text:
\begin{document}
\title{Real splitting and stability index for algebras with involution}
\author{Vincent Astier and Thomas Unger}

\date{}
\maketitle

\begin{abstract}

For central simple algebras with involution over a formally real field, ``splitting'' by real closures and maximal ordered subfields is investigated, and
another presentation of $H$-signatures of hermitian forms is given.
The stability index for central simple algebras with involution is defined and studied.
\smallskip

\noindent\textbf{Key words.} Central simple algebra,
involution, formally real field, splitting, hermitian form, $H$-signature,   stability index
\smallskip

\noindent\textbf{2010 MSC.} 16K20, 11E39, 13J30

\end{abstract}

\section{Introduction}

It is well-known that any central simple algebra $A$ possesses a splitting field, i.e. a field extension $L$ of its centre $Z$ such that $A\ox_Z L$ is Brauer equivalent to $L$. For example, any
algebraic closure of $Z$ is a splitting field of $A$. By a theorem of Wedderburn, $A$ is isomorphic to a matrix algebra over a central division $Z$-algebra $D$. Any maximal subfield of $D$ is a splitting field of $D$ and thus a splitting field of $A$.

In this paper we study the ``real splitting'' of an algebra with involution, motivated by the theory of $H$-signatures, developed in \cite{A-U-Kneb} and \cite{A-U-prime}.
More precisely: let us assume that $A$ is a simple $F$-algebra,
 equipped with an $F$-linear involution $\s$, where $F$ is a formally real field.
 Let $\vt$ be an involution on $D$ of the same kind as $\s$.
Let $P$ be an ordering of $F$ and let $F_P$ be a real closure of $F$ at $P$. By a theorem of Frobenius, $A\ox_F F_P$ is Brauer equivalent to one of $F_P$, $F_P(\sqrt{-1})$ or $(-1,-1)_{F_P}$ whenever $A\ox_F F_P$ is simple. In analogy with the first paragraph,
it is a natural question to ask if there exists a maximal ordered extension $(L,Q)$ of $(F,P)$, contained in $D$, which behaves like $F_P$ in the sense that
$D\ox_F L$ (and thus $A\ox_F L$)
is Brauer equivalent to one of $L$, $L(\sqrt{-d})$ or $(-a,-b)_L$ with $a,b,d \in Q$ whenever $D\ox_F L$ is simple.
We answer this question in the affirmative and in addition obtain more precise results about the nature of the field $L$, which take into account the involution  $\vt$.

We use this result to give another presentation of the theory of $H$-signatures and also introduce and study  the stability index of algebras with involution.

\section{Notation}\label{sec:notation}

We give a brief overview of the notation used in this paper and refer to the standard references
\cite{Knus}, \cite{BOI}, \cite{Lam} and \cite{Sch} as well as \cite{A-U-Kneb} and \cite{A-U-prime}
for the details.

For a ring $A$ and an involution $\s$ on $A$, we denote
the set of symmetric elements of $A$ with respect to $\s$ by $\Sym(A,\s) =\{a\in A\mid \s(a)=a\}$.

Let $F$ be a formally real field with space of orderings $X_F$ and Witt ring $W(F)$. For an ordering $P\in X_F$ we denote by $F_P$ a real closure of $F$ at $P$. By an
\emph{$F$-algebra with involution} we mean a pair $(A,\s)$ where $A$ is a
finite-dimensional $F$-algebra with centre $Z(A)$, equipped with an involution  $\s:A\to A$, such that $F = Z(A) \cap \Sym(A,\s)$ and which is assumed to be
either simple (if $Z(A)$ is a field) or a direct product of two simple
algebras (if $Z(A) = F \times F$). Observe that $\dim_F Z(A) =: \kappa
\le 2$.
If $A$ is a division algebra, we call $(A,\s)$ an \df{$F$-division algebra with involution}.
By \cite[Prop.~2.14]{BOI} we may assume that $\s$ is the exchange involution when $Z(A)=F\x F$.

When $\kappa=1$, we say that
$\s$ is \df{of the first kind}. When $\kappa=2$, we say that $\s$ is \df{of the second kind} (or of \df{unitary type}).  Note that $\s|_{Z(A)}$ is the non-trivial $F$-automorphism of $Z(A)$ in this case.
Assume $\kappa=1$ and $\dim_FA=m^2\in \N$. Then $\s$ is either of \df{orthogonal type}
(if $\dim_F \Sym(A,\s)= m(m+1)/2$)
 or  of \df{symplectic type} (if $\dim_F \Sym(A,\s)= m(m-1)/2$).

It follows from the structure theory of $F$-algebras with involution  that $A$ is isomorphic to a full matrix algebra $M_n(D)$ for a unique $n\in \N$ and an $F$-division algebra $D$
(unique up to isomorphism) which is equipped with an involution $\vt$ of the same kind as $\s$, cf. \cite[Thm.~3.1]{BOI}. We denote Brauer equivalence by $\sim$.

Let $(A,\s)$ and $(B,\tau)$ be $F$-algebras with involution of the same kind. If $A$ and $B$ are Brauer equivalent, then $(A,\s)$ and $(B,\tau)$ are Morita equivalent, cf. \cite[Ex.~1.4]{Dejaiffe-98}.

For $\ve\in\{-1,1\}$ we write $W_\ve(A,\s)$ for the \df{Witt group} of Witt equivalence classes of $\ve$-hermitian forms $h:M\x M\to A$, defined on
finitely generated right $A$-modules $M$. All forms in this paper are assumed to be non-singular and are identified with their Witt equivalence classes.
We write $+$ for the sum in both $W(F)$ and $W_\ve(A,\s)$. The group
$W_\ve(A,\s)$ is a $W(F)$-module and we denote the product of $q\in W(F)$ and $h\in W_\ve(A,\s)$ by $q\cdot h$.

Assume that $(A,\s)$ and $(B,\tau)$ are Morita equivalent $F$-algebras with involution of the same kind. It follows from \cite[Thm.~9.3.5]{Knus} (for full details, see \cite[Chap.~2]{GB}) that
there exists an isomorphism $W_\ve(A,\s) \simeq W_{\ve\ve_0} (B,\tau)$, where $\ve_0=1$ if $\s$ and $\tau$ are both orthogonal or both symplectic, and $\ve_0=-1$ otherwise. If $\s$ and $\tau$ are both unitary, then the isomorphism holds for any
$\ve_0\in \{-1, 1\}$, cf. \cite[Lemma~2.1$(iii)$]{A-U-Kneb}.

In the context of signatures  later on (Section~\ref{sec-signatures}), we will consider non-trivial morphisms from $W(A\ox_F F_P, \s\ox\id)$ to $\Z$ and therefore need  to know when  $W(A\ox_F F_P, \s\ox\id)$ is torsion, which motivates the following definition.

\begin{defi}\label{nil}
Let $(A,\s)$ be an $F$-algebra with involution. We define the set of
\emph{nil-orderings} of $(A,\s)$ as follows
\[\Nil[A,\s]:=\{ P\in X_F \mid W(A\ox_F F_P, \s\ox \id) \text{ is torsion}\}.\]
\end{defi}

For convenience we also introduce
\[\wt X_F:=X_F \sm \Nil[A,\s],\]
which does not indicate the dependency on $(A,\s)$ in order to avoid cumbersome notation.

Let  $R$ be a real closed field and let $(A,\s)$ be an $R$-algebra with involution. It follows from well-known theorems of Wedderburn and Frobenius  and an application of Morita theory that $W(A,\s)$
is isomorphic to one of the following Witt groups:
\begin{equation}\label{list}
\begin{aligned}
W(R,\id) \simeq W_{\pm 1}\bigl(R(\sqrt{-1}), -\bigr)   \simeq W\bigl((-1,-1)_R, -\bigr) &\simeq \Z \\
W_{-1}(R,\id) \simeq W_{\pm 1}(R\x R, \swap)  \simeq W_{\pm 1}((-1,-1)_R\x (-1,-1)_R, \swap) &\simeq 0\\
W_{-1}\bigl((-1,-1)_R, -\bigr)&\simeq \Z/2\Z
\end{aligned}
\end{equation}
where  $-$ denotes conjugation and $\swap$ denotes the exchange involution, cf. \cite[Lemma~2.1 and \S3.1]{A-U-Kneb}.

\begin{prop}\label{nil-prop}
Let $(A,\s)$ be an $F$-algebra with involution.
\begin{enumerate}[$(1)$]
\item We have
\[\Nil[A,\s]:=\{ P\in X_F \mid W(A\ox_F F_P, \s\ox \id) \text{ is isomorphic to $0$ or $\Z/2\Z$}\}.\]

\item Let $(B,\tau)$ be an $F$-algebra with involution such that $A\sim B$ and $\s$ and $\tau$ are of the same type. Then $\Nil[B,\tau]=\Nil[A,\s]$.

\item Let $L$ be an algebraic extension of $F$ and let $P \in X_F$. For every extension $Q$ of $P$ to $L$ we have  $Q\in \Nil[A\ox_F L,\s\ox\id]$ if and only if $P\in \Nil[A,\s]$.

\item Let $P\in X_F$. Then $P \in \Nil[A,\s]$ if and only if  any morphism from $W(A\ox_F F_P, \s\ox\id)$ to $\Z$ is identically zero.

\end{enumerate}
\end{prop}

\begin{proof}
(1) Let $P\in X_F$.
The statement follows from considering the list \eqref{list} with $R=F_P$.

(2) Let $P\in X_F$. By the assumption and Morita theory, $W(A\ox_F F_P,\s\ox\id)\simeq
W(B\ox_F F_P, \tau\ox\id)$.

