A priori, one needs to consider infinitely many unitaries, but as it turns out, finitely many will suffice. Indeed, choose
$\delta>0$ of continuity for $\alpha$ and $\beta$ with respect to the finite set $F$ and the tolerance $\frac{\ep}{3}$. Choose $N$ in $\N$ such
that $\frac{1}{k_N}<\delta$ and choose unitaries $u_1,\ldots,u_{k_N}$ in $A$ such that
$$\left\|u_j^*\alpha_{e^{2\pi i j/k_N}}(a)u_j-\beta_{e^{2\pi i j/k_n}}(a)\right\|<\frac{\ep}{3}$$
for all $a$ in $F$ and all $j=1,\ldots,k_N$. An easy application of the triangle inequality then shows that for $\zt$ in $\T$, if $j\in\{1,\ldots,k_N\}$
satisfies $|\zt-e^{2\pi i j/k_N}|<\dt$, then
$$\left\|u_j^*\alpha_{\zt}(a)u_j-\beta_\zt(a)\right\|<\ep$$
for all $a$ in $F$, as desired. For each $\zt$ in $\T$, we choose $j$ in $\{1,\ldots,N\}$ such that $e^{2\pi ij/k_N}$ is in the clockwise arc
between $\zt$ and $1$, and set $u_\zt=u_j$.\\
\indent Next, we need to choose, for each $n$ in $\N$, orthogonal projections $p_0,\ldots,p_{k_n-1}$ as in the definition of the Rokhlin property
for the action $\alpha|_{k_n}$, for the compact set
$$\bigcup\limits_{\zt\in\T}\alpha_\zt(F)\cup\beta_\zt(F)\cup\{u_1,\ldots,u_{k_N}\}$$
and tolerance $\ep$.
Since the restriction of a finite group action with the \Rp\ to any subgroup again has the \Rp\ by Proposition \ref{finite gp Rp properties}, and
upon combining actions
on $\Z_j$ and $\Z_\ell$ to an action of $\Z_{j \ell}$ whenever $(j,\ell)=1$, we may assume that $k_n$ divides $k_{n+1}$ for all $n$ in $\N$. Moreover,
if $k_m=rk_n$ for some positive integers $m,r$ and $n$, we may also assume that the sum of the first $r$ Rokhlin projections corresponding to
$\alpha|_{k_m}$ add up to exactly the first Rokhlin projection corresponding to $\alpha|_{k_n}$, since projections of this form satisfy the
desired conditions.\\
\indent Finally, with these choice of unitaries and projections, the unitary associated to $\alpha|_{k_n}$ with $n\geq N$, for the finite set $F$
and the tolerance $\ep$, is a perturbation of the almost unitary
$$w_n=\sum\limits_{j=0}^{k_n}p_ju_{e^{2\pi ij/k_n}}.$$
Now, the unitaries $u_{e^{2\pi ij/k_n}}$ are almost always the same, and when this happens, the corresponding projections add up to the projections
corresponding to $\alpha|_{k_N}$ by construction. It follows from Izumi's construction then that for $n,m\geq N$, the unitaries obtained from
Lemma 3.3 in \cite{izumi finite group actions I} for $\alpha|_{k_n}$ and $\beta|_{k_n}$ (for the fixed finite set $F$ and tolerance $\ep$) can be
chosen to be the same. Since $\bigcup\limits_{n\geq N}\Z_n\subseteq \T$ is dense, the result follows.
\end{proof}

We are thankful to Luis Santiago for finding a mistake in an earlier version of the following result.

