text:
\begin{document}
\global\long\def\cosetlat{\mathcal{C}}

\global\long\def\sglat{\mathcal{L}}

\global\long\def\rec{\tilde{\chi}}

\global\long\def\Aut{\operatorname{\mathsf{Aut}}}

\global\long\def\Out{\operatorname{\mathsf{Out}}}

\global\long\def\Inn{\operatorname{\mathsf{Inn}}}

\global\long\def\hol{\operatorname{\mathsf{Hol}}}

\global\long\def\diag{\mathrm{diag}}

\global\long\def\ff{\mathbb{F}}

\global\long\def\zz{\mathbb{Z}}

\global\long\def\normalin{\mathrel{\lhd}}

\global\long\def\normalineq{\mathrel{\unlhd}}

\global\long\def\semidirect{\rtimes}

\global\long\def\comma{{,}}

\global\long\def\join{\mathbin{\ast}}

\global\long\def\homol{\widetilde{H}}

\global\long\def\LL{\mathcal{L}}

\global\long\def\kone{\mathcal{K}_{1}}

\global\long\def\ktwo{\mathcal{K}_{2}}

\global\long\def\kboth{\kone\cup\ktwo}

\title[Coset posets are not contractible]{Order complexes of coset posets of finite groups are not contractible}

\author{John Shareshian and Russ Woodroofe}

\thanks{The first author was supported in part by NSF Grants DMS-0902142
and DMS-1202337.}

\address{Department of Mathematics, Washington University in St.~Louis, St.~Louis,
MO, 63130}

\email{shareshi@math.wustl.edu}

\address{Department of Mathematics \& Statistics, Mississippi State University,
Starkville, MS 39762}

\email{rwoodroofe@math.msstate.edu}
\begin{abstract}
We show that the order complex of the poset of all cosets of all proper
subgroups of a finite group $G$ is never $\ff_{2}$-acyclic and therefore
never contractible. This settles a question of K.~S.~Brown.
\end{abstract}
\maketitle

\section{Introduction}

We settle a question asked by K.~S.~Brown in \cite{Brown:2000}.
For a group $G$, $\cosetlat(G)$ will denote the poset of all cosets
of all proper subgroups of $G$, ordered by inclusion. For a poset
$P$, $\Delta P$ will denote the order complex of $P$. Other terms
used but not defined in this introduction are defined in Section~\ref{sec:Pre}.
\begin{thm}
\label{thm:Main} If $G$ is a finite group, then $\Delta\cosetlat(G)$
is not $\ff_{2}$-acyclic, and therefore is not contractible.
\end{thm}
With some explicitly stated exceptions, the groups, partially ordered
sets and simplicial complexes considered herein are assumed to be
finite. We assume some familiarity with topological combinatorics
(see for example \cite{Bjorner:1995,Wachs:2007}), along with the
rudiments of algebraic topology (see for example \cite{Hatcher:2002,Munkres:1984})
and group theory (see for example \cite{Aschbacher:2000,Dixon/Mortimer:1996}).

\subsection{History and motivation}

The topology of $\Delta\cosetlat(G)$ was studied by Brown in \cite{Brown:2000}.
More general coset complexes were studied from a somewhat different
point of view by Abels and Holz in \cite{Abels/Holz:1993}. However,
from our perspective (and that of Brown), the story begins with the
work of P.~Hall, who in \cite{Hall:1936} introduced generalized
Möbius inversion in order to enumerate generating sequences. Hall
considered the probability $P_{G}(k)$ that a $k$-tuple $(g_{1},\ldots,g_{k})$
of elements of a group $G$, chosen uniformly with replacement, includes
a generating set for $G$. He showed that
\[
P_{G}(k)=\sum_{H\leq G}\mu(H,G)[G:H]^{-k},
\]
where $\mu$ is the Möbius function on the subgroup lattice of $G$.
(We mention that Weisner introduced generalized Möbius inversion independently
in \cite{Weisner:1935a}. See \cite[Chapter 3]{Stanley:2012} for
a comprehensive discussion of this theory.)

Bouc observed that $-P_{G}(-1)$ is the reduced Euler characteristic
$\rec(\Delta\cosetlat(G))$. Indeed, Hall showed in \cite{Hall:1936}
that if $\widehat{P}$ is obtained from $P$ by adding a minimum element
$\hat{0}$ and a maximum element $\hat{1}$, then
\[
\rec(\Delta P)=\mu_{\widehat{P}}(\hat{0},\hat{1}).
\]
A straightforward computation shows that
\[
\mu_{\widehat{\cosetlat(G)}}(\hat{0},\hat{1})=-P_{G}(-1).
\]
This led to Brown's work, in which he obtained divisibility results
for $P_{G}(-1)$ using group actions on $\Delta\cosetlat(G)$.

Brown found no group $G$ for which $P_{G}(-1)=0$. As the reduced
Euler characteristic of a contractible complex is zero, the question
of contractibility arises naturally. Previous progress on this question
involved showing that $P_{G}(-1)\neq0$. Gaschütz showed in \cite[Satz 2]{Gaschutz:1959}
that $P_{G}(-1)\neq0$ when $G$ is solvable. (Brown refined this
result by calculating the homotopy type of $\Delta\cosetlat(G)$ for
a solvable group $G$ in \cite[Proposition 11]{Brown:2000}.) Patassini
proved $P_{G}(-1)\neq0$ for many almost simple groups $G$ in \cite{Patassini:2009,Patassini:2011}.
He obtained further results for some groups with minimal normal subgroups
that are products of alternating groups in \cite{Patassini:2013}.
The question of whether $P_{G}(-1)$ is nonzero for all (finite) $G$
remains open.

Abels and Holz consider in \cite{Abels/Holz:1993} a more general
class of posets. Let $G$ be a (possibly infinite) group and $\mathcal{H}$
be a collection of proper subgroups of $G$ that is closed under intersection.
Abels and Holz study the order complex of the poset $\cosetlat_{\mathcal{H}}(G)$
of all cosets of all subgroups in $\mathcal{H}$. In their Theorem~2.4,
they describe relations between connectivity properties of $\Delta\cosetlat_{\mathcal{H}}(G)$
and the structure of $G$. Our Theorem~\ref{thm:Main} says that
$\Delta\cosetlat_{\mathcal{H}}(G)$ is not infinitely connected when
$G$ is finite and $\mathcal{H}$ contains all proper subgroups of
$G$. In contrast, Ramras in \cite[Remark 2.4]{Ramras:2005} noticed
that $\Delta\cosetlat(G)$ is contractible when $G$ is not finitely
generated.

Some other papers on the topology of $\Delta\cosetlat(G)$ are \cite{Torres-Giese:2012,Woodroofe:2007,Woodroofe:2009b}.

\subsection{A brief description of our proof}

Our proof has three main ingredients, namely, a ``join theorem''
of Brown, the Classification of Finite Simple Groups, and P.~A.~Smith
Theory.

Brown showed that, given a group $G$ and normal subgroup $N$, there
is a subposet $\cosetlat(G,N)$ of $\cosetlat(G)$ such that $\Delta\cosetlat(G)$
is homotopy equivalent to the join $\Delta\cosetlat(G,N)\join\Delta\cosetlat(G/N)$.
This result allows us to use induction on $|G|$. We complete the
proof by showing that $\Delta\cosetlat(G,N)$ is not $\ff_{2}$-acyclic
when $N$ is a minimal normal subgroup of $G$. Such a subgroup is
a direct product of pairwise isomorphic simple groups.

In order to show $\Delta\cosetlat(G,N)$ is not $\ff_{2}$-acyclic,
we use Smith Theory and the Classification. For each possible minimal
normal subgroup $N$, we describe a group $E$ such that $E$ acts
on $\cosetlat(G,N)$ with no fixed point. Using results of Smith and
Oliver, we choose $E$ so as to preclude a fixed-point-free action
on an $\ff_{2}$-acyclic complex.

\subsection{Further comments}

\subsubsection{}

Theorem~\ref{thm:Main} stands in clear contrast to other results
on order complexes of posets naturally associated to finite groups.
Consider the poset $\sglat(G)$ of nontrivial proper subgroups of
$G$, ordered by inclusion. The complex $\Delta\sglat(G)$ is contractible
for many groups, including all those with nontrivial Frattini subgroup.

Next, let $p$ be a prime. Consider the subposet $\mathcal{S}_{p}(G)$
of $\mathcal{L}(G)$ consisting of all $p$-subgroups and the subposet
$\mathcal{A}_{p}(G)$ of $\mathcal{S}_{p}(G)$ consisting of all elementary
abelian $p$-subgroups. The order complexes $\Delta\mathcal{S}_{p}(G)$
and $\Delta\mathcal{A}_{p}(G)$ were first studied, respectively,
by Brown in \cite{Brown:1974,Brown:1975} and Quillen in \cite{Quillen:1978}.
These two complexes are homotopy equivalent. They are contractible
when $G$ has a nontrivial normal $p$-subgroup. The converse of this
last statement is a well-known conjecture of Quillen, see \cite[Conjecture 2.9]{Quillen:1978}.

\subsubsection{}

The identity $-P_{G}(-1)=\rec(\Delta\cosetlat(G))$ can be considered
to be an example of the phenomenon of \emph{combinatorial reciprocity}.
Often some objects of interest are counted by evaluating an appropriate
function at positive integers. Combinatorial reciprocity occurs when
evaluation of the same function at negative integers counts some closely
related objects. Combinatorial reciprocity is discussed in \cite{Beck:2012,Stanley:1974a,Stanley:1974b,Stanley:2012}.
At this point we know of no interesting interpretation of $P_{G}(-n)$
for integers $n>1$. More generally, one can evaluate $P_{G}$ at
any complex number $s$. The study of $P_{G}(s)$ as a complex function
was initiated by Boston in \cite{Boston:1996} and Mann in \cite{Mann:1996},
and was continued by various authors. See for example \cite{Damian/Lucchini:2007,Damian/Lucchini:2014,Damian/Lucchini/Morini:2004,Detomi/Lucchini:2003,Patassini:2009,Patassini:2011b,Patassini:2013b,Shareshian:1998}.

\subsubsection{}

In Lemma~\ref{lem:UniversalGeneration}~(3) below, we note that
if $L$ is a finite simple group of Lie type or a sporadic simple
group, then there exists some odd prime $p$ such that $L=\langle P,R\rangle$
whenever $P$ is a Sylow $p$-subgroup of $L$ and $R$ is a Sylow
$2$-subgroup of $L$. This property need not hold when $L$ is an
alternating group. However, one can ask whether for each $n$ there
exist primes $p=p(n)$ and $r=r(n)$ such that $\left\langle P,R\right\rangle =A_{n}$
whenever $P$ is a Sylow $p$-subgroup and $R$ is a Sylow $r$-subgroup
of $A_{n}$. This interesting question remains open. It is related
to a question raised by Dolfi, Guralnick, Herzog and Praeger in \cite[Section 6]{Dolfi/Guralnick/Herzog/Praeger:2012}.
These authors ask whether for each $n$ there exist conjugacy classes
$C,D$ in $A_{n}$ consisting of elements of prime-power order, such
that $\left\langle c,d\right\rangle =A_{n}$ for all $(c,d)\in C\times D$.
A positive answer to their question immediately implies a positive
answer to ours. We will address related questions in a forthcoming
paper.

\subsubsection{}

To our knowledge, the first use of Smith Theory in combinatorics appears
in work of Kahn, Saks and Sturtevant. In the paper \cite{Kahn/Saks/Sturtevant:1984},
these authors use the work of Smith and Oliver mentioned above to
obtain a striking result about computational complexity.

\subsection{Contents of the paper}

In Section~\ref{sec:Pre}, we introduce some basic facts and definitions.
In Section~\ref{sec:MainThm}, we reduce the proof of Theorem~\ref{thm:Main}
to some claims about nonabelian finite simple groups. In the remaining
sections, we use the Classification to prove these claims. In Sections~\ref{sec:Alternating}
and \ref{sec:A7}, we prove the required result for alternating groups.
The group $A_{7}$ requires more care than the other alternating groups.
In Section~\ref{sec:LieSporadic}, we handle sporadic groups and
groups of Lie type.

\section*{Acknowledgements}

We thank Anders Björner, Bob Guralnick, Andrea Lucchini, Richard Lyons,
and Massimiliano Patassini for helpful discussions and correspondence.
The first author was inspired to think about this problem during a
visit to the Università degli Studi di Padova in Spring 2011. He thanks
the mathematics department there, and in particular Andrea Lucchini
and Eloisa Detomi, for their gracious hospitality. The second author
is grateful to the University of Primorska for their hospitality (with
support from the European Social Fund and the Ministry of Education,
Science and Sport of the Republic of Slovenia) during his visit of
Spring 2014.

\section{\label{sec:Pre}Preliminaries}

Here we introduce some basic definitions and facts. A reader who is
familiar with topological combinatorics and group theory can skip
this section safely, and refer to it as necessary.

\subsection{Groups and cosets}

As is standard, we write $K^{g}$ for $g^{-1}Kg$ whenever $K\subseteq G$
and $g\in G$, and write $x^{g}$ for $g^{-1}xg$. Similarly, we write
$K^{\alpha}$ and $x^{\alpha}$ for the images of $K$ and $x$ under
an automorphism $\alpha$ of $G$.

When referring to a coset, we mean a right coset. This causes no loss
of generality, as every coset of every subgroup of $G$ is a right
coset of some subgroup. Indeed, $xH=H^{x^{-1}}x$. It is not hard
to see that every coset is a right coset of a unique subgroup.

\subsection{\label{sec:PreSimplicial}Simplicial complexes}

An \textit{abstract simplicial complex} is a collection $\Delta$
of sets (called \textit{faces}) such that if $S\in\Delta$ and $T\subseteq S$
then $T\in\Delta$. We make no distinction between an abstract simplicial
complex and its geometric realization.

Let $P$ be a (finite) poset. The \textit{order complex} $\Delta P$
is the simplicial complex whose $k$-dimensional faces are the chains
of length $k$ (size $k+1$) in $P$.

If $\Delta$ and $\Gamma$ are simplicial complexes on disjoint vertex
sets, the \textit{join} $\Delta\join\Gamma$ is the complex whose
faces are all sets $S\cup T$ such that $S\in\Delta$ and $T\in\Gamma$.

Associated to a simplicial complex $\Delta$ and a ring $R$ are the
\textit{reduced simplicial homology groups} $\homol_{i}(\Delta;R)$,
as described (for example) in \cite{Munkres:1984}. A complex $\Delta$
is called $R$-\textit{acyclic} if $\homol_{i}(\Delta;R)=0$ for every
integer $i$. Every contractible complex is $R$-acyclic for all $R$.
Every nonempty $R$-acyclic complex has at least one nonempty face.
Indeed, $\homol_{-1}(\{\emptyset\};R)\cong R$, hence the complex
$\{\emptyset\}$ is not acyclic over any ring $R$. The simplicial
complexes that we consider all contain the empty face $\emptyset$.

\section{\label{sec:MainThm}Proof of Theorem~\ref{thm:Main}}

Here we prove Theorem~\ref{thm:Main}, although we defer the proofs
of some key lemmas on simple groups to later sections. Let us first
collect some main ingredients in the proof.