Statement (3)  follows from the observation that $(A\ox_F L)\ox_L L_Q \simeq A\ox_F F_P$ and
(4)  follows from \cite[Thm.~4.1]{LU1}.
\end{proof}

We remark that by Proposition~\ref{nil-prop}  our exposition
of nil-orderings in this paper is equivalent to those in \cite{A-U-Kneb} and \cite{A-U-prime}.

\begin{defi}\label{real}
Let $(D,\vt)$ be an $F$-division algebra with involution and let $P\in X_F$. We say that $(D,\vt)$ is \emph{$(F,P)$-real} (or simply \emph{$F$-real} in case $F$ has a unique ordering) if
\[(D,\vt)\in \{(F,\id), (F(\sqrt{-d}), -),  \bigl((-a,-b)_F,-\bigr)\},\]
where $a,b,d \in P$ and $-$ denotes conjugation.
\end{defi}

\begin{lemma}\label{lem2.5}
Let $(D,\vt)$ be an $F$-division algebra with involution such that $\deg D\leq 2$ and let $P\in X_F$. The following statements are equivalent:
\begin{enumerate}[$(1)$]
\item $(D,\vt)$ is $(F,P)$-real;
\item $(D\ox_F F_P, \vt\ox\id)$ is $F_P$-real;
\item $P\not\in \Nil[D,\vt]$.
\end{enumerate}
\end{lemma}

\begin{proof} (1)$\Rightarrow$(2) is clear. (2)$\Rightarrow$(3): $W (D\ox_F F_P, \vt\ox\id)$ is not torsion (cf. \cite[\S3.1]{A-U-Kneb}) and thus $P\not\in \Nil[D,\vt]$.
(3)$\Rightarrow$(1) follows from an examination of \eqref{list}, Definition~\ref{real} and Proposition~\ref{nil-prop}(1).
\end{proof}

\section{Real Splitting and Maximal Ordered Extensions}

In this section, we study the ``real splitting'' behaviour of an $F$-algebra with involution $(A,\s)$.
Up to Brauer equivalence, it suffices to consider the underlying $F$-division algebra
$D$, which is equipped with an involution $\vt$ of the same kind as $\s$, as observed in Section~\ref{sec:notation}. Throughout the paper,  $Z$ will denote
the centre of $D$. Note that when $\vt$ is of the first kind, $Z=F$ and
the degree of $D$, $\deg D$, is a $2$-power, cf. \cite[Cor., p.~154]{Plat-Yan}.

\begin{lemma}\label{quat}
Let $D$ be a division algebra with centre $Z$ and let
$Z\subseteq K \subseteq M$ be subfields of $D$,
where $M$ is maximal.   Then $D\ox_Z K$ is Brauer equivalent to a quaternion
division algebra over $K$ if and only if $[M:K]=2$.
\end{lemma}

\begin{proof} Let $k=[K:Z]$ and $\ell=[M:K]$, then by \cite[Chap.~1, \S2.9, Prop., p.~139]{Plat-Yan},
\[D\ox_Z K \simeq C_D(K) \ox_{K} M_k(K),\]
where $C_D(K)$ denotes the centralizer of $K$ in $D$.
Thus,
$\dim_{Z} D = \dim_{K} C_D(K) \cdot k^2$.
Also,
$D\ox_Z M \simeq  M_{k\ell}(M)$
since $M$ is a splitting field of $D$, cf. \cite[Thm.~2, p.~139]{Plat-Yan}.
Thus,
$\dim_{Z} D = (\ell k)^2$.
It follows that
$\dim_{K} C_D(K) =\ell^2$
and thus $C_D(K)$ is a quaternion algebra over $K$ if and only if $\ell=2$.
Since $D\ox_Z K$ is Brauer equivalent to $C_D(K)$, the result follows.
\end{proof}

\begin{prop}\label{power-of-2}
  Let $(K,P)$ be an ordered field and $L\supseteq K$ a  finite field extension with
  $[L:K]$ a power of $2$ and such that $(K,P)$ has no proper ordered extension
  in $L$. Then $L$ and $K_P$ are linearly disjoint over $K$. In particular,
  $L\ox_K K_P$ is a field and $[L:K]\leq 2$.
\end{prop}

\begin{proof}
  We also use $P$ to denote the set of positive elements of $K_P$.  We know that (see for instance \cite[Thm.~1.2.2]{BCR})
  $K_P = \bigcup_{i < \lambda} K_i$ where $\lambda$ is an ordinal, $K_0 = K$
  and, for each $i < \lambda$,
  \begin{enumerate}[(1)]
    \item $K_{i+1} = K_i(\alpha_i)$ where $\alpha_i$ is a square root of an
      element in $K_i \cap P$ or is a root of a polynomial of odd degree with
      coefficients in $K_i$.
    \item If $i$ is a limit ordinal then $K_i = \bigcup_{j < i} K_j$.
  \end{enumerate}
  The proof will follow this construction of $K_P$ by transfinite induction.

  \begin{fact}\label{add-root}
    Let $K \subseteq K' \subseteq K_P$ be such that $L$ and $K'$ are
    linearly disjoint over $K$. Let $\alpha \in K_P$ be either the square root of an
    element of $P \cap K'$ or a root of an odd degree polynomial in $K'[X]$.
    Then every subset of $L$ that is linearly independent over $K'$ is linearly independent over
    $K'(\alpha)$ .
    In particular $L$ and $K'(\alpha)$ are linearly disjoint over $K$.
      \end{fact}

\noindent \emph{Proof of the Fact}: We only prove the first statement. The second one clearly
  follows from it since $L$ and $K'$ are linearly disjoint over $K$. We consider two cases, according to $\alpha$.

Case 1: $\alpha = \sqrt{\beta}$ with $\beta \in K' \cap P$. Let $\ell_1,
      \ldots, \ell_t \in L$ be linearly independent over $K'$, and assume
      $a_1\ell_1 + \cdots +a_t\ell_t = 0$ for some $a_1, \ldots, a_t \in
      K'(\alpha)$. Write $a_i = b_i + c_i\sqrt{\beta}$ with $b_i, c_i \in K'$.
      Then
      \begin{equation}\label{not-in-L}
        (b_1\ell_1 + \cdots + b_t\ell_t) + (c_1\ell_1 + \cdots +
        c_t\ell_t)\sqrt{\beta} = 0.
      \end{equation}
      But $\sqrt{\beta} \not \in L$, otherwise we would have $K(\sqrt{\beta})
      \subseteq L$, and $K(\sqrt{\beta})$ would be a proper ordered extension
      of $(K,P)$ in $L$, contradiction. Since $\sqrt{\beta} \not \in L$, its
      minimal polynomial has degree at least $2$ and \eqref{not-in-L} implies
      $b_1\ell_1 + \cdots + b_t\ell_t = 0$ and $c_1\ell_1 + \cdots + c_t\ell_t =
      0$, which yields $b_1 = \cdots = b_t = c_1 = \cdots = c_t = 0$.

Case 2: $\alpha$ is a root of some polynomial of odd degree in $K'[X]$. Since
      $K'$ and $L$ are linearly disjoint over $K$, $L \ox_K K'$ is a field \cite[Prop.~20.2]{Mo_book}
      and
      $[L \ox_K K' : K'] = [L:K]$ is a power of $2$. Since $[K'(\alpha):K']$ is
      odd, the field extensions $L \ox_K K'$ and $K'(\alpha)$ of $K'$ have
      relatively prime degrees, so are linearly disjoint over $K'$
      \cite[Ex.~20.5]{Mo_book}.
      Let $\ell_1, \ldots, \ell_t \in L$ be linearly independent over $K'$. Then
      $\ell_1, \ldots, \ell_t$, considered as elements of $L \ox_F K' = LK'$, are
      linearly independent over $K'$ and by the observation above, are linearly
      independent over $K'(\alpha)$. This concludes the proof of the fact.
\smallskip

  Using Fact \ref{add-root}, we obtain that if $L$ and $K_i$ are linearly disjoint over
  $K$, then $L$ and $K_{i+1}$ are linearly disjoint over $K$.
  If $\mu \leq \lambda$ is a limit ordinal, and $L$ and $K_i$ are linearly disjoint
  over $K$ for every $i < \mu$, then it is immediate that $L$ and $\bigcup_{i < \mu}
  K_i$ are also linearly disjoint over $K$.

  It follows that $L\ox_K K_P$ is a field, containing $K_P$. Since $K_P$  is real closed, $[L\ox_K K_P:K_P]\leq 2$  and we conclude that
  $[L:K]\leq 2$.
\end{proof}

\begin{lemma}\label{Z}
  Let $(D, \vt)$ be an $F$-division algebra with involution
   of the second kind, and let $P\in X_F
  \sm\Nil[D,\vt]$. Then $P$ does not extend to $Z$.
\end{lemma}
\begin{proof}
  Assume that $P$ extends to $Z$. Write $Z = F(\sqrt{\alpha})$ with $\alpha \in P$.
  Then the centre of $D \ox_F F_P$ is isomorphic to $Z\ox_F F_P$ (cf. \cite[Lemma~12.4c]{Pierce}) which is isomorphic to
  $F_P \x F_P$. Therefore $D \ox_F F_P$ is an $F_P$-algebra which is not simple.
  By \cite[Lemma~2.1$(iv)$]{A-U-Kneb} $W(D\ox_F F_P, \vt_P)=0$, implying that $P \in \Nil[D,\vt]$,  a   contradiction.
 \end{proof}

\subsection{Involutions of the first kind}

\begin{prop}\label{first-kind-prop}
  Let $D$ be a central $F$-division algebra whose degree is a power of two.
Let $P\in X_F$ and let
  $(L,Q)$ be a maximal ordered extension of $(F,P)$ in $D$. Then one of the
  following holds
  \begin{enumerate}[$(1)$]
    \item $L$ is a  maximal subfield of $D$ and $D \ox_F L \sim L$;
    \item There exists $d\in Q$ such that $L(\sqrt{-d})$ is a proper extension of $L$ and
     a maximal subfield  of $D$. Furthermore, $D \ox_F L$ is Brauer equivalent to a quaternion division algebra $(-d,-c)_L$ for some $c\in L$.
  \end{enumerate}
\end{prop}

\begin{proof}
  If $L$ is a maximal subfield of $D$, then $D \ox_F L$ is Brauer equivalent to
  $L$, so we assume from now on that $L$ is not a maximal subfield of $D$. Let
  $L \subseteq M$ with $M$ a maximal subfield of $D$ and let $L_Q$ be a real
  closure of $L$ at $Q$. Since  $[M:L]$
  is a power of two, it follows by Proposition \ref{power-of-2} that $[M:L] = 2$, and thus $M =
  L(\sqrt{-d})$ for some $d \in Q$.  By Lemma
  \ref{quat}, $D \ox_F L$ is Brauer equivalent to a quaternion division
  algebra
  $H$ over $L$. Now $D \ox_F L(\sqrt{-d}) \simeq D \ox_F L \ox_L L(\sqrt{-d}) \simeq M_t(H)
  \ox_L L(\sqrt{-d}) \simeq M_t(H \ox_L L(\sqrt{-d}))$ for some $t\in \N$, and since $L(\sqrt{-d})$
  is a maximal subfield of $D$ we also have $D \ox_F L(\sqrt{-d}) \sim L(\sqrt{-d})$, so $H$
  splits over $L(\sqrt{-d})$, and the result follows by \cite[Chap.~III. Thm.~ 4.1]{Lam}.
\end{proof}

\begin{thm} \label{first-kind}
Let $(D, \vt)$ be an $F$-division algebra with involution  of the first kind. Let $P\in X_F\sm \Nil[D,\vt]$ and let
  $(L,Q)$ be a maximal ordered extension of $(F,P)$ in $D$.
  \begin{enumerate}[$(1)$]
    \item If $\vt$ is orthogonal, then $L$ is a maximal subfield of $D$ and $D \ox_F L \sim L$;
    \item If $\vt$ is symplectic, there exists $d\in Q$ such that $L(\sqrt{-d})$ is a proper extension of $L$ and
     a maximal subfield  of $D$. Furthermore, $D \ox_F L$ is Brauer equivalent to a quaternion division algebra $(-d,-c)_L$ with $c\in Q$.
  \end{enumerate}
\end{thm}

\begin{proof} Since $\vt$ is of the first kind, $\deg D$ is a power of two.

