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\title[SNU crossed products course: Lecture 1]{Seoul National University
short course:
An introduction to the structure of crossed product C*-algebras.}
\subtitle{Lecture~1: What is a crossed product?}
\author{N.~Christopher Phillips}
\institute[U.~of Oregon]{University of Oregon}
\date{12~December 2009}
\begin{document}
\maketitle
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\begin{frame}
\frametitle{Comments}
There is a related set of notes posted on the web.
See the link at:
\vspace{1ex}
http://www.uoregon.edu/$\sim$ncp/Courses/LisbonCrossedProducts/
LisbonCrossedProducts.html
\vspace{1ex}
This is accessible from my home page:
\vspace{1ex}
http://www.uoregon.edu/$\sim$ncp
\vspace{1ex}
The notes contain most of what I will say during these lectures,
and much more besides.
\pause
Also see the slides from the Lisbon course,
four links on the same website.
There is a great deal of overlap.
\pause
\vspace{2ex}
Please let me know of any misprints, mistakes, etc.\ found in
the notes, slides, etc.
I will also post the slides from these lectures on my website,
at:
\vspace{1ex}
http://www.uoregon.edu/$\sim$ncp/Courses/SeoulCrossedProducts/
SeoulCrossedProducts.html
\vspace{1ex}
(also available from a link on my home page).
% \pause
% \vspace{2ex}
%
% Sign up for the operator algebraist email directory,
% by emailing:
% {\tt{ncp@uoregon.edu}}.
% \pause
\end{frame}
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\begin{frame}
\frametitle{Outline}
There are many directions in the theory of crossed products.
These lectures are biased towards the general problem of
classifying crossed products,
\pause
in cases in which they are expected to be simple.
\pause
(However, we will not get very far in that direction.)
\pause
See the end of Section~1 of the notes for other directions.
\pause
\vspace{5ex}
A brief outline of the lectures:
\pause
\begin{itemize}
\item
Introductory material, basic definitions, and examples of group actions.
\pause
\item
Construction of the \cp\ of an action by a discrete group.
\pause
\item
Examples of some elementary computations of crossed products.
\pause
\item
Simplicity of crossed products by \mh s.
\pause
\item
Toward the classification of crossed products by \mh s.
\pause
\item
Actions of $\Z^d$: an outline of the subgroupoid subalgebra method.
\end{itemize}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Actions of groups on C*-algebras}
\begin{dfn}\label{D:Action}
Let $G$ be a locally compact group, and let $A$ be a \ca.
\pause
An {\emph{action of $G$ on $A$}}
\pause
is a \hm\ $\af \colon G \to \Aut (A),$
\pause
usually written $g \mapsto \af_g,$
\pause
such that, for every $a \in A,$ the
function $g \mapsto \af_g (a),$
from $G$ to $A,$ is norm \ct.
\end{dfn}
\pause
\vspace{2ex}
On a von Neumann algebra,
substitute the $\sm$-weak operator topology for the norm topology.
\pause
\vspace{2ex}
The continuity condition
is the analog of requiring that a unitary representation
of $G$ on a Hilbert space be \ct\ in the strong operator topology.
It is usually much too strong a condition to
require that $g \mapsto \af_g$ be a norm \ct\ map from
$G$ to the bounded operators on $A.$
\pause
\vspace{2ex}
Of course, if $G$ is discrete,
it doesn't matter.
In this course, we will concentrate on discrete~$G.$
% \pause
\end{frame}
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\begin{frame}
\frametitle{We will construct crossed products}
Given $\af \colon G \to \Aut (A),$
we will construct
a crossed product \ca\ $C^* (G, A, \af)$
\pause
and a reduced crossed product \ca\ $C^*_{\mathrm{r}} (G, A, \af).$
\pause
(There are many other commonly used notations.
See Remark~3.16 in the notes.)