\begin{thm}\label{classification if many restrictions have the Rp} Let $A$ be a unital, separable, nuclear, $\Ot$-absorbing \ca, and let $\alpha$
and $\beta$ be actions of $\T$ on $A$. Assume that there is an increasing sequence $(k_n)_{n\in\N}$ such that $\alpha|_{k_n}$ and $\beta|_{k_n}$ have
the \Rp\ for all $n\in\N$. Then there is an approximately inner automorphism $\theta\in \Aut(A)$ such that
$$\alpha_\zeta=\theta\circ\beta_\zeta\circ\theta^{-1}$$
for all $\zeta\in\T$ \ifo $\alpha$ and $\beta$ induce the action on the lattice of ideals of $A$.\end{thm}
\begin{proof} Since approximately inner automorphisms induce the identity map on the lattice of ideals, if there exists an automorphism $\theta$ as in the statement, it follows that $\alpha$ and $\beta$ induce the same action on the lattice of ideals.\\
\indent Conversely, assume that $\alpha$ and $\beta$ induce the same action on the lattice of ideals of $A$, so that $\alpha_g$ and $\beta_g$ are approximately unitarily equivalent by the results in \cite{kirchberg}. Choose an increasing family of compact subsets $F_n\subseteq A$ for $n\in\N$ whose union is dense in $A$. Set $\alpha^{(1)}=\alpha$ and $\beta^{(1)}=\beta$. Choose a positive integer $n_1$ and a unitary $u_1$ in $A$ such that the conclusion of Lemma \ref{conjugacy with unitaries that almost commute} is satisfied with $\alpha^{(1)}|_{k_{n_1}}$ and $\beta^{(1)}|_{k_{n_1}}$, with
$$F=F_1'=\bigcup\limits_{\zeta\in\T}\left(\alpha_\zeta(F_1)\cup\beta_\zeta(F_1)\right)$$
and $\varepsilon=1$. For $\zt\in\T$, set $\alpha_\zt^{(2)}=\Ad(u_1)\circ\alpha_\zt\circ\Ad(u_1^*)$ and let $$F_2'=\bigcup\limits_{\zt\in\T}\alpha^{(2)}_{\zt}(F_2\cup F_1'\cup \Ad(u_1)(F_1'))\cup \bigcup\limits_{\zt\in\T}\beta^{(1)}_\zt(F_2\cup F_1'\cup \Ad(u_1)(F_1')).$$
Choose a positive integer $n_2$ with $n_2>n_1$, and a unitary $v_1$ in $A$ such that the conclusion of Lemma \ref{conjugacy with unitaries that almost commute} is satisfied with $\beta^{(1)}|_{k_{n_2}}$ and $\alpha^{(2)}|_{k_{n_2}}$ in place of $\alpha$ and
$\beta$, with $F=F_2'$ and $\varepsilon=\frac{1}{2}$. For $\zt\in\T$, set
$$\beta_\zt^{(2)}=\Ad(v_1)\circ\beta_\zt\circ\Ad(v_1^*)$$
and let
$$F_3'=\bigcup\limits_{\zt\in\T}\alpha^{(2)}_{\zt}(F_2\cup F_1'\cup \Ad(v_1)(F_1'))\cup \bigcup\limits_{\zt\in\T}\beta^{(2)}_\zt(F_2\cup F_1'\cup \Ad(v_1)(F_1')).$$
Continuing this process, we can inductively construct an increasing sequence $(n_m)_{m\in\N}$, a family of compact subsets $F_m'\subseteq A$ for $m$ in $\N$, two sequences $(u_m)_{m\in\N}$ and $(v_m)_{m\in\N}$ of unitaries in $A$, and two families $(\alpha^{(m)})_{m\in\N}$ and $(\beta^{(m)})_{m\in\N}$ of circle actions on $A$, that with $\alpha^{(1)}=\alpha$ and $\beta^{(1)}=\beta$ are given by