 (1) Let $\vt$ be orthogonal. Assume that $L$ is not a maximal subfield of $D$. Then $D\ox_F L \sim (-d,-c)_L$
for some $c\in L$ and $d\in Q$ by Proposition~\ref{first-kind-prop}. Since $\vt$ is orthogonal and $P\not\in \Nil[D,\vt]$, the algebra $D\ox_F F_P$ is split, which forces $-c\in Q$. Consider the
field extension $L(\sqrt{-c})$, which is a proper extension of $L$ since the quaternion algebra $(-d,-c)_L$ is division. The ordering $Q$ extends to an ordering $R$ of $L(\sqrt{-c})$ since $-c\in Q$. We now observe that
$[L(\sqrt{-c}) : F]=[L(\sqrt{-d}) : F]=\deg D$,
since $L(\sqrt{-d})$ is a maximal subfield of $D$ (cf. \cite[Thm.~2, p.~139]{Plat-Yan}), and that
\[D\ox_F L(\sqrt{-c}) \simeq (D\ox_F L)\ox_L L(\sqrt{-c}) \sim (-d,-c)_L \ox_L L(\sqrt{-c}) \sim L(\sqrt{-c}).\]
Hence, by \cite[Cor.~13.3, p.~241]{Pierce}, $L(\sqrt{-c})$ is $F$-isomorphic to a subfield $K$ of $D$. We denote this isomorphism by $f$. Observe that $f(R)$ is an ordering of $K$.
By the Skolem-Noether theorem (cf. \cite[p.~230]{Pierce}), there exists $a\in D^\x$ such that $f|_L =\Int(a)|_L$. It follows that $\Int(a^{-1}) (K)$ is a subfield of $D$ and a proper extension of $L$. Furthermore, the ordering $\Int(a^{-1}) (f(R))$ of $\Int(a^{-1}) (K)$ extends
$ \Int(a^{-1}) (f( Q))=Q$, contradicting the choice of $(L, Q)$ as a maximal ordered extension of $(F,P)$ in $D$.
We conclude that $L$ is a maximal subfield of $D$ and that $D \ox_F L \sim L$.

(2) Let $\vt$ be symplectic.  Assume that $L$ is a maximal subfield of $D$.
Then $D\ox_F L \sim L$.
Thus
$\vt\ox\id$ is a symplectic involution on the split algebra $D\ox_F F_P$, which
implies that $W(D\ox_F F_P, \vt\ox\id)\simeq W_{-1}(F_P, \id)$. It follows from Proposition~\ref{nil-prop}(1) that  $P\in \Nil[D,\vt]$, a contradiction. Thus, by Proposition~\ref{first-kind-prop}, $D\ox_F L \sim (-d,-c)_L$ for some $c\in L$ and $d\in Q$. As above,
since $P\not\in \Nil[D,\vt]$, $D\ox_F F_P$ is not split, forcing $c\in Q$.
\end{proof}

\subsection{Involutions of the second kind: the $2$-power degree case}

\begin{lemma}\label{second-kind-2a}
Let $D$ be an $F$-division algebra with centre $Z$ such that $[Z:F]=2$. Assume that the degree of $D$ is a power of $2$.
Let $P\in X_F$ be such that $P$ does not extend to $Z$ and let $(L,Q)$ be a maximal
  ordered extension of $(F,P)$ in $D$. Then $D \ox_F L$ is Brauer equivalent to
  $L(\sqrt{-d})$ for some $d \in P$ with $\sqrt{-d} \not\in L$.
\end{lemma}

\begin{proof}
By hypothesis, $Z=F(\sqrt{-d})$ for some $d\in P$.
   It follows that $L(\sqrt{-d})$ is a
  proper field extension of $L$. We claim that $L(\sqrt{-d})$ is a maximal
  subfield of $D$. Indeed, let $N$ be a maximal  subfield of $D$ such
  that $L(\sqrt{-d}) \subseteq N$. Since $\dim_Z D$ is a power of
  $2$ and $(L,Q)$ has no ordered extension in $N$, we can apply Proposition \ref{power-of-2}, and we obtain $[N:L]=2$, so $N=L(\sqrt{-d})$ is a maximal subfield of $D$.
  Therefore,  we have $D \ox_F L \simeq (D \ox_Z Z) \ox_F L
  \simeq D \ox_Z (F(\sqrt{-d}) \ox_F L) \simeq D \ox_Z L(\sqrt{-d}) \sim
  L(\sqrt{-d})$.
\end{proof}

\begin{thm}\label{second-kind-2}
  Let $(D, \vt)$ be an $F$-division algebra with involution of the second kind. Assume that the degree of $D$ is a power of $2$.  Let $P\in X_F \sm\Nil[D,\vt]$ and let $(L,Q)$ be a maximal
  ordered extension of $(F,P)$ in $D$. Then $D \ox_F L$ is Brauer equivalent to
  $L(\sqrt{-d})$ for some $d \in P$ with $\sqrt{-d} \not\in L$.
\end{thm}

\begin{proof} By Lemma~\ref{Z}, $P$ does not extend to the centre  of $D$ and we conclude with Lemma~\ref{second-kind-2a}.
\end{proof}

\begin{prop}\label{second-kind-2-split}
Let $D$ be an $F$-division algebra with centre $Z$ such that $[Z:F]=2$. Assume that the degree of $D$ is a power of $2$.
Let $P\in X_F$ be such that $P$ extends to an ordering $P'$ of $Z$ and let $(L,Q)$ be a maximal ordered extension of $(Z,P')$ in $D$.
Then  $D\ox_F L$ is isomorphic to $M_r(L\x L)$ or to $M_s\bigl ((a,b)_L \x (a,b)_L \bigr)$ for some $r$, $s \in\N$ and $a,b\in L^\x$.
\end{prop}

\begin{proof} Let $\alpha\in P$ be such that $Z=F(\sqrt{\alpha})$. Let $N$ be a maximal field extension of $L$ in $D$. By Proposition~\ref{power-of-2}, $[N:L]=1$ or $2$. Hence $D\ox_Z L$ is isomorphic to either $M_r(L)$ or $M_s\bigl( (a,b)_L\bigr)$ for some $r$, $s\in \N$ and  $a,b\in L^\x$ by Lemma~\ref{quat}.
Then
\begin{align*}
D\ox_F L &\simeq (D\ox_Z Z)\ox_F L\\
&\simeq D\ox_Z (Z\ox_F L)\\
&\simeq D\ox_Z \bigl( F[x]/(x^2-\alpha)\ox_F L \bigr)\\
&\simeq D\ox_Z  L[x]/(x^2-\alpha)\\
&\simeq D\ox_Z (L\x L)\\
&\simeq (D\ox_Z L)\x (D\ox_Z L)
\end{align*}
and the result follows.
 \end{proof}

\subsection{Involutions of the second kind: the odd degree case}

\begin{lemma}\label{involution}
 Let $(D, \vt)$ be an $F$-division algebra with involution of the second kind.
  Let $P\in X_F$ be such that $P$ does not extend to $Z$ and write $Z = F(\sqrt{-d})$ with $\vt(\sqrt{-d})=-\sqrt{-d}$ and $d \in P$.
  Let $L$ be a field extension of $F$ in $D$ such that $\sqrt{-d} \not \in L$.
  Then there is an $F$-linear involution $\sigma : D \rightarrow D$ such that $\sigma$
  is the identity on $L$ and $\sigma(\sqrt{-d}) = -\sqrt{-d}$.
\end{lemma}
\begin{proof}
  We consider
  \[      \xi :  L(\sqrt{-d})  \longrightarrow  D,\quad
            \ell_1 + \ell_2 \sqrt{-d}  \longmapsto  \vt(\ell_1) + \vt(\ell_2)\sqrt{-d}.\]
  A direct computation shows that $\xi$ is a morphism of rings and is
  $Z$-linear, so a morphism of $Z$-algebras from the simple $Z$-algebra
  $L(\sqrt{-d})$ to the central simple $Z$-algebra $D$. By the Skolem-Noether
  theorem (see \cite[p.~230]{Pierce}) there is $a \in D^\x$ such
  that $\xi(x) = a^{-1}xa$ for every $x \in L(\sqrt{-d})$.

  We define $\sigma := \Int(a^{-1}) \circ \vartheta$. This is an involution on $D$,
  and, for $\ell \in L$, $\sigma(\ell) = a \vartheta(\ell) a^{-1} = a \xi(\ell) a^{-1} = a
  a^{-1} \ell a a^{-1} = \ell$, so $\sigma$ is the identity on $L$. Finally
  $\sigma(\sqrt{-d}) = a \vartheta(\sqrt{-d}) a^{-1} = a (-\sqrt{-d}) a^{-1} =
  -\sqrt{-d}$ (since $\sqrt{-d}$ is in the centre of $D$).
\end{proof}

\begin{thm}\label{second-kind-odd}
 Let $(D, \vt)$ be an $F$-division algebra with involution of the second kind. Assume that the degree of $D$ is odd.
  Let $P\in X_F \sm\Nil[D,\vt]$,
  and write $Z = F(\sqrt{-d})$ with $\vt(\sqrt{-d})=-\sqrt{-d}$ and $d \in P$ \textup{(}cf. Lemma~\ref{Z}\textup{)}.
 Let $(L,Q)$ be a maximal ordered extension
  of $(F,P)$ in $D$. Then $L(\sqrt{-d})$ is a maximal subfield of $D$ and $D
  \ox_F L \sim L(\sqrt{-d})$.
\end{thm}

\begin{proof}
  Since $(L,Q)$ is an ordered extension of $(F,P)$, we have $\sqrt{-d} \not \in
  L$, and by Lemma \ref{involution} there is an $F$-involution $\sigma$ on $D$
  such that $\sigma$ is the identity on $L$ and $\sigma(\sqrt{-d}) =
  -\sqrt{-d}$. The following  is inspired by an argument in the proof of \cite[Prop.~11.22]{BOI}.
   Let $M = L(\sqrt{-d})$,
  and assume that $M$ is not a maximal subfield of $D$.