\subsection{\label{sec:BrownJoin}Brown's Join Theorem for $\Delta\protect\cosetlat(G)$}

Given a normal subgroup $N$ of $G$, we define the \textit{relative
coset poset }\textit{\emph{to be}}
\[
\cosetlat(G,N):=\{Hx\in\cosetlat(G):HN=G\}.
\]
The next result, due to Brown, is key to our proof.
\begin{thm}[{Brown's Join Theorem \cite[Proposition 10]{Brown:2000}}]
 \label{thm:BrownJoin} If $G$ is a group and $N$ is a normal subgroup
of $G$, then $\Delta\cosetlat(G)$ is homotopy equivalent to $\Delta\cosetlat(G/N)\join\Delta\cosetlat(G,N)$.\end{thm}
\begin{cor}
\label{cor:kun} Let $p$ be a prime. Let $N$ be a normal subgroup
of $G$. The complex $\Delta\cosetlat(G)$ is $\ff_{p}$-acyclic if
and only if at least one of $\Delta\cosetlat(G,N)$ and $\Delta\cosetlat(G/N)$
is $\ff_{p}$-acyclic. \end{cor}
\begin{proof}
The result follows immediately from Theorem~\ref{thm:BrownJoin}
and the Künneth Formula for joins (see for example \cite[(9.12)]{Bjorner:1995}).
\end{proof}
We see now that Theorem~\ref{thm:Main} follows quickly from the
next result.
\begin{thm}
\label{thm:MainIndStep} If $N$ is a minimal normal subgroup of $G$,
then $\Delta\cosetlat(G,N)$ is not $\ff_{2}$-acyclic. \end{thm}
\begin{proof}[Proof (of Theorem~\ref{thm:Main}, assuming Theorem~\ref{thm:MainIndStep}).]
 We proceed by induction on the order of $G$. If $G=1$, then $\Delta\cosetlat(G)=\{\emptyset\}$.
Now assume $\left|G\right|>1$, and let $N$ be a minimal normal subgroup
of $G$. The complex $\Delta\cosetlat(G/N)$ is not $\ff_{2}$-acyclic
by inductive hypothesis and $\Delta\cosetlat(G,N)$ is not $\ff_{2}$-acyclic
by Theorem~\ref{thm:MainIndStep}. Theorem~\ref{thm:Main} now follows
from Corollary~\ref{cor:kun}.
\end{proof}
It remains to prove Theorem~\ref{thm:MainIndStep}. In the rest of
Section~\ref{sec:MainThm}, we show how to reduce the proof to certain
claims about finite simple groups.

\subsection{\label{sec:SmithTheory}Group actions and Smith theory}

In order to prove Theorem~\ref{thm:MainIndStep}, we use Smith Theory.

Given a group $E$ acting by automorphisms (order preserving bijections)
on a poset $Q$, we write $Q^{E}$ for the fixed point set
\[
Q^{E}:=\{q\in Q:q^{g}=q\mbox{ for all }g\in E\}.
\]
The action of $E$ on $Q$ induces a simplicial action of $E$ on
$\Delta Q$.

Work of Smith in \cite{Smith:1941} and of Oliver in \cite{Oliver:1975}
shows that, given a prime $p$, certain groups cannot act without
fixed points on $\ff_{p}$-acyclic complexes. (A clear summary of
this work appears in \cite[Section 1]{Oliver:1975}.) Applying their
results to actions on order complexes, we obtain immediately the next
result.
\begin{thm}[Smith \cite{Smith:1941}, Oliver \cite{Oliver:1975}]
\label{thm:smith} Let $p$ and $r$ be primes. Let $Q$ be a poset
such that $\Delta Q$ is $\ff_{p}$-acyclic. Let $E$ be a group admitting
a normal series $P\normalineq H\normalineq E$ such that
\begin{enumerate}
\item $P$ is a $p$-group,
\item $H/P$ is cyclic, and
\item $E/H$ is an $r$-group.
\end{enumerate}
If $E$ acts on $Q$ by automorphisms, then $Q^{E}\neq\emptyset$. \end{thm}
\begin{rem}
It is not necessary that the primes $p,r$ in Theorem~\ref{thm:smith}
be distinct.
\end{rem}

\begin{rem}
\label{rem:SmithAppl}If $E\cong P\times K$, where $P$ is a $p$-group
and $K$ is either cyclic or an $r$-group, then $E$ satisfies conditions
(1)-(3) of Theorem~\ref{thm:smith}. The same holds for $E\cong(P\times K)\semidirect R$,
where $P$ is a $p$-group, $R$ is an $r$-group, and $K$ is cyclic.
\end{rem}
We will apply Theorem~\ref{thm:smith} to $\cosetlat(G,N)$. There
are two actions on $\cosetlat(G)$ that we wish to consider. The first
action is that of $G\times G$ by left and right translation. That
is,
\begin{equation}
Hx\cdot(g,h)=g^{-1}Hxh=H^{g}g^{-1}xh\,\,\mbox{for }(g,h)\in G\times G.\label{eq:action1}
\end{equation}
 The second action is by $\Aut(G)$, where
\begin{equation}
\left(Hx\right)^{\alpha}=H^{\alpha}x^{\alpha}\,\,\mbox{for }\alpha\in\Aut(G).\label{eq:action2}
\end{equation}
The component-wise action of $\Aut(G)$ on $G\times G$ gives rise
to the semidirect product $A:=(G\times G)\semidirect\Aut(G)$. The
actions described in (\ref{eq:action1}) and (\ref{eq:action2}) combine
to form a well-defined action of $A$ on $\cosetlat(G)$, with $(g,h,\alpha)$
mapping $Hx$ to
\[
\left(Hx\cdot(g,h)\right)^{\alpha}=(g^{-1}Hxh)^{\alpha}=(Hx)^{\alpha}\cdot(g^{\alpha},h^{\alpha}).
\]

\begin{rem}
If $\left|G\right|>1$, then the action of $A$ has a nontrivial kernel
$N$. The quotient $A/N$ is called the \emph{holomorph} of $G$.
The kernel of this action will be of no concern to us.
\end{rem}
In all but one of our arguments, we will use subgroups of $(G\times G)\semidirect\Aut(G)$
that are contained in $G\times G$. These subgroups will be of the
form $P\times K$ with $P,K\leq G$. When we mention an action of
such a subgroup on $\cosetlat(G)$, we always mean that $P$ acts
by left translation and $K$ acts by right translation.

Suppose $N\normalineq G$. If $HN=G$ then $H^{g}N=G$ for all $g\in G$.
It follows that $P\times K$ acts on $\cosetlat(G,N)$. Note that
$P\times1$ fixes $Hx$ if and only if $P\leq H$ and $1\times K$
fixes $Hx$ if and only if $K^{x^{-1}}\leq H$. The next result follows.
\begin{lem}
\label{lem:LRactionFixed} A subgroup $P\times K$ of $G\times G$
fixes $Hx\in\cosetlat(G)$ if and only if $\langle P,K^{x^{-1}}\rangle\leq H$.
\end{lem}

\subsection{Minimal normal subgroups}

Along with Smith theory, we use the Classification of Finite Simple
Groups to prove Theorem~\ref{thm:MainIndStep}. Suppose $G$ is nontrivial,
and $N$ is a minimal normal subgroup of $G$. There exist some positive
integer $t$ and some simple group $L$ such that $N$ is isomorphic
with the direct product of $t$ copies of $L$. (See for example \cite[Theorem 4.3A(iii)]{Dixon/Mortimer:1996}.)
In this situation, we abuse notation by writing $N=L^{t}$ and representing
an element of $N$ as a $t$-tuple of elements of $L$.

The case where the simple group $L$ is cyclic of prime order was
already handled by Brown.
\begin{lem}
\label{lem:amns}If $G$ has an abelian minimal normal subgroup $N$,
then $\Delta\cosetlat(G,N)$ is not $\ff_{2}$-acyclic.\end{lem}
\begin{proof}
As noted in \cite[Proposition 9]{Brown:2000}, the poset $\cosetlat(G,N)$
is an antichain of size divisible by $|N|$. Therefore, $\Delta\cosetlat(G,N)$
is not connected if it contains a nonempty face. It follows that one
of $\homol_{-1}(\Delta\cosetlat(G,N);\ff_{2})$ or $\homol_{0}(\Delta\cosetlat(G,N);\ff_{2})$
is nontrivial.
\end{proof}
We turn now to the case where $N=L^{t}$ with $L$ nonabelian simple.
A subgroup $K\leq L$ can be embedded in $N$ diagonally, as follows.
\begin{defn}
Given $N=L^{t}$ and $K\leq L$, we define
\[
K^{\diag}:=\{(k,\ldots,k):k\in K\}\leq N.
\]

\end{defn}
The next definition is key for finding useful group actions on $\Delta\cosetlat(G,N)$.
\begin{defn}
\label{def:Univpgen}Let $G$ be a group, let $H\leq G$ and let $p$
be a prime. We say that $H$ \textit{universally} $p$-\textit{generates}
$G$ if $\langle H,P\rangle=G$ whenever $P$ is a Sylow $p$-subgroup
of $G$. \end{defn}
\begin{rem}
\label{rem:UnivSylpr} Let $p,r$ be primes. A Sylow $r$-subgroup
of $G$ universally $p$-generates $G$ if and only if every maximal
subgroup of $G$ has index divisible by at least one of $p$ and $r$.
\end{rem}
The importance of Definition~\ref{def:Univpgen} is apparent from
the following lemma.
\begin{lem}
\label{lem:UnivpgenFixedpts}Let $G$ be a group and let $p$ be a
prime. Let $N\unlhd G$. If $K\leq N$ universally $p$-generates
$N$ and $P$ is any Sylow $p$-subgroup of $N$, then $\cosetlat(G,N)^{P\times K}=\emptyset$. \end{lem}
\begin{proof}
Assume for contradiction that $Hx\in\cosetlat(G,N)^{P\times K}$.
By Lemma~\ref{lem:LRactionFixed}, $H$ contains both $P$ and $K^{x^{-1}}$.
As $K$ universally $p$-generates $N$, so does $K^{x^{-1}}$. Therefore,
$H$ contains $N$. This is impossible, as $H<G$ and $HN=G$.\end{proof}
\begin{lem}
\label{lem:UnivProd} Let $L$ be a simple group, let $p$ be a prime,
and let $t$ be a positive integer. If a proper subgroup $K<L$ universally
$p$-generates $L$, then $K^{\diag}$ universally $p$-generates
$N:=L^{t}$. \end{lem}
\begin{proof}
Let $P$ be a Sylow $p$-subgroup of $N$. By assumption, $N=\prod_{i=1}^{t}L_{i}$
with each $L_{i}\cong L$. It is not hard to see that $P=\prod_{i=1}^{t}(P\cap L_{i})$.
Moreover, $P\cap L_{i}$ is a Sylow $p$-subgroup of $L_{i}$ for
each $i\in[t]$. The standard projection of $\langle P,K^{\diag}\rangle$
onto $L_{i}$ thus contains both $P\cap L_{i}$ and $K$ and is therefore
all of $L_{i}$. It follows now from (the conjugacy part of) Sylow's
Theorem that $\langle P,K^{\diag}\rangle$ contains every Sylow $p$-subgroup
of $L_{i}$.

As $K<L$ universally $p$-generates $L$, $P\cap L_{i}$ is nontrivial.
As $L$ is simple, it follows that the Sylow $p$-subgroups of $L_{i}$
together generate $L_{i}$. Hence, $L_{i}\leq\langle P,K^{\diag}\rangle$
for each $i\in[t]$. \end{proof}
\begin{cor}
\label{cor:EmptyfixProd} Let $L$ be a simple group, let $p$ be
a prime, and let $t$ be a positive integer. Let $G$ be a group with
normal subgroup $N=L^{t}$, and let $P$ be a Sylow $p$-subgroup
of $N$. If a proper subgroup $K<L$ universally $p$-generates $L$,
then $\cosetlat(G,N)^{P\times K^{\diag}}=\emptyset$. \end{cor}
\begin{proof}
This follows immediately from Lemmas \ref{lem:UnivpgenFixedpts} and
\ref{lem:UnivProd}.\end{proof}
\begin{cor}
\label{cor:UnivCyclicorPp}Let $N,G,L,p$ and $K$ be as in Corollary~\ref{cor:EmptyfixProd}.
If $K$ is either cyclic or of prime-power order, then $\Delta\cosetlat(G,N)$
is not $\ff_{p}$-acyclic. \end{cor}
\begin{proof}
This follows directly from Corollary~\ref{cor:EmptyfixProd} and
Theorem~\ref{thm:smith}.
\end{proof}
Our strategy is now clear. We go through the list of nonabelian simple
groups, as provided by the Classification. For each such group $L$,
we look for some $K<L$ that universally $2$-generates $L$ and is
either cyclic or of prime-power order. This strategy fails only when
$L=A_{7}$, in which case we use a slight extension of Corollary~\ref{cor:UnivCyclicorPp}.

Every nonabelian finite simple group is, up to isomorphism, an alternating
group $A_{n}$ with $n\geq5$, a group of Lie type, or one of twenty
six sporadic groups. Note that the small alternating groups $A_{5}\cong PSL_{2}(5)$,
$A_{6}\cong PSL_{2}(9)$ and $A_{8}\cong PSL_{4}(2)$ are all isomorphic
with simple groups of Lie type. (See for example \cite[Theorem 2.2.10]{Gorenstein/Lyons/Solomon:1998}.)
\begin{lem}
\label{lem:UniversalGeneration} If $L$ is simple and $L\not\cong A_{7}$,
then there is some $K<L$ that is either cyclic or of prime-power
order, and that universally $2$-generates $L$.

Indeed, the following claims hold.
\begin{enumerate}
\item If $L=A_{n}$ with $n\geq9$ odd and $h\in L$ is an $n$-cycle, then
$\langle h\rangle$ universally $2$-generates $L$.
\item If $L=A_{n}$ with $n\geq10$ even and $h\in L$ is an $(n-1)$-cycle,
then $\langle h\rangle$ universally $2$-generates $L$.
\item If $L$ is a sporadic simple group or a simple group of Lie type,
then there is some odd prime $p$ such that a Sylow $p$-subgroup
of $L$ universally $2$-generates $L$.
\end{enumerate}
\end{lem}
We will prove Claims~(1) and (2) in Section~\ref{sec:Alternating},
and Claim~(3) in Section~\ref{sec:LieSporadic}.

We examine $A_{7}$ in Section~\ref{sec:A7}, where we prove the
following result.
\begin{lem}
\label{lem:a7} If $G$ has a minimal normal subgroup $N\cong A_{7}^{t}$,
then $\Delta\cosetlat(G,N)$ is not $\ff_{2}$-acyclic.
\end{lem}
Theorem~\ref{thm:MainIndStep} (and so Theorem~\ref{thm:Main})
follows from Lemma~\ref{lem:amns}, Corollary~\ref{cor:UnivCyclicorPp},
Lemma~\ref{lem:UniversalGeneration}, and Lemma~\ref{lem:a7}.

\section{\label{sec:Alternating}Alternating groups of high degree}

Here we prove Claims~(1) and (2) of Lemma~\ref{lem:UniversalGeneration}.