$$\alpha^{(m+1)}=\Ad(u_m)\circ\alpha^{(m)}\circ\Ad(u_m^*) \ \ \ \mbox{ and } \ \ \ \beta^{(m+1)}=\Ad(v_m)\circ\beta^{(m)}\circ\Ad(v_m^*),$$
and such that, with $F_1'=\bigcup\limits_{\zeta\in\T}\left(\alpha_\zeta(F_1)\cup\beta_\zeta(F_1)\right)$, we have
\begin{align*} F'_{2m+1}=&\bigcup\limits_{\zeta\in \T}\alpha_\zeta^{(m+1)}(F_{2m+1}\cup F_{2m}'\cup \Ad(v_{m}v_{m-1}\cdots v_1)(F_{2m}')) \\
&\cup \bigcup\limits_{\zeta\in \T}\beta_\zeta^{(m)}(F_{2m+1}\cup F_{2m}'\cup \Ad(v_{m}v_{m-1}\cdots v_1)(F_{2m}'))\\
F'_{2m+2}=&\bigcup\limits_{\zeta\in \T}\alpha_\zeta^{(m+1)}(F_{2m+2}\cup F_{2m+1}'\cup \Ad(u_mu_{m-1}\cdots u_1)(F_{2m}')) \\
&\cup \bigcup\limits_{\zeta\in \T}\beta_\zeta^{(m+1)}(F_{2m+2}\cup F_{2m+1}'\cup \Ad(u_{m}u_{m-1}\cdots u_1)(F_{2m}'))\end{align*}
\begin{align*}\|\beta_\zeta^{(m)}(a)-\alpha_\zeta^{(m+1)}(a)\|&<\frac{1}{2^{m}} \ \ \mbox{ for } a\in F'_{2m} \mbox{ and } \zeta\in \Z_{k_{n_m}}, \\
\|\beta_\zeta^{(m+1)}(a)-\alpha_\zeta^{(m+1)}(a)\|&<\frac{1}{k_{n_{m+1}}} \ \ \mbox{ for } a\in F'_{2m+1} \mbox{ and } \zeta\in \Z_{k_{n_{m+1}}},\end{align*}
\begin{align*}\|v_{m}a-av_{m}\|&<\frac{1}{2^{m}}+\sup_{\zeta\in \Z_{k_{n_m}}} \|\beta_\zeta^{(m)}(a)-\alpha^{(m+1)}_\zeta(a)\|<\frac{1}{k_{n_{m-1}}} \ \ \mbox{ for }  a\in F'_{2m}, \ \mbox{ and }\\
\|u_{m+1}a-au_{m+1}\|&<\frac{1}{2^{m+1}}+\sup_{\zeta\in \Z_{k_{n_{m+1}}}} \|\beta_\zeta^{(m+1)}(a)-\alpha^{(m+1)}_\zeta(a)\|<\frac{1}{2^{m}} \ \ \mbox{ for } a\in F'_{2m+1},\end{align*}
for $m$ in $\N$.\\
\indent By Lemma \ref{limit actions exist}, there exist approximately inner automorphisms $\mu$ and $\nu$ of $A$ such that for all $\zeta\in\T$,
$$\lim\limits_{m\to\I} \alpha_\zeta^{(m)}=\mu\circ\alpha_\zeta\circ\mu^{-1} \ \ \ \mbox{ and } \ \ \ \lim\limits_{m\to\I} \beta_\zeta^{(m)}=\nu\circ\beta_\zeta\circ\nu^{-1}.$$
The conditions of Theorem \ref{equivariant approximate intertwining} are satisfied with the direct limit decomposition given by $A_m=B_m=A$ and $i_m=j_m=\varphi_m=\psi_m=\id_A$ for all $m$ in $\N$, and $G_m=\Z_{k_{n_m}}\subseteq \T$ for all $m$ in $\N$. The automorphism $\phi\colon A\to A$ provided by the theorem is just the identity map on $A$. In particular, it follows that
$$\mu\circ\alpha_\zeta\circ\mu^{-1} = \nu\circ\beta_\zeta\circ\nu^{-1}$$
for all $\zeta\in\T$, and by setting $\theta=\mu^{-1}\circ\nu$, the result follows. \end{proof}