  Claim: Then there is $x \in C_D(M) \setminus M$ such that $\sigma(x) = x$.\\
  Proof of the claim: By construction of $\sigma$ we have $\sigma(M) = M$  and it follows
  easily that $\sigma(C_D(M)) \subseteq C_D(M)$.
    Since $M$ is not maximal, there is $u \in C_D(M) \setminus M$, cf.~\cite[p.~236, Cor.~b]{Pierce}.   Let $u_1 =
  (u+\sigma(u))/2$ and $u_2 = (u-\sigma(u))/2$. Since $\sigma(C_D(M)) \subseteq
  C_D(M)$ we have $u_1, u_2 \in C_D(M)$. Obviously $u = u_1 + u_2$, with
  $\sigma(u_1) = u_1$ and $\sigma(u_2) = -u_2$. If $u_1 \not \in M$, then we can
  take $x = u_1$. If $u_1 \in M$, then $u_2 \not \in M$, and we can take $x =
  u_2 \sqrt{-d}$, which establishes the claim.
  \medskip

  Consider the field $L(x)$. By choice of $x$ we have $L \subsetneq L(x)
  \subseteq \Sym(D,\sigma)$. In particular $\sqrt{-d} \not \in L(x)$, and the
  diagram
  \[\xymatrix{
      & L(x,\sqrt{-d}) & \\
      L(x) \ar@{-}[ur]^2 \ar@{-}[dr] & & Z=F(\sqrt{-d}) \ar@{-}[ul]_{k \text{ odd}} \ar@{-}[dl]^{2} \\
    & F &
  }\]
  gives us that $[L(x):F] = k$ is odd. It follows that $[L(x):L]$ is odd,
  and in particular the ordering $Q$ of $L$
  extends to $L(x)$, a contradiction to the choice of $(L,Q)$. Thus $L(\sqrt{-d})$ is a maximal subfield of $D$ and $D\ox_F L\simeq D\ox_Z (Z\ox_F L) \simeq D\ox_Z L(\sqrt{-d}) \sim L(\sqrt{-d})$.
\end{proof}

\begin{prop}
 Let $(D, \vt)$ be an $F$-division algebra with involution of the second kind. Assume that the degree of $D$ is odd.
 Let $P\in X_F$ be such that $P$ extends to $Z$ and let $L$ be a maximal subfield of $D$. Then $P$ extends to $L$ and $D\ox_F L\simeq M_r(L\x L)$ for some $r\in\N$.
\end{prop}

\begin{proof}
Since $P$ extends to $Z$, we have that $Z=F(\sqrt{\alpha})$ for some $\alpha \in P$.
Since $[L:Z]$ is odd, $P$ extends to $L$. Also, $D\ox_Z L\simeq M_r(L)$ for some $r\in\N$ since $L$ is maximal in $D$.
Then
\begin{align*}
D\ox_F L &\simeq (D\ox_Z Z) \ox_F L\\
&\simeq D\ox_Z (F(\sqrt{\alpha})\ox_F L)\\
&\simeq D\ox_Z ( F[x]/(x^2-\alpha) \ox_F L)\\
&\simeq D\ox_Z  L[x]/(x^2-\alpha)\\
&\simeq D\ox_Z  (L\x L)\\
&\simeq M_r(L)\x M_r(L).\qedhere
\end{align*}
\end{proof}

\subsection{Involutions of the second kind: the arbitrary degree case}

\begin{thm}\label{second-kind-arb}
 Let $(D, \vt)$ be an $F$-division algebra with involution of the second kind.
 Let $P\in X_F$ be such that $P$ does not extend to $Z$
 and write $Z=F(\sqrt{-d})$ for some $d\in P$.
  There exists  a maximal ordered extension $(L,Q)$
  of $(F,P)$ in $D$ such that  $L(\sqrt{-d})$ is a maximal subfield of $D$ and $D
  \ox_F L \sim L(\sqrt{-d})$.
 \end{thm}

\begin{proof} We may write $D\simeq D_1\ox_Z D_2$ where $D_1$ is an $F$-division algebra of odd degree with centre $Z$ and $D_2$ is an $F$-division algebra of $2$-power degree with centre $Z$, cf. \cite[Primary Decomposition Theorem, p.~261]{Pierce}. (Note though  that the involution
$\vt$ is not necessarily decomposable.)

Let $(L_2, P')$ be a maximal ordered extension of $(F,P)$ inside $D_2$.
By Theorem~\ref{second-kind-2}, $L_2(\sqrt{-d})$ is a maximal subfield of $D_2$ and
\[D_2\ox_F L_2 \simeq D_2\ox_Z L_2(\sqrt{-d})  \simeq M_k( L_2(\sqrt{-d}))\]
for some nonzero integer $k$. Hence
\begin{align*}
D\ox_F L_2
&\simeq D_1\ox_Z M_k( L_2(\sqrt{-d}))\\
&\simeq D_1\ox_Z M_k( L_2 \ox_F F(\sqrt{-d}))\\
&\simeq D_1\ox_Z M_k(\underbrace{F(\sqrt{-d})}_{Z}) \ox_F  L_2 \\
&\simeq M_k(D_1) \ox_F L_2\\
&\simeq M_k(D_1 \ox_F L_2).
\end{align*}
Since $[L_2(\sqrt{-d}):Z]$ is coprime to $\deg D_1$ and $D_1 \ox_F L_2 \simeq D_1\ox_Z L_2(\sqrt{-d})$, we have that $D_1 \ox_F L_2$ is still a division algebra (cf. \cite[Prop.~13.4(vi), p.~243]{Pierce}), which is of (odd) degree $\deg D_1$ over its   centre $L_2(\sqrt{-d})$ (cf. \cite[Lemma~12.4c, p.~225]{Pierce}). Since it is Brauer equivalent to $D\ox_F L_2$ it possesses an involution $\tau$ of the second kind
with fixed field $L_2$ (cf. \cite[Thm.~3.1]{BOI}).

Let $(L, Q)$ be a maximal ordered extension of $(L_2, P')$ inside $D_1 \ox_F L_2$. By Theorem~\ref{second-kind-odd}, $L(\sqrt{-d})$ is a maximal subfield of $D_1 \ox_F L_2$ and
\[D_1\ox_F L\simeq (D_1 \ox_F L_2)\ox_{L_2} L \sim L(\sqrt{-d}).\]
Hence
\begin{align*}
D\ox_F L &\simeq (D\ox_F L_2) \ox_{L_2} L\\
&\simeq M_k(D_1 \ox_F L_2) \ox_{L_2} L\\
& \simeq M_k(D_1\ox_F L)\\
&\sim L(\sqrt{-d}).
\end{align*}
Note that $L$ is a subfield of $D$ since
$L\subseteq D_1\ox_F L_2\simeq D_1 \ox_Z L_2(\sqrt{-d}) \subseteq D_1\ox_Z D_2\simeq D$.
Thus $L(\sqrt{-d})$ is a subfield of $D$ since $\sqrt{-d} \in D$.
Finally,
\[ [L(\sqrt{-d}) :Z] = [L(\sqrt{-d}) : L_2(\sqrt{-d})] [L_2(\sqrt{-d}) : Z] =\deg D_1 \cdot \deg D_2=\deg D\]
since $L(\sqrt{-d})$ is a maximal subfield of $D_1\ox_F L_2\simeq D_1\ox_Z L_2(\sqrt{-d})$ and $L_2(\sqrt{-d})$ is a maximal subfield of $D_2$. Hence $L(\sqrt{-d})$ is a maximal subfield of $D$ by   \cite[Cor., p.~139]{Plat-Yan}.
It follows that $(L,Q)$ is a maximal ordered extension of $(F,P)$ inside $D$, for if $(M, R)$ were any proper ordered extension of $(L,Q)$, $M$ would be maximal and would contain $Z$, which is a contradiction since $d\in P$ and $\sqrt{-d} \in Z$.
\end{proof}

\begin{cor}\label{second-kind-arb-cor}
 Let $(D, \vt)$ be an $F$-division algebra with involution of the second kind.
 Let $P\in X_F \sm\Nil[D,\vt]$ and write $Z=F(\sqrt{-d})$ for some $d\in P$  \textup{(}cf. Lemma~\ref{Z}\textup{)}.
  There exists  a maximal ordered extension $(L,Q)$
  of $(F,P)$ in $D$ such that  $L(\sqrt{-d})$ is a maximal subfield of $D$ and $D
  \ox_F L \sim L(\sqrt{-d})$.
 \end{cor}

\begin{proof} By Lemma~\ref{Z}, $P$ does not extend to the centre  of $D$ and we conclude with Theorem~\ref{second-kind-arb}.
\end{proof}

\begin{prop}\label{second-kind-arb-b}
 Let $(D, \vt)$ be an $F$-division algebra with involution of the second kind.
Let $P\in X_F$ be such that $P$ extends to an ordering $P'$ on $Z$.
There exists  a maximal ordered extension $(L,Q)$
  of $(Z,P')$ in $D$
such that  $D\ox_F L$ is isomorphic to $M_r(L\x L)$ or to $M_s\bigl ((a,b)_L \x (a,b)_L \bigr)$ for some $r$, $s \in\N$ and $a,b\in L^\x$.

\end{prop}

\begin{proof} We may write $D\simeq D_1\ox_Z D_2$ where $D_1$ is an $F$-division algebra of odd degree with centre $Z$ and $D_2$ is an $F$-division algebra of $2$-power degree with centre $Z$.
Let $L_1$ be a maximal subfield of $D_1$. Then $[L_1:Z]$ is odd.
Let $(L_2, Q_2)$ be a maximal ordered field extension of $(Z, P')$ in $D_2$. By Proposition~\ref{second-kind-2-split} and its proof,
$L_2$ is either a maximal subfield of $D_2$ or a subfield of index $2$ in a maximal subfield of $D_2$.
Since $[L_1:Z]$ and $[L_2:Z]$ are relatively prime, $L_1\ox_Z L_2$ is a field and thus a subfield of $D_1\ox_Z D_2$. Since it is an extension of odd degree of $1\ox_Z L_2$, it carries an ordering $Q_3$ that extends $Q_2$.
 Let $(L,Q)$ be a maximal ordered field extension of $(L_1\ox_Z L_2, Q_3)$ in $D$. If $L_2$ is  maximal, then $L=L_1\ox_Z L_2$ is a maximal subfield of $D$ since $[L:Z]=\deg D$. Otherwise, $L$ has index at most $2$ in a maximal subfield of $D$.
Thus $D\ox_Z L\simeq M_r(L)$ (when $L$ is maximal) or $D\ox_Z L\simeq M_s((a,b)_{L})$ for certain $a,b\in L^\x$ (otherwise) by Lemma~\ref{quat}.
Then
\begin{align*}
D\ox_F L &\simeq D\ox_Z (Z\ox_F L)\\
&\simeq D\ox_Z (L\x L)\\
&\simeq (D\ox_Z L) \x (D\ox_Z L)
\end{align*}
and the result follows.
\end{proof}

\begin{thm}
Let $(D, \vt)$ be an $F$-division algebra with involution of the second kind.  Let $\alpha\in F$ be such that $Z=F(\sqrt{\alpha})$. Then
\[\Nil[D,\vt]= H(\alpha)=\{P\in X_F \mid P\text{ extends to } Z\},\]
where $H(\alpha)$ denotes the usual Harrison set.
\end{thm}

\begin{proof}
We will show that $X_F\sm\Nil[D,\vt]=H(-\alpha)$.
By Lemma~\ref{Z}, $P\not\in \Nil[D,\vt]$ implies $P\in H(-\alpha)$.  Now let $P\in \Nil[D,\vt]$. Assume $P\in H(-\alpha)$. Then $P$ does not extend to $Z$. Let $(L,Q)$ be  as in Theorem~\ref{second-kind-arb} and note that $\alpha \not\in Q$.
Then   $D\ox_F L\simeq M_r(L(\sqrt{\alpha}))$ for some $r\in\N$.
Let $L_Q$ be a real closure of $L$ at $Q$.
Then $\sqrt{\alpha}\not\in L_Q$ and
\begin{align*}
D\ox_F L_Q  & \simeq (D\ox_F L) \ox_L L_Q\\
&\simeq M_r(L(\sqrt{\alpha}) \ox_L L_Q)\\
&\simeq M_r(L_Q(\sqrt{\alpha}))\\
&\simeq M_r(L_Q(\sqrt{-1})).
\end{align*}
Since $F_P \simeq L_Q$, it follows that $D\ox_F F_P \simeq M_r(F_P(\sqrt{-1}))$
 and thus from \eqref{list} and Proposition~\ref{nil-prop}(1)
that $P\not\in \Nil[D,\vt]$.
\end{proof}

Note that therefore $\Nil[D,\vt]$ is clopen in $X_F$. This was already proved in a different way in
\cite[Cor.~6.5]{A-U-Kneb}.