\pause
\vspace{2ex}
If $A$ is unital and $G$ is discrete, it is a suitable
completion of the algebraic skew group ring $A [G],$
with multiplication determined by $g a g^{-1} = \af_g (a)$
for $g \in G$ and $a \in A.$
% \pause
\end{frame}
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\begin{frame}
\frametitle{Motivation for group actions on C*-algebras and their
crossed products}
Let $G$ be a locally compact group obtained as a
semidirect product $G = N \rtimes H.$
The action of $H$ on $N$ gives actions of $H$ on the full and reduced
group \ca s
$C^* (N)$ and $C^*_{\mathrm{r}} (N),$
and one has
$C^* (G) \cong C^* (H, \, C^* (N))$ and
$C^*_{\mathrm{r}} (G) \cong
C^*_{\mathrm{r}} (H, \, C^*_{\mathrm{r}} (N)).$
\pause
\vspace{2ex}
Probably the most important group action is time evolution:
if a \ca\ $A$ is supposed to represent the possible states
of a physical system in some manner,
then there should be an action $\af \colon \R \to \Aut (A)$
which describes the time evolution of the system.
Actions of $\Z,$
which are easier to study,
can be thought of as ``discrete time evolution''.
\pause
\vspace{2ex}
Crossed products are a common way of constructing simple \ca s.
We will see some examples later.
% \pause
\end{frame}
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\begin{frame}
\frametitle{Motivation for group actions on C*-algebras and their
crossed products (continued)}
If one has a \hme\ $h$ of a locally compact Hausdorff space~$X,$
the crossed product $C^* (\Z, X, h)$
sometimes carries considerable information
about the dynamics of $h.$
The best known example is the result of Giordano, Putnam, and Skau
on \mh s of the Cantor set:
isomorphism of the \tgca s is equivalent to strong orbit equivalence
of the \hme s.
\pause
\vspace{2ex}
For compact groups,
equivariant indices take values on the equivariant K-theory
of a suitable \ca\ with an action of the group.
When the group is not compact,
one usually needs instead the K-theory of the crossed product \ca,
or of the reduced crossed product \ca.
(When the group is compact, this is the same thing.)
\pause
\vspace{1ex}
In other situations as well,
the K-theory of the full or reduced \cp\ is the appropriate
substitute for equivariant K-theory.
% \pause
\end{frame}
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\begin{frame}
\frametitle{The commutative case}
\begin{definition}
A \ct\ action of a topological group $G$ on
a topological space $X$ is a \ct\ function $G \times X \to X,$
usually written $(g, x) \mapsto g \cdot x$ or $(g, x) \mapsto g x,$
such that $(g h) x = g (h x)$ for all $g, h \in G$ and $x \in X$
and $1 \cdot x = x$ for all $x \in X.$
\end{definition}
\pause
\vspace{2ex}
For a \ct\ action of a locally compact group $G$ on a \lchs~$X,$
there is a corresponding action $\af \colon G \to \Aut (C_0 (X)),$
given by $\af_g (f) (x) = f (g^{-1} x).$
\pause
\vspace{2ex}
(If $G$ is not abelian, the inverse is necessary
to get $\af_g \circ \af_h = \af_{g h}$ rather than $\af_{h g}.$)
\pause
\vspace{2ex}
One should check that these formulas
determine a one to one correspondence between \ct\ actions of $G$
on $X$ and \ct\ actions of $G$ on $C_0 (X).$
\pause
(The main point is to check that the continuity conditions match.)
% \pause
\end{frame}
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\begin{frame}
\frametitle{Examples of group actions on C*-algebras}
We will give some examples of actions of a group~$G$ on \ca s.
(Not all of them give interesting \cp s.)
\pause
\vspace{2ex}
We start with examples of group actions on locally compact spaces~$X,$
which give rise to examples of group actions on
commutative \ca s.
\pause
\vspace{2ex}
We will discuss some of their crossed products later,
but in some of the examples we state the results immediately.
\pause
As one goes through the commutative examples,
note that a closed orbit of the form $G x \cong G / H$
gives rise to a quotient of the \cp\ isomorphic to
$K (L^2 (G / H)) \otimes C^* (H).$
\pause
\vspace{2ex}
There are more examples in the notes,
and there is more detail on these in the Lisbon slides.
% \pause
\end{frame}
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\begin{frame}
\frametitle{Examples of actions on compact spaces}
\begin{itemize}
\item
$G$ is arbitrary, $X$ is a point, and the action is trivial.