The following is the main result of this work.

\begin{thm}\label{classification on Ot} Let $A$ be a unital, separable, nuclear, $\Ot$-absorbing \ca, and let $\alpha$ and $\beta$ be actions of $\T$ on $A$ with the \Rp. Then there is an approximately inner automorphism $\theta\in \Aut(A)$ such that
$$\alpha_\zeta=\theta\circ\beta_\zeta\circ\theta^{-1}$$
for all $\zeta\in\T$ \ifo $\alpha$ and $\beta$ induce the action on the lattice of ideals of $A$.\end{thm}
\begin{proof} Given $n\in\N$, the restrictions $\alpha|_n$ and $\beta|_n$ have the \Rp\ by Theorem \ref{restriction of Rp has Rp}. The result now follows from Theorem \ref{classification if many restrictions have the Rp}.\end{proof}

Since $\Ot$ is simple, we conclude the following.

\begin{cor} Let $\alpha$ and $\beta$ be actions of the circle $\T$ on $\Ot$ with the \Rp. Then there is an approximately inner automorphism $\theta\in\Aut(\Ot)$ such that $\alpha_\zeta=\theta\circ\beta_\zeta\circ\theta^{-1}$ for all $\zeta\in\T$.\end{cor}

We point out that the same result was obtained in Corollary 5.5 of \cite{gardella Kirchberg 2}, using completely different methods.\\
\ \\
\indent We present some consequences of Theorem \ref{classification if many restrictions have the Rp} and Theorem
\ref{classification on Ot}. The first one asserts that the \Rp\ for a circle action on an $\Ot$-absorbing \ca\ is actually
equivalent to the \Rp\ for all (or just infinitely many) restrictions to finite cyclic groups. This is really special to $\Ot$-absorbing
algebras, since none of the implications hold in full generality, even for Kirchberg algebras in the UCT class. See Example
\ref{eg restriction on pi ca} and Example \ref{all restrictions have Rp, but action does not}.

\begin{thm}\label{equivalences restrictions} Let $A$ be a separable, $\Ot$-absorbing \uca, and let $\alpha\colon\T\to\Aut(A)$ be a continuous action of the circle on $A$. Then \tfae:
\be\item The action $\alpha$ has the \Rp;
\item For every $n\in\N$, the restriction of $\alpha$ to $\Z_n$ has the \Rp;

\item For infinitely many integers $n$, the restriction of $\alpha$ to $\Z_n$ has the \Rp.
\ee \end{thm}
\begin{proof} That (1) implies (2) is the a consequence of Theorem \ref{restriction of Rp has Rp}. That (2) implies (3) is obvious. We will show that (3) implies (1).\\
\indent Assume that there exists a sequence $(k_n)_{n\in\N}$ in $\N$ such that $\alpha|_{k_n}$ has the \Rp\ for all $n\in\N$. We claim that there is a circle action $\beta\colon\T\to\Aut(A)$ with the \Rp\ that induces the same action on the lattice of ideals of $A$ as $\alpha$. Indeed, let $\gamma$ be the unique (up to conjugacy) circle action on $\Ot$ with the \Rp, and set $\delta=\alpha\otimes\gamma\colon \T\to\Aut(A\otimes\Ot)$. Then $\delta$ has the \Rp\ by Proposition \ref{Properties RP}. Since $A$ absorbs $\Ot$, and $\Ot$ is strongly self-absorbing, there is an isomorphism $\varphi\colon A\otimes\Ot \to A$ such that the map $A \to A$ given by $a\mapsto\varphi(a\otimes 1_{\Ot})$ is approximately unitarily equivalent to $\id_A$ (see Theorem 7.2.2 in \cite{Rordam classification}). Now, $\zeta\mapsto\varphi\circ \delta_\zeta\circ \varphi^{-1}$ defines a continuous action $\beta\colon \T\to\Aut(A)$ with the \Rp, and we claim that it induces the same action on the lattice of ideals as $\alpha$. Indeed, given $a$ in $A$ and $\zt\in\T$, choose a sequence $(u_n)_{n\in\N}$ of unitaries in $A$ such that
$$\lim\limits_{n\to\I}\|u_nxu_n^*-\varphi(x\otimes 1_\Ot)\|=0$$
whenever $x\in \{a,\alpha_{\zt}(a)\}$. We have
\begin{align*} \beta_\zt(a)&=\varphi\circ\delta_\zt\circ\varphi^{-1}(a)\\
&=\lim\limits_{n\to\I} \varphi \circ\delta_\zt\left(\varphi^{-1}(u_n)(a\otimes 1_{\Ot})(\varphi^{-1}(u_n)^*)\right)\\
&= \lim\limits_{n\to\I} \varphi\left(\delta_\zt(\varphi^{-1}(u_n))\alpha_\zt(a)\delta_\zt(\varphi^{-1}(u_n)^*)\right)\\
&= \lim\limits_{n\to\I} \varphi\left(\delta_\zt(\varphi^{-1}(u_n))\right)u_n\alpha_\zt(a)u_n^*\varphi\left(\delta_\zt(\varphi^{-1}(u_n))\right)^*.\end{align*}
It follows that $\beta_\zt(a)$ is in the ideal generated by $\alpha_\zt(a)$. Similarly, one shows that $\alpha_\zt(a)$ is in the ideal generated by $\beta_\zt(a)$, and thus these two ideals agree. \\
\indent Now, Theorem \ref{classification if many restrictions have the Rp} shows that $\alpha$ and $\beta$ are conjugate, and since $\beta$ has the \Rp, it follows that $\alpha$ does as well.\end{proof}