Our proof involves the standard division of subgroups of $S_{n}$
into three classes. Such a subgroup $H$ is \textit{transitive} if
for each $i,j\in[n]$ there is some $x\in H$ such that $ix=j$, and
\textit{intransitive} otherwise. A transitive subgroup $H$ is \textit{imprimitive}
if there is some partition $\pi=\{\pi_{1},\ldots,\pi_{\ell}\}$ of
$[n]$ into subsets, such that $1<\ell<n$, and such that for each
$x\in H$ and each $i\in[\ell]$, there exists some $j\in[\ell]$
with $\pi_{i}x=\pi_{j}$. In this case, we say that $H$ \textit{stabilizes}
$\pi$. A subgroup $H$ is \textit{primitive} if $H$ is transitive
but not imprimitive. So, each subgroup of $S_{n}$ is intransitive,
imprimitive or primitive. We begin with a classical result of Jordan.
\begin{thm}[{Jordan \cite{Jordan:1875}. See also \cite[Example 3.3.1]{Dixon/Mortimer:1996}}]
\label{thm:jordan} If $n\geq9$, then every primitive subgroup of
$S_{n}$ containing an element with exactly $n-4$ fixed points contains
$A_{n}$.
\end{thm}
When $n\geq4$, every Sylow $2$-subgroup of $A_{n}$ contains an
element with exactly $n-4$ fixed points, namely, the product of two
disjoint transpositions.
\begin{cor}
\label{cor:primitive} If $n\geq9$, then no primitive proper subgroup
of $A_{n}$ contains a Sylow $2$-subgroup of $A_{n}$.
\end{cor}
Suppose that the transitive subgroup $H\leq S_{n}$ stabilizes the
partition $\pi=\{\pi_{1},\ldots,\pi_{\ell}\}$ of $[n]$, with $1<\ell<n$.
The transitivity of $H$ forces $|\pi_{i}|=|\pi_{j}|$ for all $i,j\in[\ell]$.
Each $\pi_{i}$ has size $d=n/\ell$. The full stabilizer $G_{\pi}$
of $\pi$ in $S_{n}$ (which contains $H$) is isomorphic with the
wreath product $S_{d}\wr S_{\ell}$, and thus has order $d!^{\ell}\ell!$.
Now $G_{\pi}\not\leq A_{n}$, as $G_{\pi}$ contains a transposition.
It follows that $G_{\pi}\cap A_{n}$ contains a Sylow $2$-subgroup
of $A_{n}$ if and only if $\frac{n!}{d!^{\ell}\ell!}$ is odd.
\begin{lem}
\label{lem:impodd} If $n$ is odd, then no imprimitive subgroup of
$A_{n}$ contains a Sylow $2$-subgroup of $A_{n}$. \end{lem}
\begin{proof}
By the preceding discussion, it suffices to show that $\frac{n!}{d!{}^{\ell}\ell!}$
is even whenever $d$ is a nontrivial proper divisor of $n$ and $\ell=n/d$.
Straightforward manipulations yield
\[
\frac{n!}{d!{}^{\ell}\ell!}=\prod_{j=1}^{\ell}{jd-1 \choose d-1}.
\]
It suffices to show any term in the product on the right is even.
We calculate
\[
{2d-1 \choose d-1}=\frac{2d-1}{d}{2d-2 \choose d-1}=\frac{2d-1}{d}\cdot2{2d-3 \choose d-1}.
\]
Since the divisor $d$ of $n$ is odd, the result follows.\end{proof}
\begin{lem}
\label{lem:impeven} If $n$ is even, then no imprimitive subgroup
of $A_{n}$ contains an $(n-1)$-cycle.\end{lem}
\begin{proof}
Assume for contradiction that the $(n-1)$-cycle $h\in A_{n}$ stabilizes
the partition $\pi=\{\pi_{1},\ldots,\pi_{\ell}\}$ with $1<\ell<n$.
Without loss of generality, the unique fixed point $j$ of $h$ lies
in $\pi_{1}$. Now $\pi_{1}h=\pi_{1}$. As $\langle h\rangle$ is
transitive on $[n]\setminus\{j\}$ and $|\pi_{1}|>1$, we obtain the
contradiction $[n]=\pi_{1}$.
\end{proof}
No intransitive subgroup of $A_{n}$ contains an $n$-cycle. Thus
Claim~(1) follows from Corollary~\ref{cor:primitive} and Lemma~\ref{lem:impodd}.
To prove Claim~(2), it remains to show that when $n\geq10$ is even,
no intransitive subgroup of $A_{n}$ contains both a Sylow $2$-subgroup
of $A_{n}$ and an $(n-1)$-cycle. It suffices to show a Sylow $2$-subgroup
$P$ of $A_{n}$ contains a fixed-point-free element. Depending on
$n\mod4$, $P$ contains a conjugate of either
\[
(1,2,3,4)(5,6)\cdots(n-1,n)\quad\mbox{or}\quad(1,2)(3,4)\cdots(n-1,n).
\]

\section{\label{sec:A7}The alternating group $A_{7}$ }

Here we prove Lemma~\ref{lem:a7}.

The conclusion of Claim~(1) of Lemma~\ref{lem:UniversalGeneration}
does not hold for $A_{7}$. Indeed, $A_{7}$ has proper primitive
subgroups that contain both a 7-cycle and a Sylow $2$-subgroup of
$A_{7}$. (Such subgroups are isomorphic to $PGL_{3}(2)$, and are
embedded in $A_{7}$ through actions on points and lines in the Fano
plane.) As a result, we cannot apply Corollary~\ref{cor:EmptyfixProd}
or Corollary~\ref{cor:UnivCyclicorPp}. However, we can still apply
Smith Theory (as in Theorem~\ref{thm:smith}) when $G$ has a minimal
normal subgroup $N=A_{7}^{t}$.

We begin by collecting some information on $A_{7}$. All these facts
are straightforward to confirm, or can be verified with \cite{GAP4.4.12}
or \cite{Atlas}.
\begin{lem}
\label{lem:A7struc}The group $A_{7}$ has the following properties.
\begin{enumerate}
\item There exist conjugacy classes $\kone$ and $\ktwo$ of subgroups of
$A_{7}$ satisfying the following conditions.

\begin{enumerate}
\item If $K\in\kboth$, then $K\cong PGL_{3}(2)$, and so $[A_{7}:K]=15$.
\item A proper subgroup $K$ of $A_{7}$ contains both a Sylow $2$-subgroup
of $A_{7}$ and a 7-cycle if and only if $K\in\kboth$.
\item If $K\in\kboth$, $P$ is a Sylow $2$-subgroup of $K$, and $R$
is a Sylow $7$-subgroup of $K$, then $\left\langle P^{r}\,:\, r\in R\right\rangle =K$.
\end{enumerate}
\item There is an involution $\varphi\in\Aut(A_{7})$ satisfying the following
conditions.

\begin{enumerate}
\item The automorphism $\varphi$ normalizes both a Sylow $2$-subgroup
and a Sylow $7$-subgroup of $A_{7}$.
\item If $K\in\kone$ then $K^{\varphi}\in\ktwo$, and if $K\in\ktwo$ then
$K^{\varphi}\in\kone$.
\end{enumerate}

\noindent Indeed, the automorphism arising from conjugation by $(1,2)(3,4)(5,6)\in S_{7}$
has the desired properties.

\end{enumerate}
\end{lem}
The next result follows directly from Lemma~\ref{lem:A7struc} and
the universal property of direct products.
\begin{lem}
\label{lem:AutomOfN}Let $N$ be the direct product $\prod_{i=1}^{t}L_{i}$
with each $L_{i}\cong A_{7}$. There is an involution $\rho\in\Aut(N)$
such that the following claims hold for each $i\in[t]$.
\begin{enumerate}
\item The automorphism $\rho$ normalizes $L_{i}$.
\item The automorphism $\rho$ normalizes both a Sylow $2$-subgroup $P_{i}$
of $L_{i}$ and a Sylow $7$-subgroup $\left\langle h_{i}\right\rangle $
of $L_{i}$.
\item If $H\leq L_{i}$ is normalized by $\rho$ and contains both $P_{i}$
and $\left\langle h_{i}\right\rangle $, then $H=L_{i}$.
\end{enumerate}
Moreover, $\rho$ normalizes both $P:=P_{1}P_{2}\cdots P_{t}$ and
$K:=\left\langle (h_{1},h_{2},\dots,h_{t})\right\rangle $.\end{lem}
\begin{rem}
The automorphism $\rho$ may be realized concretely by embedding $N\cong A_{7}^{t}$
in $S_{7}^{t}$, setting $x:=(1,2)(3,4)(5,6)\in S_{7}$, and conjugating
by the element $(x,x,\dots,x)\in S_{7}^{t}$.
\end{rem}
The next lemma is a special case of a theorem of P.~Jin in \cite{Jin:2007}.
It also can be proved directly using standard facts from the cohomology
of groups (see \cite[Chapter IV]{Brown:1994}, particularly Corollary
IV.6.8 therein).

We write $\Inn(M)$ for the inner automorphism group of a group $M$
and $\Out(M)$ for the outer automorphism group $\Aut(M)/\Inn(M)$.
\begin{lem}[{See \cite[Corollary C]{Jin:2007}}]
\label{lem:ExtendAut} Let $N\normalineq G$ with $Z(N)=1$ and let
$\rho\in\Aut(N)$. If the coset $\Inn(N)\rho$ is in $Z(\Out(N))$,
then there exists some $\theta\in\Aut(G)$ such that
\begin{enumerate}
\item $\left|\theta\right|=\left|\rho\right|$,
\item $\theta$ normalizes $N$, and
\item the restriction $\theta_{N}$ of $\theta$ to $N$ is $\rho$.
\end{enumerate}
\end{lem}
Applying Lemma~\ref{lem:ExtendAut} to the involution $\rho$ described
in Lemma~\ref{lem:AutomOfN}, we obtain the following corollary.
\begin{cor}
\label{lem:A7toAutG}Suppose that the group $G$ has a normal subgroup
$N=\prod_{i=1}^{t}L_{i}$ with each $L_{i}\cong A_{7}$. If $\rho\in\Aut(N)$
is as in Lemma~\ref{lem:AutomOfN}, then there exists an involution
$\theta\in\Aut(G)$ such that $\theta$ normalizes $N$ and $\theta_{N}=\rho$.\end{cor}
\begin{proof}
Note first that $\Aut(A_{7})\cong S_{7}$ (see for example \cite[Section 8.2]{Dixon/Mortimer:1996}).
It follows that $\Aut(N)\cong S_{7}\wr S_{t}$ (this is \cite[Exercise 4.3.9]{Dixon/Mortimer:1996}).
As $Z(N)=1$, it follows in turn that $\Inn(N)\cong N$ and $\Out(N)\cong\mathbb{Z}_{2}\wr S_{t}$.
Therefore $\Out(N)$ has a central element $z$ of order $2$.

Let $\varphi\in\Aut(N)$. The coset $\Inn(N)\varphi$ is equal to
$z$ if and only if the conditions
\begin{itemize}
\item [(a)] $\varphi$ normalizes $L_{i}$, and
\item [(b)] the restriction of $\varphi$ to $L_{i}$ is not in $\Inn(L_{i})$
\end{itemize}
are satisfied for each $i\in[t]$. The automorphism $\rho$ meets
conditions (a) and (b). The conclusion now follows from Lemma~\ref{lem:ExtendAut}.
\end{proof}
We are ready to complete the proof of Lemma~\ref{lem:a7}. Suppose
that the group $G$ has a minimal normal subgroup $N=\prod_{i=1}^{t}L_{i}$
with each $L_{i}\cong A_{7}$. The automorphism $\theta$ obtained
in Corollary~\ref{lem:A7toAutG} normalizes both groups $P,K$ described
in Lemma~\ref{lem:AutomOfN}. Using the componentwise action of $\theta$
on $P\times K$, we obtain the semidirect product
\[
E:=(P\times K)\semidirect\left\langle \theta\right\rangle \leq(G\times G)\semidirect\Aut(G).
\]
 The group $E$ acts on $\cosetlat(G)$ as described in Section~\ref{sec:SmithTheory}.
Since $\theta$ normalizes $N$, this action restricts to $\cosetlat(G,N)$.
Since $\left|\theta\right|=2$, the group $E$ meets the conditions
of Theorem~\ref{thm:smith} (as discussed in Remark \ref{rem:SmithAppl}).
It thus suffices to show that $\cosetlat(G,N)^{E}=\emptyset$.

Assume for contradiction that $Hx\in\cosetlat(G,N)^{E}$. Then $H$
contains $\left\langle P,K^{x^{-1}}\right\rangle $ and is normalized
by $\theta$. In particular, for each $i\in[t]$, the intersection
$H\cap L_{i}$ is normalized by $\theta$ and contains $P_{i}$. Moreover,
the projection of $H\cap N$ to $L_{i}$ (which contains $H\cap L_{i}$
as a normal subgroup) is normalized by $\theta$ and contains $\left\langle h_{i}\right\rangle $.
By Lemma~\ref{lem:AutomOfN}~(3), this projection is $L_{i}$. As
$A_{7}$ is simple and $H\cap L_{i}$ is nontrivial, it follows that
$H\cap L_{i}=L_{i}$. Therefore $N\leq H$. This is impossible, as
$H<G$ and $HN=G$.

\section{\label{sec:LieSporadic}Simple groups of Lie type and sporadic groups}

Here we prove Claim~(3) of Lemma~\ref{lem:UniversalGeneration}.
We refer the reader to \cite[Chapter 2]{Gorenstein/Lyons/Solomon:1998}
for an introduction to the finite simple groups of Lie type, with
original references. Each such group is determined by an irreducible
crystallographic root system $\Sigma$, a (possibly trivial) automorphism
$\sigma$ of the Dynkin diagram of $\Sigma$ and a finite field $\ff_{q}$
of order $q$. If $\sigma$ has order $d$, we say that the \textit{type}
of the associated simple group is $^{d}\Sigma(q)$, suppressing $d$
when $d=1$. Much of what we need has already been proved by Damian
and Lucchini in \cite[Section 4]{Damian/Lucchini:2007}. We summarize
their results as follows.
\begin{thm}[Damian and Lucchini \cite{Damian/Lucchini:2007}]
 \label{thm:DamianLucchini} If $L$ is a finite simple group of
Lie type or a sporadic simple group, then one of the following conditions
holds.
\begin{enumerate}
\item [(a)] There is some cyclic subgroup $C\leq L$ of prime order that
universally $2$-generates $L$.
\item [(b)] The group $L$ is of Lie type $B_{n}(q)$ $(n\geq3)$, $D_{n}(q)$
$(n\geq4)$, or $G_{2}(q)$, and $q$ is odd.
\item [(c)] The group $L$ is of Lie type $A_{5}(2)$, $C_{3}(2)$, $D_{4}(2)$
or $^{2}A_{3}(2)$.
\item [(d)] The group $L$ is the McLaughlin sporadic group $McL$.
\end{enumerate}
\end{thm}
In fact, Damian and Lucchini give further restrictions on $L$ when
condition (b) of Theorem~\ref{thm:DamianLucchini} holds, but we
will not need these.

As every Sylow $p$-subgroup of a group $G$ contains a conjugate
of every element of order $p$ in $G$, it remains to examine the
groups listed in cases (b), (c) and (d) of Theorem~\ref{thm:DamianLucchini}.

Suppose $L$ is a simple group of type $^{d}\Sigma(q)$ and $q$ is
a power of the prime $p$. We say that $L$ has \textit{characteristic}
$p$ and call a subgroup $M\leq L$ \textit{parabolic} if $M$ contains
the normalizer of some Sylow $p$-subgroup of $L$. The following
result is attributed to Tits by Seitz in \cite[(1.6)]{Seitz:1973}.
\begin{lem}[Tits]
\label{lem:tits} Let $L$ be a simple group of Lie type in characteristic
$p$, and let $P$ be a Sylow $p$-subgroup of $L$. Every maximal
subgroup of $L$ containing $P$ is parabolic.
\end{lem}
The groups of types $A_{5}(2)$, $C_{3}(2)$, $D_{4}(2)$ and $^{2}A_{3}(2)$
are all classical groups. The orders of parabolic subgroups of classical
groups are known, and can be found in \cite{Kleidman/Liebeck:1990}.
It is straightforward to confirm that the index of each parabolic
subgroup of each of the four given groups is divisible by three. The
same holds for the index of each maximal subgroup of odd index in
$McL$, as can be confirmed by consulting \cite{Atlas}. We obtain
the following result.
\begin{lem}
\label{lem:cd} If the simple group $L$ is listed in case (c) or
case (d) of Theorem~\ref{thm:DamianLucchini}, then $L$ is universally
$2$-generated by any of its Sylow $3$-subgroups.
\end{lem}
It remains to handle case (b). Key to the work of Damian and Lucchini
in \cite{Damian/Lucchini:2007} is a result of Liebeck and Saxl in
\cite{Liebeck/Saxl:1985}, in which the authors describe all primitive
permutation groups of odd degree. (See also the paper \cite{Kantor:1987}
of Kantor, in particular Lemma~2.3 therein.) Such a description necessarily
includes a list of all pairs $(M,L)$ such that $L$ is a finite simple
group and $M$ is a maximal subgroup of odd index in $L$. Consulting
this list, we obtain the following result.
\begin{lem}[{Liebeck and Saxl \cite{Liebeck/Saxl:1985}; see also \cite[Lemma 2.3]{Kantor:1987}}]
\label{lem:oddpar} If $L$ is a simple group of Lie type in odd
characteristic and some parabolic subgroup of $L$ contains a Sylow
$2$-subgroup of $L$, then the type of $L$ is one of $A_{n}(q)$
or $E_{6}(q)$.
\end{lem}
Consulting \cite[Theorem 2.2.10]{Gorenstein/Lyons/Solomon:1998},
we see that, assuming the lower bounds on $n$ listed in case (b),
there exists no isomorphism between a group of type $B_{n}(q)$, $D_{n}(q)$
or $G_{2}(q)$ with $q$ odd and a group of type $A_{n}(q)$ or $E_{6}(q)$.
Thus Lemmas~\ref{lem:tits} and \ref{lem:oddpar} together complete
our proof of Claim~(3).