We finish this section with the following natural question:
Let $(D, \vt)$ be an $F$-division algebra with involution of the second kind, let $P\in X_F$ and let $(L,Q)$ be any maximal ordered extension of $(F,P)$ in $D$. Do the conclusions of Theorem~\ref{second-kind-arb} and Proposition~\ref{second-kind-arb-b} hold for $D\ox_F L$? We are currently unable to provide an answer.

\section{$H$-Signatures Revisited}\label{sec-signatures}

Using results obtained in \cite{A-U-Kneb} and \cite{A-U-prime} we give a self-contained presentation of $H$-signatures of hermitian forms  in the following paragraphs.
Let $(A,\s)$ be an $F$-algebra with involution and let $P\in \wt X_F$.
Using Proposition~\ref{nil-prop}(1), Lemma~\ref{lem2.5} and \eqref{list} we
obtain the sequence of group morphisms
\begin{equation}\label{seq}
\xymatrix{
W(A,\s) \ar[r]^--{r_P} & W(A\ox_F F_P, \s\ox\id) \ar[r]^--{\mu_P}_--{\simeq} & W(D_P,\vt_P)\ar[r]^--{\rho_P}
& W(F_P)\ar[r]^--{\sign}  & \Z,
}
\end{equation}
where $r_P$ is the canonical restriction map, $(D_P, \vt_P)$ is an $F_P$-real division algebra with involution, $\mu_P$ is an isomorphism induced by Morita theory, $\rho_P$ is defined by
$\rho_P(\eta)(x):=\eta(x,x)$ for all $\eta \in W(D_P,\vt_P)$ (cf. \cite{Jacobson-40}) and $\sign$ is the usual signature of quadratic forms at the unique ordering of $F_P$.

In \cite[\S3.2]{A-U-Kneb} we showed that  the map $|\sign\circ\rho_P\circ \mu_P\circ r_P|$ does not depend on the choice of $F_P$ and $\mu_P$. In \cite[Prop.~3.2]{A-U-prime} we showed that
there exists a hermitian form $H_0\in W(A,\s)$ such that for all $P\in \wt X_F$,
\begin{equation}\label{sign_form}
\sign (\rho_P\circ\mu_P )(H_0\ox F_P) \not= 0.
\end{equation}
We call $H_0$ a \emph{reference form} for $(A,\s)$.

\begin{defi} Let $H_0\in W(A,\s)$ be a reference form for $(A,\s)$, let $h\in W(A,\s)$, let $P\in \wt X_F$, let $\ell=\dim_{F_P} D_P$ and let $\delta_P^{H_0} = \sgn\Bigl(\sign (\rho_P\circ\mu_P )(H_0\ox F_P)\Bigr) \in \{-1,1\}$.
We define the \emph{$H$-signature of $h$ at $P\in X_F$}, $\sign_P^{H_0} h$, by
\[\sign_P^{H_0} h:=\frac{1}{\ell} \delta_P^{H_0} \sign (\rho_P\circ\mu_P )(h\ox F_P) \]
whenever $P\in \wt X_F$ and $\sign_P^{H_0} h :=0$ if $P\in\Nil[A,\s]$, cf. Proposition~\ref{nil-prop}(4).
\end{defi}

\begin{remark}\label{going-up}\mbox{}
\begin{enumerate}[(1)]
\item We showed in \cite[\S3.3]{A-U-Kneb} that $\sign_P^{H_0}$ only depends on $P$ and $H_0$ and in \cite[\S7]{A-U-prime} that  $H$-signatures  correspond canonically to a natural class of morphisms from $W(A,\s)$ to $\Z$.
\item If $L$ is a finite extension of $F$, it follows from \eqref{sign_form} that $H_0\ox_F L$ is a reference form for $(A\ox_F L, \s\ox\id)$. Moreover, if $R$ is an ordering on $L$ that extends $P\in X_F$, then
\[\sign_R^{H_0\ox L} (h\ox L) = \sign_P^{H_0} h\]
for all $h\in W(A,\s)$.
\item Let $P\in \wt X_F$. If $H_1$ is another reference form for $(A,\s)$, then easy computations show that
\[\sign_P^{H_0} h=\delta_P^{H_0} \delta_P^{H_1} \sign_P^{H_1} h \]
for all $h\in W(A,\s)$ and
\[\delta_P^{H_0} \delta_P^{H_1} = \sgn(\sign_P^{H_1}H_0 ) = \sgn(\sign_P^{H_0} H_1).\]
\end{enumerate}
\end{remark}

\begin{lemma}\label{Jacobson}
Let $P\in X_F$, let $(D,\vt)$ be an $F$-division algebra with involution which is $(F,P)$-real and let $\ell= \dim_F D$.  Consider the reference form $H=\la 1\ra_\vt$ and
the group morphism
$\rho: W(D,\vt)\too W(F)$,
where $\rho(h)(x):=h(x,x)$.   Then
\[\sign_P^H h=\frac{1}{\ell} \sign_P \rho(h).\]
\end{lemma}

\begin{proof} Observe that $H$ is indeed a reference form.
Let $F_P$ denote a real closure of $F$ at $P$ and note that $P\not\in\Nil[D,\vt]$ by
Lemma~\ref{lem2.5}.
Consider the diagram

\begin{equation}
\begin{split}
\xymatrix{
W(D,\vt) \ar[d]_--{\rho} \ar[r]^--{r_P} & W(D\ox_F F_P, \vt\ox\id) \ar[r]^--{\id} & W(D\ox_F F_P, \vt\ox\id)
\ar[d]^--{\rho_P} \\
W(F)\ar[rr]^--{r'_P} & & W(F_P)\ar[r]^--{\sign}  & \Z
}
\end{split}
\end{equation}
where we used the notation from the sequence \eqref{seq}. The square on the left commutes:
Let $h\in W(D,\vt)$.
Then $\rho(h)\ox F_P = \rho_P(h\ox F_P)$ since
\[(\rho(h)\ox F_P) (x\ox1)=\rho(h)(x)\ox 1 =h(x,x)\ox 1= (h\ox F_P) (x\ox 1, x\ox 1)=\rho_P(h\ox F_P)(x\ox 1).\]
It follows that
\begin{align*}
\sign^H_P h
&= \frac{1}{\ell} \ve_P \sign (\rho_P\circ\id)(h\ox F_P)\\
&= \frac{1}{\ell} \sign \rho(h)\ox F_P\\
&= \frac{1}{\ell} \sign_P \rho(h),
\end{align*}
where $\ve_P  = \sgn\Bigl(\sign (\rho_P\circ\id )(H\ox F_P)\Bigr)=  \sgn\bigl(\sign (\ell\x \la 1\ra)\bigr)=   1$.
\end{proof}

\begin{lemma}\label{sign-surj}
 Let $P\in \wt X_F$ and let $(D,\vt)$ be an $(F,P)$-real division algebra with involution. Let $H$ be a reference form for $(D,\vt)$.  The signature map
\[ \sign_P^H :W(D,\vt) \too \Z\]
is surjective.
\end{lemma}

\begin{proof} Let $h=\la 1\ra_\vt$. Then $\sign_P^H h = \pm 1$ by Remark~\ref{going-up}(3) and Lemma~\ref{Jacobson}.
\end{proof}

\section{Stability Index of Algebras with Involution}

In this section, we fix an $F$-algebra with involution $(A,\s)$ and a reference form $H$ for $(A,\s)$. Let $C(X_F,\Z)$ denote the ring of continuous functions from $X_F$ (equipped with the Harrison topology)  to $\Z$ (equipped with the discrete topology). For every $h \in W(A,\s)$ we denote by $\sign^H h$ the total signature map from $X_F$ to $\Z$ and remark that it is a continuous map, cf. \cite[Thm.~7.2]{A-U-Kneb}.
If $h \in W(A,\s)$, we
have $\sign^H h = 0$ on $\Nil[A,\s]$.
Therefore it is convenient to introduce
the  notation
\[\wt C(X_F, \Z) := \{f \in C(X_F, \Z) \mid f=0 \text{ on } \Nil[A,\s]\}.\]
Note that $\wt C(X_F, \Z)$ depends on the Brauer class of $A$ and the type of $\s$, but indicating this would make the
notation cumbersome. Since $(A,\s)$ is fixed, no confusion should arise.
For $P\in X_F$ and  a field extension $L$ of $F$, we define
\[X_L/P:=\{Q \in X_L \mid P \subseteq Q\}.\]

\begin{lemma}\label{power-of-2-local}
   Let $P \in \widetilde
  X_F$.  Then there exists a hermitian form $h_P \in W(A,\sigma)$ and a positive
  integer $\ell_P$ such that $\sign^H_P h_P = 2^{\ell_P}$.
\end{lemma}

\begin{proof}
By Theorem~\ref{first-kind} and Corollary~\ref{second-kind-arb-cor} there exists an ordered field extension $(L,Q)$ of $(F,P)$ with $L\subseteq A$ and such that  $(A\ox_F L, \s\ox\id)$ is Morita equivalent to an $(L,Q)$-real division algebra with involution. Note that  $[L:F]$ is finite.  Let $Q_0 \in
  X_L/P$. By Lemma~\ref{sign-surj} and \cite[Thm.~4.2]{A-U-prime}
  there exists a hermitian form $h$ over $(A
  \otimes_F L, \s \otimes \id)$ such that $\sign^{H\ox L}_{Q_0} h = 1$.
  Since $X_L/P$ is finite, there exist $a_1, \ldots, a_r \in L$ such that $\{Q_0\}
  = \{Q \in X_L/P \mid a_1, \ldots, a_r \in Q\}$. Using the notation from \cite[\S 5.3]{A-U-Kneb},
  let $h_P := \trs_{A\ox_F L}\bigl( \la\!\la a_1,
  \ldots, a_r \ra\!\ra \cdot h\bigr)$. It follows from the Knebusch trace formula
  \cite[Thm. 8.1]{A-U-Kneb}
  that
  \begin{align*}
    \sign^H_P h_P &= \sum_{Q \in X_L/P} \sign^{H \otimes L}_Q \bigl( \la\!\la a_1,
      \ldots, a_r \ra\!\ra \cdot h \bigr) \\
      &= 2^r \sign^{H \otimes L}_{Q_0} h\\
      &= 2^r.\qedhere
  \end{align*}
  \end{proof}

\begin{lemma}\label{h-naught}
  There exists a
  hermitian form $h_0 \in W(A,\sigma)$ and a positive integer $k_0$ such that
  $\sign^H_P h_0 = 2^{k_0}$ for every $P \in \widetilde X_F$.
\end{lemma}

\begin{proof}
  By Lemma \ref{power-of-2-local}, for every $P \in \widetilde X_F$ there exists $\ell_P \in \N$,
  $U_P$ clopen in $\wt X_F$ containing $P$, and $h_P \in W(A,\s)$ such that $\sign^H h_P = 2^{\ell_P}$
  on $U_P$ (simply take $U_P = (\sign^H h_P)^{-1}(2^{\ell_P})$).