While the direction `(1) implies (2)' in Theorem \ref{equivalences restrictions} holds for any separable unital \ca\ absorbing the universal UHF-algebra $\mathcal{Q}$, the direction
`(3) implies (1)' fails in this context, even if the algebra moreover absorbs $\OI$. See Example \ref{all restrictions have Rp, but action does not} and Example \ref{all restrictions have Rp, but action does not on pi}.\\
\ \\
\indent The following is an application of Theorem \ref{equivalences restrictions}.

\begin{cor}\label{pointwise outer tensor with trivial} Let $A$ be a unital, separable, nuclear, simple \ca, and let $\gamma\colon \T\to\Aut(A)$ be a pointwise outer action. Choose an isomorphism $\varphi\colon A\otimes\Ot\to\Ot$ and let $\alpha\colon\T\to\Aut(\Ot)$ be given by $\alpha_\zt=\varphi\circ(\gamma_\zt\otimes\id_\Ot)\circ\varphi^{-1}$ for all $\zt\in\T$. Then $\alpha$ has the \Rp. \end{cor}
\begin{proof} We will show that $\alpha|_n$ has the \Rp\ for every $n\in\N$. The result will then follow from Theorem \ref{equivalences restrictions}. \\
\indent Given $n\in\N$, the restriction $\gamma|_n$ is pointwise outer, and $\alpha|_n$ is conjugate, via $\varphi$, to the tensor product of $\gamma|_n$ and the trivial action of $\Z_n$ on $\Ot$. It follows from Lemma 4.2 in \cite{izumi finite group actions I} that $\gamma|_n\otimes\id_{\Ot}$ has the \Rp, and this proves the result.\end{proof}

We return to Example \ref{nice example on Ot without proof}.

\begin{prop} \label{nice example on Ot with proof} Let $\gamma \colon \T\to\Aut(\Ot)$ be the gauge action, this is, the action determined by $\gamma_\zt(s_j)=\zt s_j$ for all $\zt\in\T$ and for $j=1,2$. Choose an isomorphism $\varphi \colon \Ot\otimes\Ot\to\Ot$ and let $\alpha \colon\T\to\Aut(\Ot)$ be given by
\begin{align*}  \alpha_\zt= \varphi\circ(\gamma_\zt\otimes\id_\Ot)\circ\varphi^{-1}\end{align*}
for all $\zt\in\T$. Then $\alpha$ has the \Rp, although neither $\gamma$ nor the trivial action on $\Ot$ have the \Rp. \end{prop}
\begin{proof} In order to show that $\alpha$ has the \Rp, we only need to show that $\gamma$ is pointwise outer by Corollary
\ref{pointwise outer tensor with trivial}. But this follows immediately from the Theorem in \cite{MT}.\\
\indent For the second part of the claim, it is clear that the trivial action on $\Ot$ does not have the \Rp. To see that $\gamma$ does not have the \Rp\ either, it is enough to notice that the crossed product $\Ot\rtimes_\gamma\T$ is a corner of $M_{2^\I}\otimes\K$, and hence is not $\Ot$-absorbing. (The Rokhlin property for $\gamma$ would contradict Theorem \ref{crossed product of restriction absorbs D}.)\end{proof}

Compare the direction `(3) implies (1)' in Theorem \ref{equivalences restrictions} with the following example.

\begin{eg}\label{all restrictions have Rp, but action does not} There are a \uca\ $A$ and a circle action on $A$ such that its restriction
to every proper subgroup has the \Rp, but the action itself does not. \\
\indent Let $A$ be the universal UHF-algebra, this is, $A=\varinjlim(M_{n!},\iota_n)$ where $\iota_n\colon M_{n!}\to M_{(n+1)!}$ is
given by $\iota_n(a)=\diag(a,\ldots,a)$ for all $a$ in $M_{n!}$. For every $n\in\N$, let $\alpha^{(n)}\colon \T\to\Aut(M_{n!})$ be given by
$$\alpha^{(n)}_\zeta=\Ad(\diag(1,\zt,\ldots,\zt^{n!-1}))$$
for all $\zeta\in\T$. Then $\iota_n\circ\alpha^{(n)}_\zeta=\alpha^{(n+1)}_\zeta\circ\iota_n$ for all $n\in\N$ and all $\zeta\in\T$,
and hence there is a direct limit action $\alpha=\varinjlim\alpha^{(n)}$ of $\T$ on $A$. This action does not have the \Rp\ by Theorem
\ref{no direct limit actions with the Rp on AF-algebras}.\\