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TITLE: Does the following relation holds for infinite sums?
QUESTION [3 upvotes]: Hey i am only wondering if it is possible that the equation below holds:
$$
\sum\limits_{i=0}^{\infty} a_i(b_i - c_i) = \sum\limits_{i=0}^{\infty} (a_ib_i - a_ic_i) =\sum\limits_{i=0}^{\infty} a_ib_i - \sum\limits_{i=0}^{\infty} a_ic_i
$$
Thanks!

REPLY [2 votes]: Recall the definition of a series is: $$\sum_{i=1}^\infty a_i=\lim_{n\to\infty}(a_1+\ldots+a_n).$$
The answer to your question follows from the fact that: $$\lim_{n\to\infty}a_n(b_n-c_n)=\lim_{n\to\infty}a_n b_n-\lim_{n\to\infty}a_n c_n,$$ whenever all the limits have a finite value.

REPLY [2 votes]: It holds when
$$\sum_{i=0}^\infty a_i b_i$$
and $$\sum_{i=0}^\infty b_i c_i $$
ar both unconditional convergent.
Or $$
\sum_{i=0}^\infty a_i (b_i - c_i)$$ is unconditional convergent.
This is in the finite dimensional the case when they are absolute convergent, that means when
$$\sum_{i=0}^\infty |a_i b_i|$$ is convergent.


TITLE: Quadratic and Cubic Differential Equation
QUESTION [0 upvotes]: I came across the following Differential Equation
$$y'(x) = a \cdot y(x) + b \cdot y(x)^2 + c \cdot y(x)^3$$ Where $a,b,c$ are numerical coefficients.
I can solve it numerically, but I would like to solve it analytically. Is there some trick or substitution that might make it easier to solve?

REPLY [2 votes]: Solve $cy^2 + by + a = 0$. Let the roots be $y_0, y_1$. Now decompose $\dfrac{y'}{y(y-y_0)(y-y_1)}$ into $A\dfrac{y'}{y} + B\dfrac{y'}{y - y_0} + C\dfrac{y'}{y - y_1}$. Each of them is easily integratabtle. Can you continue?


TITLE: Every Hamiltonian group contains a subgroup isomorphic to $Q_8$
QUESTION [2 upvotes]: I read somewhere that every Hamiltonian group (a non abelian group with every subgroup normal) contains a subgroup isomorphic to quaternion group.
But I cannot find its proof anywhere on net or in book. This does not look easy when I tried. In which book/notes can I found a proof of this statement

REPLY [1 votes]: For a reference for proof you can see Group Theory
By W. R. Scott. In page 253.
(Baer) A group $G$ is Hamiltonian iff $G$ is a direct sum of $A,B,D$ where $A$ is a quaternion group, $B$ is a elementary abelian $2$-group, and $D$ is a periodic abelian group with all elements are odd order.


TITLE: Why are there two Fundamental Theorems of Calculus?
QUESTION [5 upvotes]: I don't know if this is an appropriate question for this site.  If not, I apologize.
A colleague of mine maintains that the Second Fundamental Theorem of Calculus shouldn't be taught as such (i.e. that it is "fundamental") but rather be thought of, and referred to as The Evaluation Theorem or The Net Change Theorem.
My question isn't really about the "should" or "shouldn't" but rather "How did with end up with two FTC's?"  What is the reasoning behind referring to both as Fundamental?
Again, if this is off-topic, I apologize and please close down the question!

REPLY [0 votes]: First you get to know the antiderivatives, with the idea is the area covered under a curve. And it can be obtain for some easy integrand f(x) (like some power formulas) that you gonna integrate by using the limits of a Riemann sum by using some algebraic proof over difference of squares.
Then you get to know the FTC 1 'evaluation' that is used to proof the area under a curve is actually the antiderivative evaluated at the end points, more on that you learn some properties of integrals and how you can swap limits of integration, split integrals, the average value has to cover the same area, and the more important the integral from '$a$' to '$a$' is zero.
The FTC 2 'integral formulas' tell you that you can express a function expresed with the integral formulas (that is you don't get a number anymore; you get different value depending on where is you final point).
$$F(x)=\int_a^x f(t)dt$$
That has lower limit of integration as initial point or initial condition as $(x,F(x))=(a,0)$; and the upper limit has a change of variables so the computer don't get confused. The lower limit is very important: if you want to have a initial point with value of 5 then you arrange as $F(x) =\int _a^x f(t) dt+5;\quad F(a)=\int_a^a f(t)dt+5=5$ and you can use more properties of integrals like this.
Then you address the reality that not all the function that you want to integrate has an actual antiderivative that you can evaluate at the endpoints; but you can take its linear approximation, quadratic approximation; you just have to get used to the notation of this formulas and how to obtain its derivatives.
Then you pass to composite function, maybe some piece wise. Then numerical integration and some method to find antiderivatives.
All that I remember I relearn in edx courses for free, after i fail in IRL courses.


TITLE: How to prove differential of product correlated Brownian Motions?
QUESTION [2 upvotes]: I was wondering how to prove/compute the differential of the product of two Brownian motions. I know how to do it in case they are independent as follows:
Suppose $dX_t= \mu_t dt +\sigma_t dW_t$ and $dY_t = v_t dt + \rho_t d \bar{W}_t$, where $W_t$ and $\bar{W}_t$ are independent, i.e. $(dW_t d\bar{W}_t= 0)$
Then I can prove $d(X_t Y_t)$ with the following trick:
I start with decomposition:
$(X_t+Y_t)^2=X_t^2+Y_t^2+ 2X_t Y_t$,
Which leads by differentiation to
$d(X_t Y_t)= \frac{1}{2}[d(\{X_t +Y_t\}^2) - d(X_t^2) - d(Y_t^2)]$
Then I apply Ito-lemma to all three parts separately as follows:
$d(X_t^2)=(2\mu_tXt+\sigma_t^2)dt+ 2\sigma_t X_tdW_t= 2X_t dX_t + \sigma_t^2dt$
$d(Y_t^2)=(2v_t Y_t+\rho_t^2)dt+ 2\rho_t Y_t d\bar{W}_t= 2Y_t dY_t+\rho_t^2 dt$
$d((X_t+Y_t)^2) = 2(X_t+Y_t)dX_t+ 2(X_t+Y_t)dY_t + (\sigma_t^2+\rho_t^2)dt$
Combined I eventually find:
$d(X_t Y_t) = Y_t dX_t + X_t dY_t$.
Now I was wondering how to do the prove in a similar fashion in the case of correlated brownian motions. Any help would be appreciated.

REPLY [2 votes]: Well it's simpler to tackle it multidimensionnal Itô with $f(x,y)=x.y$,
this gives you :
$$d(XY)_t=f_{x}(X_t,Y_t)dX_t +f_{y}(X_t,Y_t)dY_t + \frac{1}{2} f_{xx}(X_t,Y_t)d<X>_t+\frac{1}{2}f_{yy}(X_t,Y_t)d<Y>_t +f_{xy}(X_t,Y_t)d<X,Y>_t$$
$$d(XY)_t=Y_t.dX_t +X_t.dY_t + \frac{1}{2}.0.d<X>_t+\frac{1}{2}.0.d<Y>_t +1.d<X,Y>_t$$
$$d(XY)_t=Y_t.dX_t +X_t.dY_t + d<X,Y>_t$$
Now $d<X,Y>_t=\sigma_t.\rho_td<B,\tilde{B}>_t=c.\sigma_t.\rho_t.dt$
where $d<B,\tilde{B}>=c.dt$ so that :
$$d(XY)_t=Y_t.dX_t +X_t.dY_t + c.\sigma_t.\rho_t.dt$$


text:
\subsection{The Contraction Hypothesis in Vector Spaces}
\label{section: CH in vector spaces higher rank}

	As before, we first construct functions in linear representations encoding divergence in $X$ and then convert our linear estimates into a height function on $X$.
    The relevant representation in this case is $\bigoplus_i \bigwedge^i \R^{d}$.

    Let $ H_0,Z \in \mrm{Lie}(G) \cong \mf{sl}(2,\R)$ be such that
    \begin{equation} \label{log g_t}
    	g_t = \exp( t H_0), \qquad u_s  \exp(sZ).
     \end{equation}
     In particular, by definition of $g_t$ and $u_s$, we have
     \[ [H_0,Z] = 2Z.  \]

    Denote by $P^+$ the set of all possible highest weights appearing in linear representations of $G$.
    From the representation theory of $\mrm{SL}(2,\R)$, the set $P^+$ of highest weights can be identified with $\N \cup \set{0}$.
    Given an arbitrary finite dimensional representation $V$ of $G$ and $\l\in P^+$, we use $V^\l$ to denote the direct sum of all irreducible subrepresentations of $V$ whose highest weight is $\l$.
    We denote by $\pi_\l:V \r V^\l$ the associated $G$-equivariant projection.

    Following~\cite{BQ-RandomWalkRecurrence}, we define two sets of exponents.
	For $i \in \set{1,\dots,d-1}$ and $\l \in P^{+}$, define
    \begin{equation}
    \d_i = i(d-i) \qquad \d_\l = \l(H_0).
    \end{equation}

    In particular, we have
    \begin{equation*}
    	\l(H_0) = 0 \Leftrightarrow \l = 0.
    \end{equation*}

    For every $\epsilon >0$ and $0<i <d$, we define a function $\vp_{\epsilon}$ on $\bigwedge^i \R^{d}$ as follows.
    For $v \in \bigwedge^i \R^{d}$, let
    \begin{align} \label{defn: phi}
    \vp_{\epsilon}(v) = \begin{cases}
    	 \min\limits_{\l \in P^+ \backslash\set{0} } \epsilon^{\frac{\d_i}{\d_\l} } \norm{ \pi_\l(v)}^{-1/\d_\l}
         				& \text{if } \norm{\pi_0(v)} < \epsilon^{\d_i}, \\
         0 & \text{otherwise.}
    \end{cases}
    \end{align}
    Here we use the convention $1/0 = \infty$.

    The following Lemma is the form in which we use Proposition~\ref{propn: expansion in linear representations} in our setting.
    \begin{lemma} \label{lemma: contraction hypothesis for phi}
    For every $\b \in (0, 1)$, there exists $D \geqslant 1$ such that for all $t,\epsilon >0$ and all $v \in \bigwedge^i \R^{d} $ with $0< i<d$,
    \begin{equation*}
    	\frac{1}{2}\int_{-1}^1 \vp_{\epsilon}^\b(g_t u_s v) \;ds \leqslant D e^{-\b   t} \vp_{\epsilon}^\b(v).
    \end{equation*}
    \end{lemma}

    \begin{proof}

    	First, we note that for all $g\in G$, $\pi_0(gv) = g\pi_0(v) = \pi_0(v)$.
        In particular, if $\vp_\epsilon(v) = 0$, then
        $\norm{\pi_0(v)} = \norm{\pi_0(g_t u_s v) }\geqslant \epsilon^{\d_i}$ for all $s$ and $t$
        and the statement follows in this case. Hence, we may assume $\vp_\epsilon (v) \neq 0$.

        Moreover, since the integral of the minimum of finitely many functions is bounded by the minimum of their integrals, it suffices to prove
        \begin{equation} \label{contraction in each weight}
        \frac{1}{2} \int_{-1}^1 \norm{  \pi_\l(g_t u_s v)}^{-\b/\d_\l} \;ds =
        	\frac{1}{2} \int_{-1}^1 \norm{ g_t u_s \pi_\l(v)}^{-\b/\d_\l} \;ds
            	\leqslant D e^{- \b  t} \norm{\pi_\l(v)}^{-\b/\d_\l}
        \end{equation}
        for each $\l \in P^{+}\backslash\set{0}$ and for some constant $D$ depending only on $\b$.

        If $\pi_\l(v) = 0$ for some $\l$, then the right-hand side of~\eqref{contraction in each weight} is $\infty$ and the claim is proved in this case.
        Now, suppose that $\pi_\l(v) \neq 0$ for some $\l \in P^+\backslash \set{0}$.

        Denote by $V =\left( \bigwedge^i \R^{d} \right)_\l$, i.e. the image of $\bigwedge^i\R^{d}$ under the projection $\pi_\l$.
        From the representation theory of $\mrm{SL}(2,\R)$, we see that the dimension of an irreducible representation with weight $\l$ is equal to $\d_\l + 1$.
        In particular, choosing $\b\in (0,1)$ allows us to apply
        Proposition~\ref{propn: expansion in linear representations} to get that
        \begin{equation*}
        	\frac{1}{2} \int_{-1}^1 \norm{ g_t u_s \pi_\l(v)}^{-\b/\d_\l} \;ds
            	\leqslant D e^{- \b t } \norm{\pi_\l(v)}^{-\b/\d_\l}.
        \end{equation*}
        This proves~\eqref{contraction in each weight} and completes the proof.
    \end{proof}

config: arxiv


TITLE: Finding minimun on a quadratic form
QUESTION [1 upvotes]: Let $b \in \mathbb{C}^n$, i'm trying to find the min for the following function:
$H(b) = b^\star P b - q^\star b - b^\star q + s$
I know it's gonna be $b = P^{-1} q$, as P is invertible (also positive semi-definite), but i've seen how to reach it without the term $b^\star q$ where $H'= Pb - q$ and it follows from solving the linear system.
But i'm not used to working with this kind of objective functions (or optimization at all), how can i work with this?

REPLY [1 votes]: You can do it by completing the square.
Try introducing an $a$ and a $c$ like this, and then solve for them to match your expression
$H(b) = (b^\star -a^\star)P(b-a) + c = b^\star P b - a^\star P b - b^\star P a - a^\star P a + c$
Comparing with your expression,
$H(b) = b^\star P b - q^\star b - b^\star q + s$
we see that to match up we need $b^\star P a = b^\star q$ and $a^\star P b = a^\star b$. Because your matrix is positive definite, then $q^\star P^{-1} = (P^{-1}q)^\star$, so we can solve both conditions simultaneously to get $a = P^{-1} q$. We also need $c = s - q^\star P^{-1} q$.
Now we have
$H(b) = (b^\star - q^\star P^{-1})P(b-P^{-1}q) + s - q^\star P^{-1} q$,
and your function is written explicitly as a square plus a constant. The minimum of this function will be when $b-P^{-1}q = 0$, with minimum value $H_{min} = s - q^\star P^{-1} q$ at that point.
It may already be clear to you why $b=P^{-1}q$ minimises the function, but if it's not yet then you can see this by diagonalising the matrix. $P$ is positive definite and hence $P = P^
\star$, and any such matrix can be diagonalised by a unitary matrix. This means that there exists an $S$ with $S^\star = S^{-1}$ such that $P = S^\star D S$, where $D$ is diagonal and every diagonal element is positive. Then
$H(b) = (b^\star - q^\star P^{-1})S^\star D S(b-P^{-1}q) + s - q^\star P^{-1} q$
Defining a new variable $y = S(b-P^{-1}q)$, then
$H(y) = y^\star D y + s - q^\star P^{-1} q = \sum_i \lambda_i |y_i|^2  + s - q^\star P^{-1} q$,
where $\lambda_i > 0$ are the eigenvalues of $P$, and $y_i \in \mathbb{C}$ are the coordinates of $y$. Hopefully it's now clear to inspect that this function is minimised by $y = 0$, which implies that $b=P^{-1}q$.