\pause
The full and reduced crossed products are
the usual full and reduced group \ca s
$C^* (G)$ and $C^*_{\mathrm{r}} (G).$
\pause
\item
$X = G,$
and the action is given by (left) translation:
$g \cdot x = g x.$
\pause
The full and reduced crossed products are both
isomorphic to $K (L^2 (G)).$
\pause
\item
If $H \S G$ is a closed subgroup,
then $G$ acts \ct ly on $G / H$ by translation.
\pause
It turns out that
$C^* (G, \, G / H) \cong K (L^2 (G / H)) \otimes C^* (H).$
Note that there is no ``twisting''.
\pause
\item
If $H \S G$ is a closed subgroup,
then $H$ acts \ct ly on $G$ by translation.
\pause
It turns out that
$C^* (H, G)$ is stably isomorphic to $K (L^2 (H)) \otimes C_0 (G / H).$
Stably, there is no ``twisting''.
\pause
\item
$X = S^1 = \{ \zt \in \C \colon | \zt | = 1 \},$
$G = \Z,$
and the action is rotation by multiples of a fixed angle $2 \pi \te.$
\pause
These are {\emph{rational rotations}} (for $\te \in \Q$) or
{\emph{irrational rotations}} (for $\te \not\in \Q$),
\pause
and the \cp s are the
well known (rational or irrational) rotation algebras.
\end{itemize}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Examples of actions on compact spaces (continued)}
\begin{itemize}
\item
Take $X = \{ 0, 1 \}^{\Z},$
with elements being described as $x = (x_n)_{n \in \Z}.$
Take $G = \Z,$
with action generated by the {\emph{shift}} \hme\ $h (x)_n = x_{n - 1}$
for $x \in X$ and $n \in \Z.$
\pause
\item
Subshifts:
In the previous example,
replace $X$ by an invariant subset.
\pause
\item
More general shifts and subshifts:
replace $\{ 0, 1 \}$ by some other \cms~$S.$
\pause
\item
Let $X = \Z_p,$ the group of $p$-adic integers.
\pause
It is a compact topological group,
and as a metric space it is homeomorphic to the Cantor set.
\pause
Let $h \colon X \to X$ be the \hme\ defined on
the dense subset $\Z$
by $h (n) = n + 1,$
and take the action of $\Z$ it generates.
\pause
Many variations are possible.
\end{itemize}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Examples of actions on compact spaces (continued)}
\begin{itemize}
\item
Take $X = S^n = \{ x \in \R^{n + 1} \colon \| x \|_2 = 1 \}.$
Multiplication by $-1$ generates an action of $\Z / 2 \Z.$
\pause
The \cp\ turns out to be isomorphic to the section algebra of a
locally trivial but nontrivial bundle over the
real projective space $\R P^n = S^n / ( \Z / 2 \Z)$
with fiber~$M_2.$
\pause
\item
Complex conjugation generates
an action of $\Z / 2 \Z$ on $S^1 \S \C.$
\pause
\item
Take $G = {\mathrm{SL}}_2 (\Z).$
It acts linearly on $\R^2$ (as a subgroup of ${\mathrm{GL}}_2 (\R)$),
fixing $\Z^2,$
\pause
so the action is well defined on
$\R^2 / \Z^2 \cong S^1 \times S^1.$
\end{itemize}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Examples of actions on noncommutative C*-algebras}
\begin{itemize}
\item
There is a trivial action of $G$ on any \ca~$A.$
\pause
The full \cp\ turns out to be $C^* (G) \otimes_{\mathrm{max}} A,$
and the reduced \cp\ turns out to be
$C^*_{\mathrm{r}} (G) \otimes_{\mathrm{min}} A.$
\pause
\item
If $A$ is unital and $u \in A$ is unitary,
let $\Ad (u)$ be the automorphism
$a \mapsto u a u^*.$
\pause
Now let $G$ be locally compact,
let $A$ be unital,
and let $g \mapsto z_g$ be a norm \ct\ group \hm\ from $G$ to the
unitary group $U (A)$ of $A.$
\pause
Then $g \mapsto \Ad (z_g)$
defines an action of $G$ on $A.$
\pause
These actions are called {\emph{inner}}.