  Therefore $\widetilde X_F = \bigcup_{P \in \widetilde X_F} U_P = \bigcup_{i=1}^n
  U_{P_i}$ since $\widetilde X_F$ is compact. By removing  the intersections
  of the sets $U_{P_i}$ we obtain $\widetilde X_F = \bigdotcup_{i=1}^r C_i$ where each
  $C_i$ is clopen and $\sign^H \eta_i = 2^{\ell_i}$ on $C_i$ for  hermitian forms
  $\eta_i \in W(A,\s)$, $i=1,\ldots, r$.

  Let $q_i \in W(F)$ be such that $\sign q_i$ is equal to $2^{s_i}$ on $C_i$
  (for some integer $s_i$) and $0$ elsewhere, cf. \cite[VIII, Lemma~6.10]{Lam}. Then
  $\sign^H(q_i\cdot \eta_i)$ is equal to $2^{s_i+\ell_i}$ on $C_i$ and $0$ elsewhere. Taking
  $k_0 = \max\{s_1+\ell_1, \ldots, s_r+\ell_r\}$ and multiplying $q_i\cdot \eta_i$ by a suitable power of $2$, we obtain a form $h_i \in
  W(A,\s)$ such that $\sign^H h_i = 2^{k_0}$ on $C_i$ and $0$ elsewhere. It
  follows that, for $h_0 := h_1 + \cdots + h_r$, $\sign^H h_0 = 2^{k_0}$ on
  $\widetilde X_F$.
\end{proof}

\begin{prop}\label{cokernel}
  Let $k_0 \in \N$ and $h_0 \in W(A,\s)$ be such that $\sign^H h_0 = 2^{k_0}$ on
  $\wt X_F$.
  \begin{enumerate}[$(1)$]
    \item Let $q \in W(F)$. Then there exists $h \in W(A,\s)$ such that
      $\sign(2^{k_0}q) = \sign^H(h)$ on $\wt X_F$.
    \item Let $f \in \widetilde C(X_F, \Z)$. Then there exists $n \in \N$ such that $2^n f \in
      \im(\sign^H)$.
  \end{enumerate}
\end{prop}

\begin{proof}
(1) We take $h = q\cdot h_0$.

 (2) We know that there exists $m \in \N$ such that $2^mf = \sign(q)$ for some
      $q \in W(F)$, cf. \cite[VIII, Lemma~6.9]{Lam}. Then $2^{m+k_0}f = \sign^H(q\cdot h_0)$.
\end{proof}

\begin{defi}\mbox{}
  \begin{enumerate}[(1)]
    \item We define  $\st^H(A,\s)$ to be the smallest nonnegative integer $k$ such that
      \[2^k\cdot \widetilde C(X_F,\Z) \subseteq \im(\sign^H)\]
      if such an integer exists, and infinity otherwise.
    \item We define $S^H(A,\s)$  to be the cokernel of the total signature map
      \[\sign^H : W(A,\s) \rightarrow \widetilde C(X_F,\Z).\]
  \end{enumerate}
\end{defi}

The following corollary follows immediately from Proposition~\ref{cokernel}.
\begin{cor}
  The group $S^H(A,\s)$ is $2$-primary torsion, and its exponent is $2^{\st(A,\s)}$ \textup{(}with the convention that $2^\infty = \infty$\textup{)}.
\end{cor}

\begin{prop}\label{stab-iso}
  Let $H'$ be another reference form for $(A,\s)$. Then $S^H(A,\s) \simeq
  S^{H'}(A,\s)$. In particular $\st^H(A,\s) = \st^{H'}(A,\s)$.
\end{prop}

\begin{proof}
  By \cite[Prop.~3.3$(iii)$]{A-U-prime}, there exists $f \in C(X_F,\{-1,1\})$ such that $\sign^H =
  f\cdot \sign^{H'}$. Define
  \[\begin{array}{rcl}
      \xi : \widetilde C(X_F,\Z) & \longrightarrow & \widetilde C(X_F,\Z)/\im \sign^H \\
      g & \longmapsto & f\cdot g + \im \sign^H
    \end{array}\]
  The map $\xi$ is a surjective morphism of groups since $f$ is invertible in $C(X_F,\Z)$. Moreover, $g \in \ker \xi$ if and only
  if $f\cdot g \in \im \sign^H = f\cdot \im \sign^{H'}$, so $\ker \xi = \im
  \sign^{H'}$ and $\xi$ induces an isomorphism from $S^{H'}(A,\s)$ to
  $S^H(A,\s)$.
\end{proof}

\begin{defi}  We call $S(A,\s)$ the \emph{stability group} of $(A,\s)$. It is well-defined up to isomorphism by Proposition~\ref{stab-iso}. We call $\st(A,\s)$ the \emph{stability index} of $(A,\s)$. \end{defi}

\begin{prop}
  Let $h_0 \in W(A,\s)$ and $k_0 \in \N$ be as in Lemma~\ref{h-naught}.
Then
\[\st(A,\s) \le \st(F)   + k_0.\]
  \end{prop}

\begin{proof}
Assume that $\st(F)$ is finite.
  Let $f \in \widetilde C(X_F, \Z)$. Then there exists $q \in W(F)$ such that $2^{\st(F)} f = \sign
  q$, and thus $\sign^H(q\cdot h_0) = 2^{\st(F) + k_0}f$.
\end{proof}

\begin{prop}
  Let $(A,\s)$ and $(B,\tau)$ be two Morita equivalent $F$-algebras with involution.  Then $S(A,\s) \simeq S(B,\tau)$ and $\st(A,\s) =
  \st(B,\tau)$.
\end{prop}
\begin{proof}
  It suffices to prove the first part of the statement, but this follows
  immediately from \cite[Thm.~ 4.2]{A-U-prime}.
\end{proof}

The following theorem extends a well-known result in quadratic form theory (cf. \cite[(1.6)]{Broecker-74}) to algebras with involution:

\begin{thm} Let $(A,\s)$ be an $F$-algebra with involution and let $W_\tor(A,\s)$ denote the torsion subgroup of $W(A,\s)$.
The sequence
\begin{equation*}
\xymatrix{
0 \ar[r] &   W_\tor(A, \sigma ) \ar[r] & W(A, \sigma) \ar[r]^--{\sign^H} &  \wt C(X_F,\mathbb{Z}) \ar[r] & S(A, \sigma )\ar[r] & 0
}
\end{equation*}
is exact. The groups $W_\tor(A, \sigma )$ and $S(A, \sigma )$ are $2$-primary torsion groups.
\end{thm}

\begin{proof} This follows from \cite[Thm.~4.1]{LU1}, \cite[Thm.~5.1]{Sch1} and the definition of $S(A,\s)$.
\end{proof}

\begin{examples} In each of the examples below, $A$ is a quaternion division algebra.
\begin{enumerate}[(1)]
\item Let $F=\R$, $A=(-1,-1)_F$ and $\s$ quaternion conjugation. Then $\im\sign^H\simeq \Z\simeq \wt C(X_F,\Z)$. Hence $S(A,\s)\simeq \{0\}$ and $\st(A,\s)=0$. Note that $\st(\R)=0$.

\item Let $F=\R$, $A=(-1,-1)_F$ and $\s$ orthogonal. Then $\im\sign^H\simeq \{0\} \simeq \wt C(X_F,\Z)$. Hence $S(A,\s)\simeq \{0\}$ and $\st(A,\s)=0$.

\item Let $F=\Q(\sqrt{2})$, $A=(-1, -\sqrt{2})_F$ and $\s$ quaternion conjugation. Then $\im\sign^H\simeq \Z\x \{0\} \simeq \wt C(X_F,\Z)$. Hence $S(A,\s)\simeq \{0\}$ and $\st(A,\s)=0$. Note that $\st(\Q(\sqrt{2}))=1$.

\item Let $F=\R(\!(x)\!)$, $A=(x, -1)_F$ and $\s$ orthogonal. Then $\im\sign^H\simeq 2\Z\x \{0\}$ and $\wt C(X_F,\Z)\simeq \Z \x \{0\}$. Hence $S(A,\s)\simeq \Z/2\Z$ and $\st(A,\s)=1$. Note that $\st(\R(\!(x)\!))=1$.
\end{enumerate}
\end{examples}

We now consider the total signature  $\sign^H h$ of a hermitian form $h\in W(A,\s)$. Since this is a continuous function, there exists an integer $k$ such that $2^k\sign^H (h)$ is the total signature of some quadratic form over $F$. In the next two results we will show that $k$ can be chosen independently of $h$.

\begin{lemma}\label{rel-part}
  There exist disjoint clopen subsets $U_1, \ldots, U_t$ of $\widetilde X_F$ and
  positive integers $n_1, \ldots, n_t $ such that $\widetilde X_F = U_1 \cup
  \cdots \cup U_t$ and for every $h \in W(A,\s)$ and $i \in \{1,\ldots, t\}$,
  there exists $q_i \in W(F)$ such that $\sign(q_i) = 2^{n_i}\sign^H(h)$ on $U_i$.
\end{lemma}

\begin{proof}
   By Theorem~\ref{first-kind} and Corollary~\ref{second-kind-arb-cor}, for every $P \in \widetilde X_F$ there exists a
   finite ordered field extension  $(L_P, R_P)$ of $(F,P)$ such that $(A
   \otimes_F L_P, \s\ox\id)$ is Morita equivalent to an $(L_P,R_P)$-real
   division algebra with involution $(D_P, \vt_P)$.