TITLE: Show that every eigenvalue to a unitary operator has absolute value 1
QUESTION [0 upvotes]: Let $A$ be a unitary operator on hilbert space $H$, i.e.
$$(Au|Av) = (u|v)$$
for all $u,v \in D_A.$.
I'm asked to show that all eigenvalues to this unitary operator has absolute value 1.
My attempt:
Let u,v be eigenvectors to $A$ and $\lambda_1, \lambda_2$ their eigenvalues.
$$(Au|Av) = (\lambda_1u|\lambda_2v) = \lambda_1\lambda_2 (u|v) = (u|v) \implies \lambda_1\lambda_2=1.$$
For the statement to be true, $\lambda_1$ must equal $\lambda_2$ but I can't see why this is true. What am I missing?

REPLY [1 votes]: Let $\lambda$ and eigenvalue of $A$. With $u$ his eigenvector.
Then $(u,u) = (Au,Au) = (\lambda u, \lambda u) = \lambda\overline{\lambda}(u,u) = |\lambda|^{2}(u,u)$
Thus $|\lambda|^{2} = 1 \implies |\lambda| = 1$


TITLE: Why vary the action with respect to the inverse metric?
QUESTION [5 upvotes]: Whenever I have read texts which employ actions that contain metric tensors, such as the Nambu-Goto, Polyakov or Einstein-Hilbert action, the equations of motion are derived by varying with respect to the inverse of the metric, i.e. indices are in superscript.
Why is this? Why not vary with respect to simply the metric tensor? Or is it just a convention/preference?

REPLY [6 votes]: Since the metric and inverse metric are related by
$$
  g^{\mu\lambda}g_{\lambda\nu} = \delta^\mu_\nu
$$
taking the variation of both sides gives
$$
  \delta g^{\mu\lambda}g_{\lambda\nu} + g^{\mu\lambda}\delta g_{\lambda\nu} =0
$$
or in other words
$$
  \delta g_{\mu\nu} = -g_{\mu\rho}g_{\nu\sigma}\delta g^{\rho\sigma}
$$
It follows that there is a one-to-one correspondence between variations in the metric and its inverse, so that a functional of the metric is stationary with respect to variations of the metric if and only if a functional of the metric is stationary with respect to its inverse.  Noting this fact, we see that we can choose to perform either variations with respect to the metric or its inverse depending on which is more convenient.
For example, the Eisntein-Hilbert action involves the Ricci scalar which can be written in terms of the inverse metric as $R = g^{\mu\nu}R_{\mu\nu}$, so varying with respect to the inverse metric becomes convenient; $\delta R = \delta g^{\mu\nu} R_{\mu\nu} + g^{\mu\nu}\delta R_{\mu\nu}$.


TITLE: Serre's proof that zeta function is meromorphic
QUESTION [9 upvotes]: I try to understand the proof of Chap. VI, n° 3.1, Prop. 10 in Serre's "A course in arithmetic" (page 70). The goal is to prove that zeta-function can be written as
\begin{align*}
\zeta(s)=\frac{1}{s-1}+\phi(s)
\end{align*}
with an holomorphic $\phi$ for any $s\in\mathbb{C}$ with $\mathfrak{Re}(s)>0$.
I understand the proof except for one detail: $\phi$ is given explicitly by $\phi=\sum_{n=1}^\infty \phi_n$ with
\begin{align*}
\phi_n(s)=\int_n^{n+1} \left(n^{-s}-t^{-s}\right) dt.
\end{align*}
To prove the convergence of $\phi$, we want to show that
\begin{align*}
\left|\phi_n(s)\right|\leq\frac{|s|}{n^{x+1}} \quad \text{with}\quad x=\mathfrak{Re}(s)\quad(*)
\end{align*}
For this Serre first notes that
\begin{align*}
\left|\phi_n(s)\right| \leq \sup_{n\leq t\leq n+1}\left| n^{-s}-t^{-s}\right|
\end{align*}
Which is clear since the range of integration is $(n+1)-n=1$. Then he sais that the derivative of $n^{-s}-t^{-s}$ is equal to $\frac{s}{t^{s+1}}$. And from this somehow follows (*). My question is: How exactly does this follow?
I noted the following statements for $s=x+iy$ where $x=\mathfrak{Re}(s)$ and $y=\mathfrak{Im}(s)$ and $f(t)=n^{-s}-t^{-s}$

Because $|t^{iy}|=|e^{iy\ln(t)}|=1$ (since it is on the unit circle) and $|t^{x+1}|=t^{x+1}$ (since $t$ and $x$ are real) we see that
\begin{align*}
\left|f'(t)\right|=\left|\frac{s}{t^{x+1+iy}}\right|=\frac{|s|}{|t^{x+1}| |t^{iy}|}=\frac{|s|}{t^{x+1}}
\end{align*}
Because $f(n)=0$ we can calculate
\begin{align*}
|f(n+1)|=|f(n+1)-f(n)|=\left|\int_n^{n+1}f'(t)dt\right|\leq\sup_{n\leq t\leq n+1}\left|f'(t)\right|=\sup_{n\leq t\leq n+1}\frac{|s|}{t^{x+1}}=\frac{|s|}{n^{x+1}}
\end{align*}

Am I close? :-)

REPLY [6 votes]: By the definition of $\phi_n$ and the Fundamental theorem of calculus we have
$$|\phi_n(s)| = \left| \int_n^{n+1} f(t) \, dt \right| \stackrel{f(n)=0}{=} \left| \int_n^{n+1} \left(\int_n^t f'(y) \, dy \right) \, dt \right| \leq \int_n^{n+1} |f'(y)| \, dy$$
Use your calculation from 1. (and note that $y \in [n,n+1]$, hence $y\geq n$).


TITLE: Proofs of inequalities used by Erdos-Renyi in their Random Graphs Paper 1
QUESTION [2 upvotes]: Please refer to this, it is Erdos-Renyi 1959 paper 1 on Random Graphs. I am currently working on this, but I am stuck on the fifth page, where they use two estimates. More specifically, here's the question:

Define $N_c=[\dfrac{1}{2}n\log n+cn]$ where $[.]$ denotes the greatest integer function, and $c$ is any arbitrary fixed real constant. Also, let $M={n\choose 2}$.
Then prove, for large $n$, the following inequalities:
$$\large{n\choose s}\dfrac{{{M - s(n-s)}\choose N_c}}{M\choose N_c}\leq \dfrac{e^{(3-2c)s}}{s!}\space\space\space\space\text{for   } s\leq\dfrac{n}{2}$$
and $$\large{n\choose s}\dfrac{{{M - s(n-s)}\choose N_c}}{M\choose N_c}\leq \dfrac{e^{(3-2c)(n-s)}}{(n-s)!}\space\space\space\space\text{for   } s\geq\dfrac{n}{2}$$
Here, $s$ is a positive integer such that $s<n-\dfrac{2N_c}{n}$.

I admit, these inequalities are a bit strange. But, I am convinced that these are simply estimates. The paper I am referring to, simply says, "by some elementary estimations, we get...", but do not mention how they get these.
I "believe" Stirling's Approximation has been used, repeatedly, along with some other crude estimates. I am particularly having some problem figuring out how those estimates have been used.
I observe that for large $n$, by applying Stirling's Approximation, $${n\choose s}\sim \dfrac{e^{-s}n^{n+1/2}}{(n-s)^{n-s+1/2}s!}---(1)$$
$${{M-s(n-s)}\choose N_c}\sim \dfrac{e^{-N_c}{(M-s(n-s))}^{M-s(n-s)+1/2}}{{(M-s(n-s)-N_c)}^{M-s(n-s)-N_c+1/2}N_c!}---(2)$$
$${M\choose N_c}\sim \dfrac{e^{-N_c}M^{M+1/2}}{(M-N_c)^{M-N_c+1/2}}---(3)$$
Then, $(1)\times (2)/(3)$ is something, whose bound I want to make equal to the bounds given.
But things are getting too complicated by now. Please remember that $M={n\choose 2}$. Am I on the right track? I tried to simplify these but things aren't working in my favor.
There are other inequalities in the paper with "similar" flavour, hence I believe that if I can understand how these inequalities are solved, then I can understand how the other inequalities have been deduced. I have been stuck on it for quite some time now, so some help is definitely more than appreciated.

REPLY [3 votes]: Let $n$ be large enough so that $N_c=\lfloor\frac{1}{2}n\ln n+cn\rfloor\ge0$ (even when $c<0$). On the other hand, the condition that $s$ is a positive integer such that $s<n-\dfrac{2N_c}{n}$ yields $N_c\le M$. Thus, $N_c\in\{0,\dots,M\}$, and so, the left-hand side of your inequalities is well defined -- assuming, by the standard convention, that $\binom{M - s(n-s)}N=0$ if $M - s(n-s)<N$.
Let $N:=N_c$, so that $c\le c_N:=-\frac12\ln n+\frac{N+1}n$. So, to prove the inequalities in question, it suffices to show that
$$(1)\qquad r_N:=r_{N,t}:=\binom nt
\dfrac{\binom{M - t(n-t)}N}{\binom{M}N}\Big/\dfrac{e^{(3-2c_N)t}}{t!}\le1
$$
for $N\in\{0,\dots,M\}$,
where $t:=s\wedge(n-s)$ -- and $c$ is gone.
If $M - t(n-t)<N$ then $r_N=0$, and hence $(1)$ is trivial.
If $M - t(n-t)\ge N$ and $N\in\{0,\dots,M-1\}$ then
$$\frac{r_{N+1}}{r_N}=\frac{M-N-t(n-t)}{M-N}\,e^{2/n}
\le\frac{M-t(n-t)}{M}\,e^{2/n}\le\frac{M-1(n-1)}{M}\,e^{2/n}=(1-2/n)e^{2/n}<1,
$$
so that $r_0\ge r_1\ge \cdots\ge r_M$. Thus, it suffices to prove $(1)$ for $N=0$.
But
$$r_{0,t}
=\frac{n!}{(n-t)!} \Big(\frac{e^{2/n-3}}{n}\Big)^t
$$
and
$$\frac{r_{0,t+1}}{r_{0,t}}=\frac{n-t}n\,e^{2/n-3}\le1,
$$
so that $r_{0,t}\le r_{0,0}=1$, which completes the proof. In fact, it is seen that $3$ can be replaced in $(1)$ (and hence in the original inequalities in question) by the smaller (and hence better) value $2/n$.


text:
\begin{document}

\large
\title{Weakly almost periodic functionals on the measure algebra}
\author{Matthew Daws}
\maketitle

\begin{abstract}
It is shown that the collection of weakly almost periodic functionals on the convolution algebra of a commutative Hopf von Neumann algebra is a C$^*$-algebra.  This implies that the weakly almost periodic functionals on $M(G)$, the measure algebra of a locally compact group $G$, is a C$^*$-subalgebra of $M(G)^* = C_0(G)^{**}$.  The proof builds upon a factorisation result, due to Young and Kaiser, for weakly compact module maps.  The main technique is to adapt some of the theory of corepresentations to the setting of general reflexive Banach spaces.

Subject classification: 43A10, 46L89 (Primary); 43A20, 43A60, 81R50 (Secondary).

\end{abstract}

\section{Introduction}

For a topological (semi-)group $G$, the space of \emph{weakly almost periodic}
functions on $G$ is the subspace of $C(G)$ consisting of those $f\in C(G)$
such that the (left) translates of $f$ form a relatively weakly-compact subset
of $C(G)$.  We denote this set by $\wap(G)$.  Then $\wap(G)$ is a unital C$^*$-subalgebra
of $C(G)$, say with character space $G^\wap$.  By continuity, we can extend the
product from $G$ to $G^\wap$, turning $G^\wap$ into a compact semigroup whose product
is separately continuous, a \emph{semitopological semigroup}.  Indeed, $G^\wap$
is the universal semitopological semigroup compactification of $G$.  See \cite{BJM}
or \cite{HS} for further details.

Now suppose that $G$ is a locally compact group, so we may form the Banach space
$L^1(G)$, which becomes a Banach algebra with the convolution product.  Then $L^\infty(G)$,
as the dual of $L^1(G)$, naturally becomes an $L^1(G)$-bimodule.  We define the
space of \emph{weakly almost periodic functionals} on $L^1(G)$, denoted by $\wap(L^1(G))$,
to be the collection of $f\in L^1(G)$ such that the map
\[ R_f:L^1(G) \rightarrow L^\infty(G), \quad a\mapsto a\cdot f \qquad (a\in L^1(G)), \]
is weakly-compact.  This is equivalent to the map $L_f:L^1(G)\rightarrow L^\infty(G),
a\mapsto f\cdot a$ being weakly-compact.
\"Ulger showed in \cite{ulger} that $\wap(L^1(G)) = \wap(G)$, where $C(G)$ is
naturally identified with a subspace of $L^\infty(G)$.  This fact also follows easily
from \cite[Lemma~6.3]{wong}, using the fact that if a set is relatively weakly compact,
then the weak- and weak$^*$-topology closures coincide.  Both these papers use simple
bounded approximate identity arguments.

The definition of $\wap(L^1(G))$ obviously generalises to any Banach algebra $\mc A$.
In general, we can say little about $\wap(\mc A)$, except for some interesting
links with the Arens products, see \cite{daws} and references therein, or
\cite[Section~1.4]{palmer} and \cite[Theorem~2.6.15]{dales}.
However, motivated by
the above example, we might expect that when $\mc A$ has a large amount of structure,
$\wap(\mc A)$ also might have extra structure.  In this paper, we shall investigate
$\wap(M(G))$, where $M(G)$ is the measure algebra over a locally compact group $G$.
In particular, we shall show that $\wap(M(G))$ is a C$^*$-subalgebra of $M(G)^*$,
where $M(G)$ is identified as the dual of $C_0(G)$, so that $M(G)^* = C_0(G)^{**}$ is
a commutative von Neumann algebra.

The central idea is to develop a theory of corepresentations on reflexive Banach spaces,
for commutative Hopf von Neumann algebras.  Our theory exactly replicates that for
Hilbert spaces, but care needs to be taken to ensure that everything works in our more
general setting.

The connection between weakly almost periodic functionals and
representations of Banach algebras goes back to Young, \cite{young}, and Kaiser,
\cite{kai}.  For $L^1(G)$, there is a correspondence between (non-degenerate)
representations of $L^1(G)$ and representations of $G$.  Using Young and Kaiser's work,
it is easy to see that weakly almost periodic functionals on $L^1(G)$ correspond to
coefficient functionals for representations of $G$ on reflexive spaces.  Then
multiplication of functions in $L^\infty(G)$ corresponds to tensoring representations.
The existence of reflexive tensor products (see \cite{as} for example) hence shows
that the product of two weakly almost periodic functionals is again weakly almost
periodic.  Of course, for $L^1(G)$, it is far easier to use \"Ulger's result, and then
argue directly that $\wap(G)$ is an algebra (which follows from Grothendieck's criteria
for weak compactness, see \cite{BJM}).  For $M(G)$, while $M(G)=L^1(X)$ for some
measure space $X$, we do not have that $X$ is a (semi)group, and so we turn to
corepresentations, which work with the algebra $M(G)^*$ directly.