\pause
The \cp s turn out to be the same as for the trivial action.
\pause
\item
An action via inner automorphisms is not necessarily an
inner action.
\pause
Let $A = M_{2},$ let $G = (\Z / 2 \Z)^{2}$ with
generators $g_{1}$ and $g_{2},$
and set
\pause
\[
\af_1 = \id_A, \,\,\,\,
\af_{g_{1}}
= \Ad \left(
\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix} \right),
\,\,\,\,
\af_{g_{2}}
= \Ad \left(
\begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right),
\,\,\,\,
\af_{g_1 g_{2}}
= \Ad \left(
\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix} \right).
\]
\pause
The point is that the implementing unitaries
for $\af_{g_1}$ and $\af_{g_2}$ commute up to a scalar,
\pause
but can't be appropriately modified to commute exactly.
\pause
The \cp\ turns out to be isomorphic to~$M_4.$
\end{itemize}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Examples of actions on C*-algebras
(continued)}
\begin{itemize}
\item
For $\te \in \R,$
let $A_{\te}$ be the rotation algebra,
\pause
the universal \ca\ generated by unitaries $u$ and $v$
satisfying $v u = \exp (2 \pi i \te) u v.$
\pause
The group $G = \SL_2 (\Z)$ acts on $A_{\te}$ by sending the
matrix
$n = \left( \begin{smallmatrix} n_{1, 1} & n_{1, 2} \\
n_{2, 1} & n_{2, 2} \end{smallmatrix} \right)$
to the automorphism
\pause
\[
\af_n (u)
= \exp (\pi i n_{1, 1} n_{2, 1} \te) u^{n_{1, 1}} v^{n_{2, 1}},
\,\,\,\,\,\,
\af_n (v)
= \exp (\pi i n_{1, 2} n_{2, 2} \te) u^{n_{1, 2}} v^{n_{2, 2}}.
\]
\pause
This is the noncommutative version of the action of
$\SL_2 (\Z)$ on $S^1 \times S^1$ above.
\pause
\item
Restrict the action of the previous example to finite subgroups.
\pause
We now know that for $\te \not\in \Q$
the \cp s are all~AF.
\pause
\item
There is an action
$\af \colon S^1 \times S^1 \to \Aut (A_{\te})$
determined by
\[
\af_{(\zt_1, \zt_2)} (u) = \zt_1 u \andeqn
\af_{(\zt_1, \zt_2)} (v) = \zt_2 v.
\]
\pause
\vspace*{-4ex}
\item
Restrict the previous action to subgroups of $S^1 \times S^1.$
\pause
For example, a single such automorphism
generates an action of~$\Z.$
\end{itemize}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Examples of actions on C*-algebras
(continued)}
\begin{itemize}
\item
Let $s_1, s_2, \ldots, s_n$ be the standard generators
of the Cuntz algebra ${\mathcal{O}}_n,$
satisfying $s_j^* s_j = 1$ for $1 \leq j \leq n$
and $\sum_{j = 1}^n s_j s_j^* = 1.$
\pause
There is an action of $(S^1)^n$ on ${\mathcal{O}}_n$
such that $\af_{(\zt_1, \zt_2, \ldots, \zt_n)} (s_j) = \zt_j s_j$
for $1 \leq j \leq n.$
\pause
\item
Regarding $(S^1)^n$ as the diagonal unitary matrices,
this action extends to an action of the unitary group
$U (M_n)$ on ${\mathcal{O}}_n.$
\pause
If $u = ( u_{j, k} )_{j, k = 1}^n \in M_n$ is unitary, then
$\af_u \in \Aut ({\mathcal{O}}_n)$
is determined by
\pause
\[
\af_u (s_j) = \sum_{k = 1}^n u_{k, j} s_k.
\]
\pause
\item
Any individual automorphism from this action gives an
action of $\Z$ on ${\mathcal{O}}_n.$
\pause
\item
The first example on this slide generalizes to give gauge
actions on graph \ca s.