\indent On the other hand, we claim that given $m\in\N$, the restriction $\alpha|_m\colon \Z_m\to\Aut(A)$ has the \Rp. So fix $m\in\N$. Then $\alpha|_m$ is the direct limit of the actions $\left(\alpha^{(n)}|_m\right)_{n\in\N}$, whose generating automorphisms are
$$\alpha^{(n)}_{e^{2\pi i/m}}=\Ad(\diag(1,e^{2\pi i/m},\ldots,e^{2\pi i(n!-1)/m})).$$
Let $F\subseteq A$ be a finite subset and let $\varepsilon>0$. Write $F=\{a_1,\ldots,a_N\}$. Since $\bigcup\limits_{n\in\N}M_{n!}$ is dense in $A$, there are $k\in\N$ and a finite subset $F'=\{b_1,\ldots,b_N\}\subseteq M_{k!}$ such that $\|a_j-b_j\|<\frac{\ep}{2}$ for all $j=1,\ldots,N$. \\
\indent Let $n\geq \max\{k,m\}$. Then the $\Z_m$-action $\alpha^{(n)}|_m$ on $M_{n!}$ is generated by the automorphism
$$\alpha^{(n)}_{e^{2\pi i/m}}=\Ad(1,e^{2\pi i/m},\ldots,e^{2\pi i(m-1)/m},\ldots,1,e^{2\pi i/m},\ldots,e^{2\pi i(m-1)/m}).$$
(There are $n!/m$ repetitions.) Denote by $e_0$ the projection
$$1_{M_{(n-1)!}}\otimes \left(
               \begin{array}{ccccccccccc}
                 \frac{1}{m} & \frac{1}{m} & \cdots & \frac{1}{m} & 0 &\cdots & 0 & 0 & 0 & \cdots & 0 \\
                 \frac{1}{m} & \frac{1}{m} & \cdots & \frac{1}{m} & 0 &\cdots & 0 & 0 & 0 & \cdots & 0 \\
                 \vdots & \vdots & \ddots & \vdots &\vdots &\cdots & \vdots & \vdots & \vdots & \cdots & \vdots \\
                 \frac{1}{m} & \frac{1}{m} & \cdots & \frac{1}{m}& 0 & \cdots & 0 & 0 & 0 & \cdots & 0 \\
                 0 & 0 & \cdots & 0 & \frac{1}{m} & \cdots & 0 & 0 & 0 & \cdots & 0\\
                 \vdots & \vdots &\cdots &\vdots & \vdots &\ddots & \vdots &\vdots &\vdots& \cdots& \vdots \\
                 0 & 0 &\cdots & 0 & 0 &\cdots & \frac{1}{m} & 0 & 0 & \cdots & 0\\
                 0 & 0 &\cdots & 0 & 0 &\cdots & 0 & \frac{1}{m} & \frac{1}{m} & \cdots & \frac{1}{m}\\
                 0 & 0 &\cdots & 0 & 0 &\cdots & 0 & \frac{1}{m} & \frac{1}{m} & \cdots & \frac{1}{m}\\
                 \vdots & \vdots &\cdots & \vdots & \vdots &\cdots &\vdots & \vdots & \vdots& \ddots & \vdots\\
                 0 & 0 &\cdots & 0 & 0  &\cdots & 0 & \frac{1}{m} & \frac{1}{m} & \cdots & \frac{1}{m}\\
                 \end{array}
             \right)$$
in $M_{n!}\subseteq A$, and for $j=1,\ldots, m-1$, set $e_j =\alpha^{(n)}_{e^{2\pi ij/m}}(e_0)\in A$. One checks that
$e_0,\ldots,e_{m-1}$ are orthogonal projections with $\sum\limits_{j=0}^{m-1} e_j=1$, and moreover that $\alpha^{(n)}_{e^{2\pi i/m}}(e_{m-1})=e_0$. \\
\indent By construction, these projections are cyclically permuted by the action $\alpha|_m$ and they sum up to one, so we only need to check that they almost commute with the given finite set. The projections $e_0,\ldots,e_{m-1}$ exactly commute with the elements of $F'$. Thus, if $k\in\{1,\ldots,N\}$ and $j\in\{0,\ldots,m-1\}$, then
\begin{align*} \|a_ke_j-e_ja_k\|&\leq \|a_ke_j-b_ke_j\|+\|b_ke_j-e_jb_k\|+\|e_jb_k-e_ja_k\|\\
&<\frac{\ep}{2}+\frac{\ep}{2}=\varepsilon,\end{align*}
and hence $\alpha|_m$ has the \Rp. \end{eg}