  Let $S_P = \{Q \in \widetilde X_F \mid Q \textrm{ extends to } L_P\}$. By
  \cite[Chap.~3, Thm.~4.4]{Sch} we have $S_P= \{Q \in \widetilde X_F \mid
  \sign_Q (\trs_{L_P/F}\la 1 \ra) > 0\}$.  The set $S_P$ is therefore a clopen
  subset of $\widetilde X_F$ containing $P$, and by compactness there are $P_1,
  \ldots, P_t \in \widetilde X_F$ such that $\widetilde X_F = S_{P_1} \cup
  \cdots \cup S_{P_t}$. It follows that there are disjoint clopen sets $S_i
  \subseteq S_{P_i}$ (for $i=1, \ldots, t$) such that $\widetilde X_F = S_1 \dotcup
  \cdots \dotcup S_t$.

Let $i \in \{1,\ldots, t\}$ and write $S:=S_i$, $L:=L_{P_i}$. It suffices to prove the lemma for $S$ instead of $\wt X_F$.
  Let $Q\in S$ and let $R \in X_{L}/Q$. We have
  morphisms of Witt groups \[W(A\ox_F L, \s\ox\id) \stackrel{\mu}{\too}
  W(D,\vt) \stackrel{\rho}{\too} W(L),\] where $(D,\vt)$ is $(L,
  R)$-real  (note that  $R\not\in\Nil[D,\vt]$ since $Q\not\in\Nil[A,\s]$ by
  Proposition~\ref{nil-prop}), $\mu$ is an isomorphism induced by Morita
  equivalence and $\rho$ is the map of Lemma~\ref{Jacobson}.  Let $H_0=\la
  1\ra_{\vt} \in W(D, \vt)$ and let $\delta_R:= \sgn(\sign_R^{\mu(H\ox
  L)} H_0) $.  Let $h\in W(A,\s)$ be arbitrary. Then, using
  Remark~\ref{going-up}, we have
  \begin{align*}
    4\sign_Q^H h &= 4\sign_R^{H\ox L} (h\ox L) \\
    &= 4\sign_R^{\mu(H\ox L)} \mu(h\ox L) & \textrm{[by  \cite[Thm.~4.2]{A-U-prime}]}\\
    &= 4\sgn(\sign_R^{\mu(H\ox L)} H_0) \sign_R^{H_0} \mu(h\ox L) \\
    &= 4\delta_R \sign_R^{H_0} \mu(h\ox L)\\
    &= \delta_R\frac{4}{\ell} \sign_R \rho(\mu(h\ox L))    & \textrm{[by Lemma~\ref{Jacobson}]}\\
    &= \delta_R \sign_R\vf,
  \end{align*}
  where $\ell=\dim_{L}D \in \{1,2,4\}$
  and $\vf:= \frac{4}{\ell}\rho(\mu(h\ox L)) \in W(L)$.

  Observe that for every $R \in X_{L}/Q$, $\sign_R^{\mu(H\ox L)} H_0 \not
  = 0$ since $H_0$ is a reference form for $(D,\vt)$.
  Since $X_{L}/Q$ is finite, there is a finite tuple $\bar a_Q \in L$ of length $\ell_Q$
  such
  that $H(\bar a_Q) \cap (X_{L}/Q)$  contains only one ordering $R_{Q}$. In
  particular
  \[\sum_{R \in X_{L}/Q} \sign_R^{\mu(H\ox L)} (\la\!\la \bar a_Q\ra\!\ra
    \cdot H_0) = 2^{\ell_Q} \sign_{R_{Q}}^{\mu(H\ox L)} H_0 \not =
  0.\]
  Define $\vf_{Q} := \la\!\la \bar a_Q \ra\!\ra \cdot \vf$. Then, by \cite[Chap.~3, Thm.~4.5]{Sch},
  \[\sign_Q (\trs_{L/F} \vf_{Q}) = \sum_{R \in X_{L}/Q} \sign_R \vf_{Q} =
  2^{\ell_Q} \sign_{R_{Q}} \vf = 2^{\ell_Q + 2}
  \delta_{R_{Q}} \sign^H_Q h,\]
  where $\trs_{L/F} : W(L) \to W(F)$ denotes the Scharlau transfer.
  Therefore the clopen subset of $X_F$
  \[U_Q := (\sign (\trs_{L/F} \vf_{Q}) - 2^{\ell_Q + 2}
  \delta_{R_{Q}} \sign^H h)^{-1}(0)\]
  contains $Q$, and thus $S = \bigcup_{Q \in X_F} U_Q$. Since $S$ is compact
  we obtain $S = U_{Q_1} \cup \cdots \cup U_{Q_r}$ for some $Q_1, \ldots, Q_r
  \in S$, and for every $Q \in U_{Q_j}$:
  \[2^{\ell_{Q_j} + 2} \delta_{R_{Q_j}} \sign^H_Q h = \sign_Q
  (\trs_{L/F} \vf_{Q_j}),\]
  and thus
  \[2^{\ell_{Q_j} + 2} \sign^H_Q h = \sign_Q
  (\delta_{R_{Q_j}} \trs_{L/F}  \vf_{Q_j}).\]
  Since  $\delta_{R_{Q_j}} \trs_{L/F} \vf_{Q_j} \in W(F)$, the result
  follows by taking clopen sets $U_j \subseteq U_{Q_j}$ such that $S = U_1
  \dotcup \cdots \dotcup U_r$.
  \end{proof}

\begin{thm}\label{rel-st}
  There exists $n_0 \in \N$ such that for every $h \in W(A,\s)$ there exists $q_h \in
  W(F)$ with $2^{n_0} \sign^H h = \sign q_h$.
  \end{thm}

\begin{proof}
  We use the terminology of Lemma~\ref{rel-part}. Let $r := n_1
  + \cdots + n_t$. Then for every $h \in W(A,\s)$ and every $i \in \{1, \ldots,
  t\}$ there exists $q_i \in W(F)$ such that $\sign q_i = 2^r \sign^H h$ on $U_i$.

  Since $t$ is finite, there exist $m \in \N$ and $p_i \in W(F)$ such that $\sign
  p_i = 2^m$ on $U_i$ and $\sign p_i = 0$ on $X_F \setminus U_i$. Therefore
  $ \sign (p_i q_i)= 2^m \sign q_i=2^{m+r} \sign^H h$ on $U_i$ and $\sign (p_i q_i) =
  0$ on $X_F \setminus U_i$, and we obtain for $i=1, \ldots, t$,
  \begin{align*}
    \sign (p_1 q_1 + \cdots + p_tq_t) &= \sign p_iq_i \text{ on } U_i \\
         &= 2^{m + r} \sign^H h \text{ on } U_i
  \end{align*}
  and thus
 \[     \sign (p_1 q_1 + \cdots + p_tq_t) = 2^{m + r} \sign^H h \text{ on } X_F\]
  (note that by construction the quadratic form $p_1 q_1 + \cdots + p_tq_t$ has
  zero signature on $\Nil[A,\s]$).
\end{proof}

\begin{defi} We define
  \[W_\rd(A,\s) := W(A,\s)/W_\tor(A,\s) \simeq \sign^H(W(A,\s))\subseteq \wt C(X_F,\Z)\]
  and call this the \emph{reduced Witt group} of $(A,\s)$.
\end{defi}

In order to compare reduced hermitian forms and reduced quadratic forms, we also introduce
\[ \wt W_\rd(F):= \{q \in W_\rd(F) \mid \sign q=0 \text{ on } \Nil[A,\s]\}.\]
Observe that $W_\rd(A,\s)$ is a $W_\rd(F)$-module and also a $\wt W_\rd(F)$-module in the natural way.

\begin{prop}
  Let $h_0 \in W(A,\s)$ and $k_0 \in \N$ be such that $\sign(h_0) = 2^{k_0}$ (cf. Lemma~\ref{h-naught}).   With notation as in Theorem~\ref{rel-st},
  the maps
  \[\wt W_\rd(F) \longrightarrow W_\rd(A,\s), \ q \longmapsto qh_0\]
  and
  \[W_\rd(A,\s) \longrightarrow \wt W_\rd(F), \ h \longmapsto q_h\]
are well-defined injective
  morphisms of $W_\rd(F)$-modules.
\end{prop}

\begin{proof}
  Identifying $W_\rd(F)$ and $W_\rd(A,\s)$ with the images of
  $\sign$ and $\sign^H$, we see that the first map is simply  multiplication
  by $2^{k_0}$.

  The second map is
  well-defined because $h_1 = h_2$ in $W_\rd(A,\s)$ is equivalent to $\sign^H h_1 =
  \sign^H h_2$, which implies $\sign q_{h_1} = \sign q_{h_2}$, so $q_{h_1} =
  q_{h_2}$ in $\wt W_\rd(F)$. It is also easy to check that it is an
  injective morphism of $W_\rd(F)$-modules.
\end{proof}

We finish this paper by pointing out some difficulties that need to be overcome in order to further the study of the stability index of algebras with involution.

In the quadratic form literature one can find important links between the stability index of the field $F$ and the powers of the fundamental ideal $I^n(F)$, which crucially depend on Pfister forms (see for example \cite[Satz~3.17]{Broecker-74}).
Although we can define $I(A,\s)$, the lack of a tensor product of hermitian forms in general   is a
serious obstacle to the development of  analogous concepts and connections for Witt groups of algebras with involution.

Another issue is the following: the quadratic Pfister form $\la 1, a\ra$ has signature $2$ on $H(a)$ and signature $0$ on $H(-a)$, which is a fundamental observation when considering   $\st(F)$. In contrast, this behaviour cannot in general be replicated with hermitian forms since
the $H$-signature of the form $\la 1\ra_\s$ may not be constant  and in addition may take values which are not in $\{-1,1\}$.