The structure of the paper is as follows.  We first introduce some notions from the theory
of tensor products of Banach spaces, in particular the projective and injective tensor
norms.  We then define what a (commutative) Hopf von Neumann algebra is, and show
carefully that $M(G)$ (as well as $L^1(G)$) fits into this abstract framework.  For the
rest of the paper, we work with commutative Hopf von Neumann algebras, the results for
$M(G)$ (and, indeed, $L^1(G)$) being immediate corollaries.  As an immediate application,
we make a quick study of almost periodic functionals.  We then turn our attention to
weakly almost periodic functionals, and build a theory of corepresentations on
reflexive Banach spaces.  The final application is then obtained by checking that the usual way
of tensoring corepresentations still works in this more general setting.

For an introduction to quantum groups from a functional analysis viewpoint, \cite{ku},
or the pair of articles \cite{kt1} and \cite{kt2}, are very readable.  A good starting
point for details about (weakly) almost periodic functionals on general Banach algebras is
\cite{dul}.

A few notes on notion.  We generally follow \cite{dales} for details about Banach
algebras.  We write $E^*$ for the dual of a Banach space $E$, and use the dual pairing
notation $\ip{\mu}{x} = \mu(x)$, for $\mu\in E^*$ and $x\in E$.
We write $\mc B(E,F)$ for the collection of bounded linear maps from $E$ to $F$, we write
$\mc B(E,E)=\mc B(E)$, and we write $T^*$ for the linear adjoint of an operator $T$.

\smallskip\textbf{Acknowledgements:}
The author would like to thank Garth Dales, for bringing this problem
to his attention, and for careful proofreading.  Thanks to Tony Lau for providing
the reference \cite{wong}.

\section{Hopf von Neumann algebras}

We start by recalling some elementary definitions and facts from the theory
of tensor products of Banach spaces.  We refer the reader to the books \cite{ryan} and
\cite{df}, or \cite[Chapter~VIII]{du}, for further details.

Let $E$ and $F$ be Banach
spaces.  The \emph{projective tensor norm}, $\|\cdot\|_\pi$, on $E\otimes F$ is defined by
\[ \|\tau\|_\pi = \inf\Big\{ \sum_{k=1}^n \|x_k\| \|y_k\| : \tau=\sum_{k=1}^n x_k
\otimes y_k \Big\} \qquad (\tau\in E\otimes F). \]
Then $E\proten F$, the \emph{projective tensor product of $E$ and $F$}, is the
completion of $E\otimes F$ with respect to $\|\cdot\|_\pi$.  The projective tensor
product has the property that any bounded, bilinear map $\psi: E\times F\rightarrow G$
admits a unique bounded linear extension $\tilde\psi: E\proten F\rightarrow G$, with
$\|\tilde\psi\| = \|\psi\|$.  For measure spaces $X$ and
$Y$, we have that $L^1(X)\proten L^1(Y) = L^1(X\times Y)$.  We identify $(E\proten F)^*$
with $\mc B(E,F^*)$ under the dual pairing
\[ \ip{T}{x\otimes y} = \ip{T(x)}{y} \qquad (T\in\mc B(E,F^*), x\in E, y\in F), \]
and using linearity and continuity.

The \emph{injective tensor norm}, $\|\cdot\|_\epsilon$, on $E\otimes F$ is defined by
regarding $E\otimes F$ as a subspace of $\mc B(E^*,F)$, where $\tau = \sum_{k=1}^n
x_k \otimes y_k$ induces the finite-rank operator
\[ E^*\rightarrow F, \quad \mu\mapsto \sum_{k=1}^n \ip{\mu}{x_k} y_k. \]
Then $E\inten F$, the \emph{injective tensor product of $E$ and $F$}, is the
completion of $E\otimes F$ with respect to $\|\cdot\|_\epsilon$.  For locally
compact Hausdorff spaces $K$ and $L$, we have that $C_0(K) \inten C_0(L) =
C_0(K\times L)$.  We write $\mc A(E,F)$ for the closure of the finite-rank operators
from $E$ to $F$; these are the \emph{approximable operators from $E$ to $F$}.
Then, almost by definition, we have that $\mc A(E,F) = E^* \inten F$.

There is a canonical norm-decreasing map $E\proten F\rightarrow E\inten F$.
By taking the adjoint, we get an injective contraction $(E\inten F)^* \rightarrow
\mc B(E,F^*)$.  The image, equipped with the norm induced by $(E\inten F)^*$,
is the space of \emph{integral operators}, $\mc I(E,F^*)$.
The map $E^*\proten E\rightarrow E^*\inten E$ is injective if and only if $E$ has the
\emph{approximation property}.  We can regard $E^*\proten E$ as a subspace of
$\mc I(E^*) = (E^*\inten E)^* = \mc A(E)^*$ by
\[ \ip{\mu\otimes x}{T} = \ip{\mu}{T(x)} \qquad (T\in\mc A(E), \mu\otimes x\in
E^*\proten E). \]
Similarly, we can regard $E\proten F$ as a subspace of $\mc I(E^*,F)$; here we use
that fact that $\mc I(E^*,F)$ is isometrically a subspace of $\mc I(E^*,F^{**})
= (E^*\inten F^*)^* = \mc A(E,F^*)^*$.  We say that $E$ has the \emph{metric approximation
property} if and only if the map $E^*\proten E\rightarrow\mc I(E^*)$ is an isometry
onto its range, or equivalent, $E\proten F \rightarrow \mc I(E^*,F)$ is an
isometry onto its range, for all $F$.  There are characterisations of the (metric)
approximation property in terms of finite-rank approximations of the identity on compact
sets.  We have that $C_0(K)$ and $L^1(X)$ have the metric approximation property for
all $K$ and $X$.

\subsection{Commutative Hopf von Neumann algebras}

A \emph{Hopf von Neumann algebra} is a von Neumann algebra $\mc M$ equipped with
a \emph{coproduct} $\Delta:\mc M\rightarrow \mc M\vnten\mc M$.  Here $\vnten$ denotes
the von Neumann tensor product.  This means that
$\Delta$ is a normal $*$-homomorphism, and that $(\Delta\otimes\id)\Delta =
(\id\otimes\Delta)\Delta$, that is, $\Delta$ is coassociative.

We shall concentrate on the case where $\mc M$ is commutative, so that $\mc M = L^\infty(X)$
for some measure space $X$.  Then $\mc M \vnten \mc M = L^\infty(X\times X)$, and so, as
$\Delta$ is normal, it drops to give a contractive map $\Delta_*:L^1(X\times X) =
L^1(X) \proten L^1(X) \rightarrow L^1(X)$.  Hence $\Delta_*$ induces a contractive bilinear
map $L^1(X)\times L^1(X) \rightarrow L^1(X)$.  Then $\Delta$ being coassociative is
equivalent to $\Delta_*$ being associative.

As both $\mc M$ and $\mc M_*$ are Banach algebras, we have natural module actions
of $\mc M$ on $\mc M_*$ and of $\mc M_*$ on $\mc M$.  For the action of $\mc M$ on
$\mc M_*$, we shall, for example, write $F\cdot a\in\mc M_*$ for $F\in\mc M$ and
$a\in\mc M_*$.  For the action of $\mc M_*$ on $\mc M$, we shall always explicitly
invoke the map $\Delta_*$ or $\Delta$.

For an example of a commutative Hopf von Neumann algebra, let $G$ be a locally compact
group, and consider the algebra $L^\infty(G)$
equipped with the coproduct $\Delta$ defined by
\[ \Delta(f)(s,t) = f(st) \qquad (f\in L^\infty(G), s,t\in G). \]
Then $\Delta_*$ induces the usual convolution product on $L^1(G)$.

A slightly less well-known example is furnished by $M(G)$.  As $M(G) = C_0(G)^*$, we see that
$M(G)$ is the predual of the commutative von Neumann algebra $C_0(G)^{**}$.  As such,
$M(G)^* = L^\infty(X)$ for some measure space $X$ (see \cite[Chapter~III]{tak}), and
so by the uniqueness of preduals, $M(G) = L^1(X)$.  Let $\Phi$ be the canonical coproduct
on $C_0(G)$, so that $\Phi$ is the $*$-homomorphism $C_0(G)\rightarrow C(G\times G)$ defined
by \[ \Phi(f)(s,t) = f(st) \qquad (f\in C_0(G), s,t\in G). \]
We identify $C(G\times G)$, the space of bounded continuous functions on $G\times G$,
with the \emph{multiplier algebra} of $C_0(G\times G)$,
and hence (see \cite[Chapter~III, Section~6]{tak}) we may identify $C(G\times G)$ with
\[ \{ x\in C_0(G\times G)^{**} : fx,xf\in C_0(G\times G) \ (f\in C_0(G\times G)) \}. \]
We can hence regard $\Phi$ as a $*$-homomorphism $C_0(G) \rightarrow C_0(G\times G)^{**}$.

We claim that $M(G) \proten M(G)$ is, isometrically, a subspace of $M(G\times G)
= C_0(G\times G)^*$.  From the above, we can identify $M(G\times G)$ with
$\mc I(C_0(G), M(G))$.  As $M(G)$ has the metric approximation property, we see
that $M(G) \proten M(G)$ is isometrically a subspace of $\mc I(M(G)^*,M(G))$, or
equivalently, by properties of the integral operators, isometrically a subspace of
$\mc I(C_0(G),M(G))$, as required.

Alternatively, for any C$^*$-algebra $\mc A$,
we could define a norm on $\mc A^*\otimes\mc A^*$ by embedding $\mc A^*\otimes\mc A^*$
into $(\mc A\otimes_{\min}\mc A)^*$.  This induces the \emph{operator space projective
tensor norm}, see \cite[Chapter~7]{er}, and as $\mc A$ has the \emph{minimal} operator
space structure, it follows that $\mc A^*$ has the \emph{maximal} structure, and
so this norm agrees with the (Banach space) projective tensor norm.

Hence $L^\infty(X)\vnten L^\infty(X) = (M(G)\proten M(G))^*$ is a quotient of
$M(G\times G)^* = C_0(G\times G)^{**}$.  We claim that this quotient map is a
$*$-homomorphism, for which it suffices to check that the kernel
\[ \{ \tau\in C_0(G\times G)^{**} : \ip{\tau}{\mu\otimes\lambda}=0 \ (\mu,\lambda
\in M(G)) \} \]
is an ideal.  Let $\mu,\lambda\in M(G)$, let $g\in C_0(G\times G)$, and let
$f=f_1\otimes f_2 \in C_0(G) \otimes C_0(G)$.  Then
\[ \ip{(\mu\otimes\lambda)\cdot f}{g} = \int f_1(s) f_2(t) g(s,t) \ d\mu(s) \ d\lambda(t), \]
so $(\mu\otimes\lambda)\cdot f$ is the measure $\mu\cdot f_1 \otimes \lambda\cdot f_2
\in M(G) \otimes M(G)$.  By continuity, we see that $(\mu\otimes\lambda)\cdot f
\in M(G) \proten M(G)$ for any $f\in C_0(G\times G)=C_0(G)\inten C_0(G)$.
Let $\tau$ be in the kernel, so that
\[ \ip{\tau\cdot(\mu\otimes\lambda)}{f} = \ip{\tau}{(\mu\otimes\lambda)\cdot f} = 0, \]
as $\tau$ kills $M(G) \proten M(G)$.  Thus $\tau\cdot(\mu\otimes\lambda)=0$ in
$C_0(G\times G)^*$.  So, for $\sigma\in C_0(G\times G)^{**}$, we see that
\[ \ip{\sigma\tau}{\mu\otimes\lambda} = \ip{\sigma}{\tau\cdot(\mu\otimes\lambda)} = 0, \]
so that $\sigma\tau$ lies in the kernel.

Hence we have the following chain of $*$-homomorphisms
\[ \xymatrix{ C_0(G) \ar[r]^{\Phi} & C_0(G\times G)^{**}
\ar[r] & L^\infty(X\times X) = C_0(G)^{**} \vnten C_0(G)^{**}, } \]
say, giving rise to a $*$-homomorphism $\Delta_0:C_0(G)\rightarrow L^\infty(X\times X)$.
There is hence a canonical extension (compare \cite[Chapter~III, Lemma~2.2]{tak})
$\Delta:L^\infty(X) \rightarrow L^\infty(X\times X)$, which is a normal $*$-homomorphism.
Indeed, the preadjoint $\Delta_*:M(G)\proten M(G)\rightarrow M(G)$ is defined by the
chain of maps
\[ \xymatrix{ M(G)\proten M(G) = L^1(X\times X) \ar[r] & L^\infty(X\times X)^*
\ar[r]^{\Delta_0^*} & C_0(G)^* = M(G). } \]
Then, for $\mu,\lambda\in M(G)$ and $f\in C_0(G)$, we see that
\begin{align*} \ip{\Delta_*(\mu\otimes\lambda)}{f} &=
\ip{\Delta_0(f)}{\mu\otimes\lambda} = \int_{G\times G} f(st) \ d\mu(s) \ d\lambda(t),
\end{align*}
so $\Delta_*$ induces the usual convolution product on $M(G)$.  We have hence shown
that $M(G)^*$ is a commutative Hopf von Neumann algebra.
Notice that throughout, we have actually only used the fact that $G$ is a
locally compact semigroup.

For a recent survey on measure algebras, see \cite{DLS}, where the authors view
$M(G)$ as a \emph{Lau algebra} (see \cite{Lau}).

\subsection{Almost periodic functionals}

For a Banach algebra $\mc A$, a functional $\mu\in\mc A^*$ is \emph{almost periodic}
if the map
\[ R_\mu: \mc A\rightarrow\mc A^*, \quad a\mapsto a\cdot\mu \qquad (a\in\mc A), \]
is compact.  We denote the collection of almost periodic functionals by $\ap(\mc A)$.
Then it is easy to see that $\ap(\mc A)$ is a closed subspace of $\mc A^*$.  Using
the viewpoint of Hopf von Neumann algebras, it is easy to see that
$\ap(M(G))$ is a C$^*$-algebra.

\begin{theorem}\label{ap_case}
Let $(L^\infty(X),\Delta)$ be a commutative Hopf von Neumann algebra, so that $L^1(X)$
becomes a Banach algebra.  Then $\ap(L^1(X))$ is a C$^*$-subalgebra of $L^\infty(X)$.
\end{theorem}
\begin{proof}
Let $F\in\ap(L^1(X))\subseteq L^\infty(X)$.  For $f\in L^1(X)$, we shall write
$f^*$ for the pointwise complex-conjugation of $f$, so that $f\mapsto f^*$ is
the preadjoint of the involution on $L^\infty(X)$.  We see that for $f,g\in L^1(X)$,
\begin{align*} \ip{R_{F^*}(f)}{g} &= \ip{F^*}{\Delta_*(g\otimes f)}
= \ip{\Delta(F)^*}{g\otimes f} = \ip{F}{\Delta_*(g^*\otimes f^*)} \\
&= \ip{R_F(f^*)}{g^*} = \ip{R_F(f^*)^*}{g}, \end{align*}
so we conclude that $R_{F^*}$ is compact if and only if $R_F$ is compact.  Hence
$\ap(L^1(X))$ is $*$-closed.