\begin{eg}\label{all restrictions have Rp, but action does not on pi}
The phenomenon exhibited in the example above is not special to UHF-algebras. Indeed, if $A$ and $\alpha$ are as in Example \ref{all restrictions have Rp, but action does not}, let $B=A\otimes\OI$ and let $\beta\colon\T\to\Aut(B)$ be given by $\beta_\zeta=\alpha_\zeta\otimes\id_{\OI}$ for all $\zeta\in\T$. Then $B$ is a unital Kirchberg algebra, the action $\beta$ does not have the \Rp, and for every $m\in\N$, the restriction $\beta|_m\colon \Z_m\to\Aut(B)$ has the \Rp.\end{eg}

\subsection{Comments on the range of the invariant}

Theorem \ref{classification on Ot} classifies circle actions with the \Rp\ on unital, separable, nuclear, $\Ot$-absorbing \ca s in terms
of their induced action on the lattice of the ideals of the algebra. As with any other classification theorem, it is natural to try to
compute the range of the invariant, and we discuss this issue below.\\
\ \\
\indent Given a \ca\ $A$, it is well known that its primitive ideal space uniquely determines its lattice of ideals and viceversa. Hence,
for a circle action on $A$, the induced action on $\Prim(A)$ is equivalent to the induced action on the lattice of ideals of $A$. In this
way, the invariant in Theorem \ref{classification on Ot} can be replaced by the induced action on $\Prim(A)$. Such a circle action on
$\Prim(A)$ is known to be continuous in full generality. We do not know, however, whether this is the only obstruction for a circle action
on $\Prim(A)$ to be induced by an action on $A$ with the \Rp. \\
\indent The question about the range of the invariant therefore becomes:

\begin{qst}\label{qst 1} Let $A$ be a unital, separable, nuclear, $\Ot$-absorbing \ca\ and let $X$ denote its primitive ideal space.
What continuous circle actions on $X$ are induced by circle actions on $A$ with the \Rp?\end{qst}

One should not expect that action on $\Prim(A)$ induced by a circle action on $A$ with the \Rp\ to be free, since even the trivial action
on $\Prim(A)$ is induced by a Rokhlin action. In fact, it is shown in the course of the proof of Theorem \ref{equivalences restrictions}
that if $A$ is a \ca\ as in Question \ref{qst 1} and
$\alpha\colon\T\to\Aut(A)$ is any continuous action, then there exists an action $\beta\colon\T\to\Aut(A)$ with the \Rp\ whose induced
action on $\Prim(A)$ coincides with that of $\alpha$. Indeed, one may tensor $\alpha$ with any Rokhlin action on $\Ot$, and use an isomorphism
$A\otimes\Ot\cong A$ to obtain $\beta$.\\
\indent It follows that the invariant has ``full" range, in some sense. This reduces the problem to the following:

\begin{qst}\label{qst 2} Let $A$ be a unital, separable, nuclear, $\Ot$-absorbing \ca\ and let $X$ denote its primitive ideal space.
What continuous circle actions on $X$ are induced by circle actions on $A$?\end{qst}