\def\cprime{$'$}


text:
\subsection{Stability result for LS-CS (dynamic CS-residual)}
The overall approach is similar to the one discussed above for modified-CS. The key difference is in the detection condition lemma, which we give below.

\begin{lemma}[Detection condition for LS-CS]
Assume that $\|w\| \le \eps$, $|T| \le S_T$ and $|\Delta| \le S_\Delta$.
Let $b:=\|x_\Delta\|_\infty$. For a $\gamma \le 1$, let $\Delta_1:=\{i \in \Delta: \gamma b \le |x_i| \le b \}$ and let $\Delta_2:= \Delta \setminus \Delta_1$. Assume that $|\Delta_1| \le S_{\Delta_1}$ and $\|x_{\Delta_2}\| \le \kappa$.
\\
All $i \in \Delta_1$ will definitely get detected at the current time if
$\delta{(2S_\Delta)} < \sqrt{2}-1$, $\delta{(S_T)} < 1/2$,
\label{detectcond_lscs}
\bea
&&   \theta{(S_T,S_\Delta)} \sqrt{S_{\Delta_1}} C''(S_T,S_\Delta) < {\gamma} \ \ \text{and} \nn \\
&& \max_{|\Delta| \le S_\Delta} \frac{ \alpha_{add} + C'(S_T,|\Delta|)\eps  + \theta{(S_T,|\Delta|)} C''(S_T,|\Delta|) \kappa }{\gamma - {\theta{(S_T,|\Delta|)}} \sqrt{S_{\Delta_1}} C''(S_T,|\Delta|)} < b \ \  \nn
\label{detcond_lscs}
\eea
where $C'(.,.)$, $C''(.,.)$ are defined in Theorem \ref{lscsbnd}.
\end{lemma}

{\em Proof: }
From Theorem \ref{lscsbnd}, if $\|w\| \le \eps$, $\delta{(2|\Delta|)} < \sqrt{2}-1$, $\delta{(|T|)} < 1/2$, then $\|x - \xhat_{CSres}\| \le C'(|T|,|\Delta|) \eps +  \theta{(|T|,|\Delta|)} C''(|T|,|\Delta|) \|x_\Delta\|$.
Using $\|x_\Delta\|  \le \sqrt{|\Delta_1|} b + \|x_{\Delta_2}\|$, fact \ref{det1} of Sec. \ref{stab_modcs} and the fact that for all $i \in \Delta_1$, $|x_i| \ge \gamma b$, we can conclude that all $i \in \Delta_1$ will get detected if $\delta{(2|\Delta|)} < \sqrt{2}-1$, $\delta{(|T|)} < 1/2$ and $\alpha_{add} + C' \eps + \theta C'' \|x_{\Delta_2}\| + \theta C'' \sqrt{|\Delta_1|} b < \gamma b$. The third inequality holds if ${\theta} \sqrt{|\Delta_1|}C'' < {\gamma}$ and $\frac{\alpha_{add} + C' \eps + \theta  C'' \|x_{\Delta_2}\|}{\gamma - \theta \sqrt{|\Delta_1|} C''} < b$.

Since we only know that $|T| \le S_T$, $|\Delta| \le S_\Delta$,  $|\Delta_1| \le S_{\Delta_1}$ and $\|x_{\Delta_2}\| \le \kappa$, we need the above inequalities to hold for all values of  $|T|,|\Delta|,|\Delta_1|,\|x_{\Delta_2}\|$ satisfying these upper bounds. This leads to the conclusion of the lemma. The left hand sides (LHS) of all the required inequalities, except the last one, are non-decreasing functions of  $|\Delta|,|T|,|\Delta_1|$ and thus we just use their upper bounds. The LHS of the last one is non-decreasing in $|T|,|\Delta_1|,\|x_{\Delta_2}\|$, but is not monotonic in $|\Delta|$ (since $C'(|T|,|\Delta|)$ is not monotonic in $|\Delta|$). Hence we explicitly maximize over $|\Delta| \le S_\Delta$.

$\blacksquare$

The stability result then follows in the same fashion as Theorem \ref{stabres_modcs}. The only difference is that instead of Lemma \ref{detectcond_modcs}, we use Lemma \ref{detectcond_lscs} applied with $S_T=S_0, S_\Delta=k_1(d_0)$, $b=d_0M/d$, $\gamma=1$, $S_{\Delta_1}=S_a$ and $\kappa = \sm(d_0)$.

\begin{theorem}[Stability of LS-CS]
Assume Signal Model \ref{sigmod2} and $\|w\| \le \eps$. If the following hold for some $1 \le d_0 \le d$,
\ben
\item  {\em (add/del thresholds) } (same condition as in Theorem \ref{stabres_modcs})
\item {\em (no. of measurements, $n$) } $n$ is large enough so that
\label{measmodel_lscs}
\ben
\item $\delta{(2k_1(d_0))} < \sqrt{2}-1$ and $\delta{(S_0+S_a + f)} < 1/2$

\item $ \theta{(S_0,k_1(d_0))} \sqrt{S_a} C''(S_0,k_1(d_0))   <  1$
\label{measmodel_lscs_2}
\een

\item {\em (SNR and $n$) } $(d_0M/d) \ge \max(\tilde{G}_1,\tilde{G}_2)$, where
\label{add_del_lscs}
\bea
&& \tilde{G}_1 \defn
\max_{|\Delta| \le k_1(d_0)} \nn \\ &&
 [\frac{ \alpha_{add} + C'(S_0,|\Delta|)\eps  + \theta{(S_0,|\Delta|)} C''(S_0,|\Delta|) \sm(d_0) }{\gamma - {\theta{(S_0,|\Delta|)}} \sqrt{S_a} C''(S_0,|\Delta|)}] \nn \\
&& \tilde{G}_2 \defn \alpha_{del} + \sqrt{2} \eps + 2{\theta{(S_0+S_a+f,k_2(d_0))}} \sm(d_0) \nn
\eea

\item {\em (initialization) }  (same condition as in Theorem \ref{stabres_modcs})
\een
then, all conclusions of Theorem \ref{stabres_modcs} and Corollary \ref{stabres_modcs_cor} hold for LS-CS, except the second claim of Corollary \ref{stabres_modcs_cor}, which is replaced by $\|x_t - \xhat_{t,\CSres}\| \le \max_{|\Delta| \le k_1(d_0)} [ C'(S_0,|\Delta|)\eps  + \theta{(S_0,|\Delta|)} C''(S_0,|\Delta|)[e(d_0)+S_a (d_0M/d)^2]]$.
\label{stabres_lscs}
\end{theorem}

\subsection{Implications of the Results and Comparisons}
Both the stability results provide sufficient conditions under which the number of misses is less than $(2d_0-2)S_a$ and the number of extras is zero.
From our discussion in the introduction, stability is meaningful only if $(2d_0-2)S_a \ll S_0$.
From \cite{icipmodcs}, we know that support changes slowly, i.e. $S_a \ll S_0$. Also, usually it is valid to assume that either $2dS_a \ll S_0$ (only a small number of coefficients are increasing or decreasing) or $M$ is large enough to ensure that $d_0 \ll d$. Either of these ensures $(2d_0-2)S_a \ll S_0$.

For modified-CS, under Signal Model \ref{sigmod2}, ``stability" is ensured under the following fairly mild assumptions: (a) the addition/deletion thresholds are appropriately set (conditions \ref{addthresh} and \ref{delthresh});  (b) the noise is bounded and the number of measurements, $n$, is large enough for condition \ref{measmodel} to hold; and (c) for a given support change rate, $S_a$, and magnitude change rate, $M/d$, the worst-case noise power, $\eps^2$ is small enough and $n$ is large enough to ensure that condition \ref{add_del} holds.

The main difference for LS-CS stability is that the detection condition is much stronger (compare Lemma \ref{detectcond_lscs} with Lemma \ref{detectcond_modcs}). This is because CS-residual error depends on $\|x_\Delta\|$, while there is no such dependence for modified-CS. Thus LS-CS needs an extra condition on $\theta{(S,S')}$ given in condition \ref{measmodel_lscs_2} of Theorem \ref{stabres_lscs}, which may not hold if $n$ is too small.
Also, when $n$ is just large enough for condition \ref{measmodel_lscs_2} to hold,  $\tilde{G}_1$ (defined in condition \ref{add_del_lscs}) may be very large and thus LS-CS will need higher signal increase rate, $M/d$, for its condition \ref{add_del_lscs} to hold.

When $n$ is large enough (for stability), it is easy to set $\alpha_{add}$ so that there are few false detections, $f$, compared to $S_0$, e.g. in our simulations, the average value of $f$ was less than 1 when $S_0=20$. Now, $f \ll S_0$ along with $(2d_0-2)S_a \ll S_0$ (discussed earlier) implies that the requirements of Theorem \ref{stabres_modcs} on $n$ (condition \ref{measmodel}) are {\em significantly weaker} than those for the corresponding CS result \cite{candes_rip} which requires $\delta(2S_0) \le \sqrt{2}-1$.
The same is true when comparing the LS-CS requirements with those for CS, as long as $M/d$ is large enough to ensure that $d_0$ is small enough for its conditions \ref{measmodel_lscs_2} and \ref{add_del_lscs} to hold.

config: arxiv


TITLE: Adding copy of row to "almost normalized" matrix does not change $X^\top X$ product
QUESTION [0 upvotes]: Good evening, I have a hard time with the following problem, I found how to start, but don't see how to continue:
Imagine we have a $D$x$N$ matrix $X$ such that all rows are normalized (zero mean and variance one), except for the last row, that has variance 2. Suppose that we also have a second matrix, say $\tilde{X}$, which is defined as a $(D+1)$x$N$ matrix with the same rows as X, except row $D$ is now normalized, and row $D + 1$ is a copy of row $D$.
So to summarize, $X$ is almost normalized, only last row has variance 2 and $\tilde{X}$ is perfectly normalized, but has two equal rows. Now I have to show that $X^\top X = \tilde{X}^\top\tilde{X}$.
I started by writting $\tilde{X}$ as $KX$ where K is a $(D+1)$x$D$ matrix with the first $D$ rows forming an identity matrix, and the last row being $\frac{1}{\sqrt{2}} * \textbf{e}_n$ where $\textbf{e}_n$ is the unit vector with a 1 in position n and zeros everywhere else, so that we have a normalized row in $\tilde{X}$. After that, I still cannot show the result...
Any help would be greatly appreciated

REPLY [1 votes]: Suppose that $X$ is a matrix with rows $r_1,r_2,\dots,r_D$ (each of size $1 \times N$). We have
$$
X^\top X = \sum_{i=1}^D r_i^\top r_i = r_D^\top r_D + \sum_{i=1}^{D-1}r_i^\top r_i.
$$
Now, the modified matrix $\tilde X$ has the same rows except that the final row $r_D$ is replaced with the normalized $\tilde r_D = r_D/\sqrt{2}$ and a row $\tilde r_{D+1} = \tilde r_D$ is added. We then find that
$$
\begin{align}
{\tilde X}^\top \tilde X &= \tilde r_{D+1}^\top \tilde r_{D+1} + \tilde r_D^\top \tilde r_D + \sum_{i=1}^{D-1}r_i^\top r_i
\\ & =
[r_D/\sqrt{2}]^\top [r_D/\sqrt{2}] + [r_D/\sqrt{2}]^\top [r_D/\sqrt{2}] +
\sum_{i=1}^{D-1}r_i^\top r_i
\\ & =
\frac 12 r_D^\top r_D + \frac 12 r_D^\top r_D + \sum_{i=1}^{D-1}r_ir_i^\top
\\ & = r_D^\top r_D + \sum_{i=1}^{D-1} r_i^\top r_i = X^\top X.
\end{align}
$$

Alternatively, note that we can write $\tilde X = RX$, where
$$
R = \pmatrix{
I_{(D-1) \times (D - 1)} & 0_{(D - 1) \times 1}\\
0_{1 \times (D-1)} & 1/\sqrt{2}\\
0_{1 \times (D-1)} & 1/\sqrt{2}}.
$$
Verify that $R^\top R = I$, so that
$$
[\tilde X]^\top [\tilde X] = [RX]^\top [RX] =
X^\top R^\top R X = X^\top I X = X^\top X.
$$