We claim that $R_F = \Delta(F)^*\kappa_{L^1(X)}$.  Indeed, for $f,g\in L^1(X)$, we have that
\begin{align*} \ip{R_F}{f\otimes g} &= \ip{R_F(f)}{g} = \ip{F}{\Delta_*(g\otimes f)}
= \ip{\Delta(F)}{g\otimes f} \\ &= \ip{\Delta(F)(g)}{f}
= \ip{\Delta(F)\kappa_{L^1(X)}(f)}{g}. \end{align*}
So $\Delta(F) = R_F^*\kappa_{L^1(X)}$, and hence $R_F$ is compact if and only if
$\Delta(F)$ is compact.
As $L^\infty(X)$ has the approximation property, it follows that $\Delta(F)$ is compact
if and only if \[ \Delta(F)\in\mc A(L^1(X),L^\infty(X)) = L^\infty(X) \inten L^\infty(X)
= L^\infty(X) \otimes_{\min} L^\infty(X) \subseteq L^\infty(X) \vnten L^\infty(X). \]
Thus, if $F,G\in\ap(L^1(X))$, then $\Delta(F),\Delta(G) \in
L^\infty(X)\otimes_{\min} L^\infty(X)$, and so $\Delta(FG) = \Delta(F)\Delta(G)
\in L^\infty(X)\otimes_{\min} L^\infty(X)$, as $L^\infty(X)\otimes_{\min} L^\infty(X)$
is an algebra.  Hence $FG\in\ap(L^1(X))$, as required.
\end{proof}

\begin{corollary}
For a locally compact group $G$, $\ap(M(G))$ is a C$^*$-subalgebra of $M(G)^*$.
\end{corollary}

\section{Weakly almost periodic functionals}

We shall make use of vector valued $L^p$ spaces; for a measure space $X$, a Banach space
$E$, and $1\leq p<\infty$, we write $L^p(X,E)$ for the space of (classes of almost
everywhere equal) Bochner $p$-integrable functions from $X$ to $E$.  Then $L^p(X)
\otimes E$ naturally maps into $L^p(X,E)$ with dense range, inducing a norm
$\Delta_p$ on $L^p(X) \otimes E$.  This norm is studied in \cite[Chapter~7]{df}.
We have that $L^1(X) \proten E = L^1(X,E)$, so that $\Delta_1 = \|\cdot\|_\pi$, the
projective tensor norm.

It is worth noting that $\Delta_p$ is not a \emph{tensor norm}, as $T\in\mc B(L^p(X))$
may fail to extend to a bounded map $T\otimes\id: L^p(X,E)\rightarrow L^p(X,E)$.

However, note that for $F\in L^\infty(X)$,
then denoting also by $F$ the multiplication operator on $L^p(X)$, it is elementary
that $F\otimes\id$ is bounded, with norm $\|F\|$, on $L^p(X,E)$.  The norm
$\Delta_p$ does satisfy the estimates
\[ \|\tau\|_\epsilon \leq \Delta_p(\tau) \leq \|\tau\|_\pi
\quad (\tau \in L^p(X)\otimes E), \]
so in particular, $\Delta_p(f\otimes x) = \|f\| \|x\|$ for $f\in L^p(X)$ and $x\in E$.

We shall henceforth restrict to the case where $E$ is reflexive.  Then $E^*$ has the
Radon-Nikod\'ym property, and so $L^p(X,E)^* = L^{p'}(X,E^*)$ for $1<p<\infty$,
where $1/p' = 1-1/p$, see \cite[Appendix~D]{df}, or \cite{du}, for further details.
We stress that even when $p=2$, the dual pairing between $L^2(X,E)$ and $L^2(X,E^*)$
is always \emph{bilinear} and not sesquilinear.

\begin{lemma}
Let $E$ be a reflexive Banach space, and let $X$ be a measure space.  The map
\[ \Lambda:\big( L^2(X) \otimes E^* \big) \times \big( L^2(X) \otimes E \big)
\rightarrow L^1(X) \otimes \big(E^*\proten E\big);
\big( f\otimes\mu , g\otimes x\big) \mapsto fg \otimes (\mu\otimes x) \]
extends to a metric surjection
\[ \Lambda:L^2(X,E^*) \proten L^2(X,E) \rightarrow L^1\big(X,E^*\proten E\big) =
L^1(X) \proten E^* \proten E. \]
Here $fg$ denotes the pointwise product, so the Cauchy-Schwarz inequality shows that
$fg\in L^1(X)$ for $f,g\in L^2(X)$.
\end{lemma}
\begin{proof}
Let $F\in L^2(X,E^*)$ and $G\in L^2(X,E)$ be simple functions, so that
there exists a disjoint partition of $X$, say $(X_k)_{k=1}^n$, and $(x_k)_{k=1}^n
\subseteq E$ and $(\mu_k)_{k=1}^n \subseteq E^*$ with
\[ F = \sum_{k=1}^n \chi_{X_k} \otimes \mu_k, \quad
G = \sum_{k=1}^n \chi_{X_k} \otimes x_k. \]
Here we write $\chi_{X_k}$ for the indicator function of $X_k$.
Hence we see that
\[ \Lambda(F\otimes G) = \sum_{k=1}^n \chi_{X_k} \otimes (\mu_k\otimes x_k), \]
which has norm
\begin{align*}
\sum_{k=1}^n |X_k| \|\mu_k\otimes x_k\| &\leq
\Big( \sum_{k=1}^n |X_k| \|\mu_k\|^2 \Big)^{1/2}
\Big( \sum_{k=1}^n |X_k| \|x_k\|^2 \Big)^{1/2} = \|F\| \|G\|.
\end{align*}
As the simple functions are dense in $L^2(X,E)$, respectively, $L^2(X,E^*)$, we conclude
that the map $\Lambda:L^2(X,E^*) \times L^2(X,E) \rightarrow L^1(X,E^*\proten E)$
is a contraction, and so extends to a contraction $L^2(X,E^*) \proten L^2(X,E)
\rightarrow L^1(X,E^*\proten E)$.

As $E$ is reflexive, we may identify $(E^*\proten E)^*$ with $\mc B(E)$.
Hence $\Lambda^*$ is a map $\mc B(L^1(X),\mc B(E)) \rightarrow \mc B(L^2(X,E))$, say
$\pi \mapsto W$, where
\[ \ip{f\otimes\mu}{W(g\otimes x)} = \ip{\mu}{\pi(fg)(x)}
\qquad (f,g\in L^2(X), \mu\in E^*, x\in E). \]
By a suitable choice of $f,g,x$ and $\mu$, we see that $\|W\| \geq \|\pi\|$,
and so we conclude that actually $\|W\| = \|\pi\|$.  Hence $\Lambda^*$
is an isometry, so $\Lambda$ must be a metric surjection, as required.
\end{proof}

For $F\in L^\infty(G)$ and $T\in\mc B(E)$, we see that $F\otimes T$ extends to a bounded
linear map on $L^2(X,E)$.  Let $L^\infty(X) \vnten \mc B(E)$ be the weak$^*$-closure
of $L^\infty(X)\otimes \mc B(E)$ inside $\mc B(L^2(X,E))$.  This is then a \emph{dual
Banach algebra}, that is, multiplication in $L^\infty(X) \vnten \mc B(E)$ is
separately weak$^*$-continuous.  See \cite[Section~8]{daws}, where similar ideas
are explored.

\begin{proposition}\label{image_of_lambda_ad}
The above lemma isometrically identifies $\mc B(L^1(X),\mc B(E))$ with a subspace
of $\mc B(L^2(X,E))$, under the mapping $\Lambda^*$.  The image of $\Lambda^*$
is precisely $L^\infty(X) \vnten \mc B(E)$.
\end{proposition}
\begin{proof}
Standard Banach space theory shows that the image of $\Lambda^*$ is equal to
\[ (\ker\Lambda)^\perp = \big\{ T\in\mc B(L^2(X,E)) : \ip{T}{\tau}=0 \
(\tau\in L^2(X,E^*) \proten L^2(X,E), \Lambda(\tau)=0) \big\}. \]
Hence the image of $\Lambda^*$ is weak$^*$-closed.  Notice that $L^\infty(X) \vnten
\mc B(E)$ is equal to $Z^\perp$, where
\[ Z = \big\{ \tau\in L^2(X,E^*) \proten L^2(X,E) : \ip{F\otimes S}{\tau}=0
\ (F\in L^\infty(X), S\in\mc B(E)) \big\}. \]
Hence we need to show that $\ker\Lambda = Z$.

Let $F\in L^\infty(X)$ and $S\in\mc B(E)$.  Then let $T=F\otimes S\in \mc B(L^2(X,E))$,
and let $\pi:L^1(X) \rightarrow \mc B(E)$ be the rank-one operator induced by
$F\otimes S$, that is, $\pi(a)=\ip{F}{a}S$ for $a\in L^1(X)$.
Then $\Lambda^*(\pi) = T$, from which it follows that $\ker\Lambda\subseteq Z$.

As $L^1(X)$ has the approximation property, for each non-zero $\sigma \in L^1(X)
\proten (E^*\proten E)$, there exists $F\in L^\infty(X)$ and $S\in\mc B(E)$ with
$\ip{F\otimes S}{\sigma}\not=0$.  Hence, if $\tau\in L^2(X,E^*) \proten L^2(X,E)$
is such that $\sigma = \Lambda(\tau) \not=0$, then there exists $T\in L^\infty(X)
\otimes \mc B(E)$ with $0\not=\ip{T}{\sigma}=\ip{\Lambda^*(T)}{\tau}$.  This
shows that $Z \subseteq \ker\Lambda$.
\end{proof}

Informally, the above proposition allows us to write
\[ \mc B(L^1(X),\mc B(E)) = \big( L^1(X) \proten (E^*\proten E) \big)^*
= L^\infty(X) \vnten \mc B(E), \]
which is reminiscent of the operator space projective tensor result that
$(\mc M_* \proten \mc N_*)^* = \mc M \vnten \mc N$, for von Neumann algebras $\mc M$
and $\mc N$, see \cite[Theorem~7.2.4]{er}.  The important point for us is that we
have turned $\mc B(L^1(X),\mc B(E))$ into an algebra.  It is multiplication in this
algebra which will ultimately give rise to the multiplication of weakly almost periodic
functionals in $L^\infty(X)$.

Now let $L^\infty(X)$ be a Hopf von Neumann algebra, so it admits a coproduct $\Delta$.
We have a map
\[ \Delta_*\otimes\id: L^1(X\times X) \proten (E^*\proten E) \rightarrow
L^1(X) \proten (E^*\proten E), \]
whose adjoint, which we denote by $\Delta\otimes\id$, is a map
\[ \Delta\otimes\id: L^\infty(X) \vnten \mc B(E) \rightarrow
L^\infty(X\times X) \vnten \mc B(E), \]
where, of course, $L^\infty(X\times X) \vnten \mc B(E)$ is a subalgebra of
$\mc B(L^2(X\times X,E))$.

\begin{lemma}\label{delta_homo}
With notation as above, $(\Delta\otimes\id)$ is a homomorphism.

\end{lemma}
\begin{proof}
Let $F\in L^\infty(X)$ and $a,b\in L^1(X)$.  As $\Delta$ is a homomorphism,
it follows that
\[ \Delta_*\big( \Delta(F) \cdot (a\otimes b) \big) =
F \cdot \Delta_*(a\otimes b). \]

Let $U\in L^\infty(X)\vnten\mc B(E)$, let $T\in\mc B(E)$, and let $V=F\otimes T$.
For $\tau\in E^*\proten E$, we see that
\begin{align*} \ip{ (\Delta\otimes\id)(UV) }{a\otimes b\otimes\tau}
&= \ip{U(F\otimes T)}{\Delta_*(a\otimes b)\otimes\tau}
= \ip{U}{F\cdot\Delta_*(a\otimes b)\otimes T\cdot\tau} \\
&= \ip{U}{ \Delta_*( \Delta(F) \cdot (a\otimes b)) \otimes T\cdot\tau} \\
&= \ip{(\Delta\otimes\id)U}{\Delta(F) \cdot (a\otimes b) \otimes T\cdot\tau} \\
&= \ip{((\Delta\otimes\id)U)((\Delta\otimes\id)V)}{a\otimes b \otimes\tau}. \end{align*}
By linearity, we conclude that $(\Delta\otimes\id)(UV) =
( (\Delta\otimes\id)U )( (\Delta\otimes\id)V )$ for all $U\in L^\infty(X)\vnten\mc B(E)$
and $V\in L^\infty(X) \otimes \mc B(E)$.  By weak$^*$-continuity, this must also hold
for $V\in L^\infty(X) \vnten \mc B(E)$.
\end{proof}

We now wish to adapt \emph{leg numbering notation} to our setup.
Given $W\in\mc B(L^2(X,E))$, define $W_{23} \in \mc B(L^2(X\times X,E))$ by
\[ W_{23}(f_1\otimes f_2\otimes x) = f_1 \otimes W(f_2\otimes x)
\qquad (f_1,f_2\in L^2(X), x\in E). \]
Using the fact that $L^2(X\times X,E) = L^2(X, L^2(X,E))$, it is easy to see that
$W\mapsto W_{23}$ is a weak$^*$-continuous, isometric mapping.  If $W=
F\otimes S$ for some $F\in L^\infty(X)$ and $S\in\mc B(E)$, then clearly
$W_{23} = 1 \otimes F \otimes S \in L^\infty(X\times X) \otimes \mc B(E)$.  By
weak$^*$-continuity, we conclude that if $W\in L^\infty(X) \vnten \mc B(E)$, then
$W_{23} \in L^\infty(X\times X) \vnten \mc B(E)$.

Let $\chi:L^2(X\times X) \rightarrow L^2(X\times X)$ be the ``swap map'', defined on
elementary tensors by $\chi(f\otimes g) = g\otimes f$.  For
$W\in L^\infty(X\times X)\vnten \mc B(E)$, it is clear that $(\chi\otimes\id)W$
and $W(\chi\otimes\id)$ both also lie in $L^\infty(X\times X)\vnten \mc B(E)$.
For $W\in L^\infty(X)\vnten\mc B(E)$, we define $W_{13} = (\chi\otimes\id) W_{23}
(\chi\otimes\id) \in L^\infty(X\times X) \vnten \mc B(E)$.

\begin{theorem}\label{when_rep}
Let $(L^\infty(X),\Delta)$ be a Hopf von Neumann algebra, and let $E$ be a reflexive
Banach space.
Let $\pi:L^1(X)\rightarrow\mc B(E)$ be a bounded linear map, giving rise to
$W \in L^\infty(X) \vnten \mc B(E)$.  Then $\pi$ is a homomorphism, with respect
to $\Delta_*$, if and only if $(\Delta\otimes\id)W = W_{13} W_{23}$.
\end{theorem}
\begin{proof}
Let $f_1,f_2,g_1,g_2\in L^2(X)$, $\mu\in E^*$ and $x\in E$.  Then
\begin{align*} \ip{f_1\otimes f_2\otimes\mu}{(\Delta\otimes\id)W (g_1\otimes g_2\otimes x)}
&= \ip{\pi}{\Delta_*(f_1g_1 \otimes f_2g_2) \otimes (\mu\otimes x)} \\
&= \ip{\mu}{ \pi( \Delta_*(f_1g_1\otimes f_2g_2)) (x) }. \end{align*}

We now come to a proof where ``Sweedler notation'' would help greatly, but we should
perhaps, at least once, give a formal proof.  Informally, we shall ``pretend'' that
$W(g_2\otimes x) = h\otimes y$.  Then
\begin{align*} \ip{\mu}{\pi(f_1g_1)\pi(f_2g_2)(x)}
&= \ip{\pi(f_1g_1)^*(\mu)}{\pi(f_2g_2)(x)}
= \ip{f_2 \otimes \pi(f_1g_1)^*(\mu)}{W(g_2\otimes x)} \\
&= \ip{f_2}{h} \ip{\mu}{\pi(f_1g_1)(y)}
= \ip{f_2}{h} \ip{f_1\otimes\mu}{W(g_1\otimes y)} \\
&= \ip{f_1\otimes f_2\otimes\mu}{W_{13}(g_1\otimes h\otimes y)} \\
&= \ip{f_1\otimes f_2\otimes\mu}{W_{13}W_{23}(g_1\otimes g_2\otimes x)}, \end{align*}
which completes the proof.