Let $A$ be a separable unital \ca\ and let $X=\Prim(A)$. Then it can be shown that $X$ is compact, second countable and, by a result of Kirchberg,
sober (meaning that the only closed irreducible subsets of $X$ are the closures of the points of $X$). There is now a characterization of those
spaces which arise as the primitive ideal space of a separable, unital \emph{nuclear} \ca\ (see \cite{kirchberg}), and for every space $X$ in this
class, there is, up to
isomorphism, a unique separable, nuclear, $\Ot$-absorbing \ca\ $A$ with $\Prim(A)\cong X$. The problem of lifting actions from $\Prim(A)$ to $A$
naturally arises in the context of $\Ot$-absorbing \ca s, since all $K$-theoretic obstructions vanish. \\
\indent For $\Z$-actions, this is always possible, as is shown in \cite{kirchberg}. For more general groups (even for $\Z_2$!), this is open, and it
is likely to be rather challenging. \\
\indent However, there is a bright spot, which we proceed to describe. If $X$ is a topological space, we denote by $\mathcal{O}(X)$ the lattice of open subsets of $X$. Recall that for a \ca\ $A$, there is a canonical bijective correspondence
$$\mathcal{O}(\Prim(A))\cong \mbox{Ideals}(A)$$
as complete lattices. Moreover, $\Prim(A)$ can be recovered from $\mathcal{O}(\Prim(A))$: its points are the irreducible closed subsets. Any open subset $U$ of $\Prim(A)$ has associated to it an ideal $A(U)$ of $A$, and if $C$ is a closed subset of $\Prim(A)$, then $C$ has associated to it the quotient
$$A(C)=A/A(\Prim(A)\setminus C).$$
When $C=\{x\}$ for some $x$ in $X$, one simply writes $A_x$ for $A_{\{x\}}$.
\begin{prop}\label{range of the invariant} Let $A$ be a unital, separable, nuclear, $\Ot$-absorbing \ca, and let $X$ denote its primitive ideal space. If $X$ is Hausdorff and finite dimensional, then a circle action on $X$ is induced by a continuous circle action on $A$ \ifo it is continuous.\end{prop}
\begin{proof} Using results of Kasparov and the facts that $X$ is Hausdorff and $A$ is separable, it follows that $A$ is a continuous field over $X$. For $x$ in $X$, denote by $U_x$ the complement in $X$ of $\{x\}$, which is open in $X$. The fiber $A_x$ can then be identified with the quotient $A/\mathcal{O}(U_x)$, which is unital, separable, simple and nuclear. Being a quotient of an $\Ot$-absorbing algebra, $A_x$ itself must absorb $\Ot$ by Corollary 3.3 in \cite{toms winter ssa}, and thus $A_x\cong \Ot$ by Theorem 3.8 in \cite{kirchberg phillips}. \\
\indent It follows that $A$ is a continuous field over $\Ot$. Since all $\Ot$-continuous fields over finite dimensional Hausdorff spaces are trivial by Theorem 1.1 in \cite{dadarlat}, there is an isomorphism $A\cong C(X)\otimes\Ot$. \\
\indent It is clear that a continuous action of $\T$ on $X$ induces a continuous action of $\T$ on $C(X)\otimes\Ot$, and therefore on $A$. Conversely, a continuous circle action on $A$ induces a continuous circle action on $C(X)\otimes\Ot$, which induces a continuous action on $X\cong\mbox{Max}(C(X)\otimes\Ot)$.\end{proof}

We make a few remarks about the assumptions of the proposition above. If $X$ is not assumed to be finite dimensional, then it is not known whether the
continuous field $A$ is trivial or even stably trivial. If $X$ is not Hausdorff, our methods break down completely since $A$ is not in general a
continuous field over $X$.\\
\indent Finally, we point out that Kaplansky has characterized those \ca s whose primitive ideal space is Hausdorff as follows. Given $a$ in $A$,
we may represent $a$ by the set $\{a(P)\}_{P\in\Prim(A)}$ consisting of the images of $a$ in $A/P$ for every primitive ideal $P$ in $A$. Then
Theorem 4.1 in \cite{kaplansky} asserts that $\Prim(A)$ is Hausdorff \ifo for every $a$ in $A$, the map $\Prim(A)\to \R$ given by $P\mapsto \|a(P)\|$
is continuous.  \\
\indent It is an interesting problem to characterize those \ca s whose primitive ideal space is finite dimensional.

config: arxiv