To make this rigorous, for $\epsilon>0$, we can find a finite sum of
elementary tensors
$\sum_k h_k \otimes y_k \in L^2(X)\otimes E$ with $\|W(g_2\otimes x) -
\sum_k h_k\otimes y_k \| < \epsilon$.  Then
\[ \Big\| \ip{f_2 \otimes \pi(f_1g_1)^*(\mu)}{W(g_2\otimes x)} -
\sum_k \ip{f_2}{h_k} \ip{\mu}{\pi(f_1g_1)(y_k)} \Big\| < \epsilon\|f_2\|
\|\pi\| \|f_1\| \|g_1\| \|\mu\|, \]
and, as above,
\[ \sum_k \ip{f_2}{h_k} \ip{\mu}{\pi(f_1g_1)(y_k)} =
\sum_k \ip{f_1\otimes f_2\otimes\mu}{W_{13}(g_1\otimes h_k\otimes y_k)}, \]
so approximating again,
\begin{align*} \Big\|\sum_k & \ip{f_1\otimes f_2\otimes\mu}{W_{13}(g_1\otimes h_k\otimes y_k)}
- \ip{f_1\otimes f_2\otimes\mu}{W_{13}W_{23}(g_1\otimes g_2\otimes x)} \Big\|
\\ &< \epsilon \|W\|\|f_1\| \|f_2\| \|\mu\| \|g_1\|, \end{align*}
and so
\begin{align*} \big| \ip{\mu}{\pi(f_1g_1)\pi(f_2g_2)(x)} &  -
\ip{f_1\otimes f_2\otimes\mu}{W_{13}W_{23}(g_1\otimes g_2\otimes x)} \big| \\
&< 2\epsilon \|W\|\|f_1\| \|f_2\| \|\mu\| \|g_1\|. \end{align*}
As $\epsilon>0$ was arbitrary, the proof is complete.
\end{proof}

\subsection{Application to weakly almost periodic elements}

The following result was first shown by Young in \cite{young}, building upon \cite{DFJP},
and was recast in terms of the real interpolation method by Kaiser in \cite{kai}
(see also the similar arguments in \cite{daws}).

\begin{theorem}\label{wap_rep}
Let $\mc A$ be a Banach algebra, and let $\mu\in\mc A^*$.  The following are equivalent:
\begin{enumerate}
\item $\mu\in\wap(\mc A)$;
\item there exists a reflexive Banach space $E$, a contractive homomorphism $\pi:
\mc A\rightarrow\mc B(E)$, and $x\in E$, $\lambda\in E^*$ such that
\[ \ip{\mu}{a} = \ip{\lambda}{\pi(a)(x)} \qquad (a\in\mc A). \]
\end{enumerate}
\end{theorem}

We shall need a way of tensoring reflexive Banach spaces in a way that gives a reflexive
Banach space.  As we do not wish to get bogged down in the details of any one specific
way to do this, so we shall make an ad hoc definition.

\begin{definition}
Let $E$ and $F$ be reflexive Banach spaces, and let $\alpha$ be some norm on
$E\otimes F$ such that:
\begin{enumerate}
\item we have that $\|\tau\|_\epsilon \leq \alpha(\tau) \leq \|\tau\|_\pi$ for
  each $\tau\in E\otimes F$;
\item the completion of $E\otimes F$ with respect to $\alpha$ is a reflexive
  Banach space, say $E\proten_\alpha F$;
\item given $T\in\mc B(E)$ and $S\in\mc B(F)$, the map $T\otimes S:E\otimes F
  \rightarrow E\otimes F$ extends to a bounded operator on $E\proten_\alpha F$
  with norm $\|T\| \|S\|$.
\end{enumerate}
Then we say that $\|\cdot\|$ is a \emph{reflexive tensor norm} on $E\otimes F$.
\end{definition}

The existence of reflexive tensor norms is shown in \cite{as}, for example.

Let $E$ and $F$ be reflexive Banach spaces, and let $\alpha$ be a reflexive tensor norm
on $E\otimes F$.  As the map $E\proten F\rightarrow E \proten_\alpha F$ is
contractive with dense range, the adjoint $(E\proten_\alpha F)^* \rightarrow
\mc B(E,F^*)$ is injective.  We write $\mc B_{\alpha'}(E,F^*)$ for the image,
and equip it with the norm coming from $(E\proten_\alpha F)^*$, so we may write
$(E\proten_\alpha F)^* = \mc B_{\alpha'}(E,F^*)$.  For some norms $\alpha$, there
exists a \emph{dual norm} $\alpha'$, which is a reflexive tensor norm on
$E^* \otimes F^*$, such that $\mc B_{\alpha'}(E,F^*) = E^* \proten_{\alpha'} F^*$.
We shall, however, not have to assume this extra condition.  For us, it suffices
to note that as $\alpha$ dominates $\|\cdot\|_\epsilon$, there is a natural
embedding of $E^*\otimes F^*$ into $(E\proten_\alpha F)^*$.

\begin{theorem}
Let $(L^\infty(X),\Delta)$ be a commutative Hopf von Neumann algebra, and use $\Delta_*$
to turn $L^1(X)$ into a Banach algebra.  Then $\wap(L^1(X))$ is a C$^*$-subalgebra of
$L^\infty(X)$.
\end{theorem}
\begin{proof}
We know that $\wap(L^1(X))$ is a closed subspace of $L^\infty(X)$.  Exactly the same
argument as in the proof of Theorem~\ref{ap_case} shows that $\wap(L^1(X))$ is $*$-closed,
so it remains only to show that $\wap(L^1(X))$ is closed under multiplication.

Let $F_1,F_2\in\wap(L^1(X))$.  By Theorem~\ref{wap_rep}, for $i=1,2$, there exists a
reflexive Banach space $E_i$, a contractive homomorphism
$\pi_i:L^1(X)\rightarrow\mc B(E_i)$, $x_i\in E_i$ and $\mu_i\in E_i^*$ such that
\[ \ip{F_i}{a} = \ip{\mu_i}{\pi_i(a)(x_i)} \qquad (a\in L^1(X)). \]
Let $\alpha$ be a reflexive tensor norm on $E_1\otimes E_2$, and define
$\hat\pi_i:L^1(X) \rightarrow \mc B(E_1\proten_\alpha E_2)$, for $i=1,2$, by
\[ \hat\pi_1(a) = \pi_1(a)\otimes\id, \quad
\hat\pi_2(a) = \id\otimes\pi_2(a) \qquad (a\in L^1(X)). \]
Then $\hat\pi_1$ and $\hat\pi_2$ are contractive homomorphisms from $L^1(X)$ to
$\mc B(E_1\proten_\alpha E_2)$, and hence give rise, respectively, to $U,V\in L^\infty(X)
\vnten \mc B(E_1\proten_\alpha E_2)$, such that
\[ (\Delta\otimes\id)U = U_{13}U_{23}, \quad
(\Delta\otimes\id)V = V_{13}V_{23}. \]

Let $W = UV \in L^\infty(X) \vnten \mc B(E_1\proten_\alpha E_2)$, and let
$\pi:L^1(X) \rightarrow \mc B(E_1\proten_\alpha E_2)$ be induced by $W$.
By Lemma~\ref{delta_homo}, we see that
\[ (\Delta\otimes\id)W = ((\Delta\otimes\id)U)((\Delta\otimes\id)V)
= U_{13} U_{23} V_{13} V_{23}. \]
We also see that
\[ W_{13} W_{23} = U_{13} V_{13} U_{23} V_{23}. \]
We claim that $U_{23} V_{13} = V_{13} U_{23}$, from which it follows, from
Theorem~\ref{when_rep}, then $\pi$ is a homomorphism.

To prove the claim, we shall again deploy Sweedler notation: the argument can be
made rigorous in the same way as in the proof of Theorem~\ref{when_rep}.
Let $f_1,f_2,g_1,g_2\in L^2(X)$, $w_1\in E_1$, $w_2\in E_2$ and $T\in\mc B_{\alpha'}(E_1,
E_2^*) = (E_1\proten_\alpha E_2)^*$.  Suppose that $U^*(f_2\otimes T) = h_1\otimes S_1$,
so that for $k\in L^2(X)$, $z_1\in E_1$ and $z_2\in E_2$, we have that
\begin{align*} \ip{h_1\otimes S_1}{k\otimes z_1\otimes z_2}
&= \ip{f_2\otimes T}{U(k\otimes z_1\otimes z_2)}
= \ip{T}{\pi_1(f_2k)(z_1)\otimes z_2} \\&= \ip{T\pi_1(f_2k)(z_1)}{z_2}. \end{align*}
Similarly, suppose that $V^*(f_1\otimes T) = h_2\otimes S_2$, so that
\begin{align*} \ip{h_2\otimes S_2}{k\otimes z_1\otimes z_2}
&= \ip{f_1\otimes T}{V(k\otimes z_1\otimes z_2)}
= \ip{T}{z_1\otimes \pi_2(f_1k)(z_2)} \\&= \ip{T(z_1)}{\pi_2(f_1k)(z_2)}. \end{align*}
Thus we see that
\begin{align*}
&\ip{f_1\otimes f_2\otimes T}{U_{23} V_{13}(g_1\otimes g_2\otimes w_1\otimes w_2)} \\
&= \ip{f_1\otimes U^*(f_2\otimes T)}{(\chi\otimes\id\otimes\id)
   (g_2\otimes V(g_1\otimes w_1\otimes w_2))} \\
&= \ip{h_1\otimes f_1\otimes S_1}{g_2\otimes V(g_1\otimes w_1\otimes w_2)}
= \ip{h_1\otimes S_1}{g_2 \otimes w_1 \otimes \pi_2(f_1g_1)(w_2)} \\
&= \ip{T\pi_1(f_2g_2)(w_1)}{\pi_2(f_1g_1)(w_2)},
\end{align*}
and also
\begin{align*}
&\ip{f_1\otimes f_2\otimes T}{V_{13} U_{23} (g_1\otimes g_2\otimes w_1\otimes w_2)} \\
&= \ip{f_2 \otimes V^*(f_1\otimes T)}{(\chi\otimes\id\otimes\id)
  (g_1\otimes U(g_2\otimes w_1\otimes w_2))} \\
&= \ip{h_2 \otimes f_2\otimes S_2}{g_1\otimes U(g_2\otimes w_1\otimes w_2)}
= \ip{h_2 \otimes S_2}{g_1 \otimes \pi_1(f_2g_2)(w_1)\otimes w_2} \\
&= \ip{T\pi_1(f_2g_2)(w_1)}{\pi_2(f_1g_1)(w_2)},
\end{align*}
which proves equality, as required.

Finally, let $x=x_1\otimes x_2 \in E_1 \proten_\alpha E_2$ and let $\mu=\mu_1\otimes\mu_2
\in (E_1 \proten_\alpha E_2)^*$.  By Theorem~\ref{wap_rep}, if $F\in L^\infty(X)$
is defined by
\[ \ip{F}{a} = \ip{\mu}{\pi(a)(x)} \qquad(a\in L^1(X)), \]
then $F\in\wap(L^\infty(X))$.  Now, for $a\in L^1(X)$, pick $f,g\in L^2(X)$ with
$fg=a$, so we see that
\begin{align*} \ip{\mu}{\pi(a)(x)}
&= \ip{f\otimes\mu_1\otimes\mu_2}{W(g\otimes x_1\otimes x_2)}
= \ip{f\otimes\mu_1\otimes\mu_2}{UV(g\otimes x_1\otimes x_2)}.
\end{align*}

Notice that for $k\in L^2(X)$, $\lambda_1\in E_1^*$ and $\lambda_2\in E_2^*$,
\begin{align*} \ip{k\otimes\lambda_1\otimes\lambda_2}{V(g\otimes x_1\otimes x_2)}
&= \ip{\lambda_1\otimes\lambda_2}{(\id\otimes\pi_2(kg))(x_1\otimes x_2)} \\
&= \ip{\lambda_1}{x_1} \ip{\lambda_2}{\pi_2(kg)(x_2)}. \end{align*}
For $T\in (E_1\proten_\alpha E_2)^* = \mc B_{\alpha'}(E_1,E_2^*)$, as
$E\proten_\alpha F$ is reflexive, we hence must have that
\[ \ip{k\otimes T}{V(g\otimes x_1\otimes x_2)}
= \ip{T(x_1)}{\pi_2(kg)(x_2)}. \]
Define a map $\theta: L^2(X)\otimes E_2 \rightarrow L^2(X) \otimes
E_1\proten_\alpha E_2$ by $\theta(k\otimes x) = \theta(k\otimes x_1\otimes x)$
for $k\in L^2(X)$ and $x\in E_2$ on elementary tensors.  A simple calculation
shows that $\theta$ extends to a contraction $L^2(X,E_2) \rightarrow
L^2(X,E_1\proten_\alpha E_2)$.  Then, for $\tau\in L^2(X,E_2)$, we have that
\[ \ip{k\otimes\lambda_1\otimes\lambda_2}{\theta(\tau)}
= \ip{\lambda_1}{x_1} \ip{k\otimes\lambda_2}{\tau}, \]
and so, similarly,
\[ \ip{k\otimes T}{\theta(\tau)} = \ip{k\otimes T(x_1)}{\tau}. \]
Let $\hat V\in L^\infty(X)\vnten\mc B(E_2)$ be defined by $\pi_2$.
Then
\[ \ip{k\otimes T}{V(g\otimes x_1\otimes x_2)}
= \ip{k\otimes T(x_1)}{\hat V(g\otimes x_2)}
= \ip{k\otimes T}{\theta \hat V(g\otimes x_2)}. \]
Thus $V(g\otimes x_1\otimes x_2)$ is in the image of $\theta$, being
equal to $\theta \hat V(g\otimes x_2)$.

Again, we use Sweedler notation, so by the previous paragraph, we may suppose
that $V(g\otimes x_1\otimes x_2) = h \otimes x_1\otimes y_2$.  Then, for
$k\in L^2(X)$, $\lambda_1\in E_1^*$ and $\lambda_2\in E_2^*$, we see that
\begin{align*} \ip{k\otimes\lambda_1\otimes\lambda_2}{h \otimes x_1\otimes y_2}
&= \ip{\lambda_1\otimes\lambda_2}{(\id\otimes\pi_2(kg))(x_1\otimes x_2)}
= \ip{\lambda_1}{x_1} \ip{\lambda_2}{\pi_2(kg)(x_2)}.
\end{align*}
Hence we have that
\[ \ip{k\otimes\lambda_2}{h\otimes y_2}
= \ip{\lambda_2}{\pi_2(kg)(x_2)}. \]
Finally, we have that
\begin{align*} \ip{\mu}{\pi(a)(x)}
&= \ip{f\otimes\mu_1\otimes\mu_2}{U(h\otimes y_1\otimes y_2)}
= \ip{\mu_1\otimes\mu_2}{(\pi_1(fh)\otimes\id)(y_1\otimes y_2)} \\
&= \ip{\mu_2}{y_2} \ip{F_1}{fh}
= \ip{F_1 f \otimes \mu_2}{h\otimes y_2} \\
&= \ip{\mu_2}{\pi_2((F_1 f)g)(x_2)}
= \ip{F_2}{(F_1 f)g} = \ip{F_2}{F_1 fg} = \ip{F_1 F_2}{a}.
\end{align*}
So we conclude that $F_1 F_2 = F \in \wap(L^1(X))$, showing that $\wap(L^1(X))$
is an algebra.
\end{proof}

\begin{corollary}
Let $G$ be a locally compact group.  Then $\wap(M(G))$ is a C$^*$-subalgebra of
$M(G)^* = C_0(G)^{**}$.
\end{corollary}