\end{itemize}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Examples of actions on C*-algebras
(continued)}
\begin{itemize}
\item
Let $A$ be the UHF algebra $\bigotimes_{n = 1}^{\I} M_{k_n},$
let $G$ be a locally compact group,
and let $\bt^{(n)} \colon G \to \Aut (M_{k_n})$ be an
action of $G$ on $M_{k_n}.$
\pause
Define an
action $\af \colon G \to \Aut (A)$
by
\[
\af_g (a_1 \otimes \cdots \otimes a_n \otimes 1 \otimes \cdots)
= \bt^{(1)}_g (a_1)
\otimes \cdots \otimes \bt^{(n)}_g (a_n)
\otimes 1 \otimes \cdots.
\]
\pause
\vspace*{-3ex}
\item
If each $\bt^{(n)}$ above is the inner action coming from a
unitary representation of $G$ on $\C^{k_n},$
then $\af$ is called a {\emph{product type action}}.
\pause
\item
As a specific example, take $G = \Z / 2 \Z,$
and for every $n$ take $k_n = 2$ and take
$\bt^{(n)}$ to be generated by
$\Ad \left(
\begin{smallmatrix} 1 & 0 \\ 0 & -1 \end{smallmatrix} \right).$
\pause
\item
Let $A$ be a unital \ca.
The {\emph{tensor flip}} is the action of $\Z / 2 \Z$
on $A \otimes_{\mathrm{max}} A$
generated by $a \otimes b \mapsto b \otimes a.$
\pause
\item
There is also a tensor flip on $A \otimes_{\mathrm{min}} A.$
\pause
\item
The symmetric group $S_n$ acts on
the $n$-fold maximal and minimal tensor products of $A$ with itself.
\pause
\item
There is also a ``tensor shift'',
a noncommutative analog,
defined on $\bigotimes_{n \in \Z} A,$
of the shift on $S^{\Z}.$
\end{itemize}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Covariant representations}
To define the \cp, we need:
\pause
\vspace{2ex}
\begin{dfn}\label{D:CvRep}
Let $\af \colon G \to \Aut (A)$
be an action of a locally compact group $G$ on a \ca~$A.$
\pause
A {\emph{covariant representation}}
of $(G, A, \af)$ on a Hilbert space $H$
is
\pause
a pair $(v, \pi)$
consisting of a unitary representation $v \colon G \to U (H)$
(the unitary group of $H$)
\pause
and a representation $\pi \colon A \to L (H)$
(the algebra of all bounded operators on $H$),
\pause
satisfying the {\emph{covariance condition}}
\[
v (g) \pi (a) v (g)^* = \pi (\af_g (a))
\]
for all $g \in G$ and $a \in A.$
\pause
It is called {\emph{nondegenerate}} if $\pi$ is nondegenerate.
\end{dfn}
\pause
\vspace{2ex}
By convention, unitary representations are strong operator \ct.
\pause
Representations of \ca s,
and of other *-algebras
are *-representations
(and, similarly, \hm s are *-\hm s).
% \pause
\end{frame}
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\begin{frame}
\frametitle{Remarks on Banach space valued integration}
The crossed product \ca\ $C^* (G, A, \af)$ is the universal \ca\ for
covariant representations of $(G, A, \af),$
\pause
in essentially the same way that the (full) group \ca\ $C^* (G)$ is
the universal \ca\ for unitary representations of $G.$
\pause
We construct it in a similar way to the group \ca.
We start with the analog of $L^1 (G).$
\pause
\vspace{2ex}
For a general \lcg,
one needs an appropriate notion of integration of Banach space valued
functions.
\pause
One must prove that twisted convolution below
is well defined, associative, distributive,
and satisfies $(a b)^* = b^* a^*.$
\pause
Once one has the appropriate notion of integration,
the proofs are similar to the proofs of the corresponding
facts about convolution in $L^1 (G).$
\pause
Integration of \cfn s with compact support is much easier
than integration of $L^1$ functions,
\pause
but the ``right'' way to do this is to define measurable
Banach space valued functions and their integrals.
\pause
This has been done;
one reference is Appendix B of the book of Williams.
Things simplify considerably if $G$ is second countable
and $A$ is separable,
but neither of these conditions is necessary.
% \pause
\end{frame}
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\begin{frame}
\frametitle{Twisted convolution}
\begin{dfn}\label{D:L1}
Let $\af \colon G \to \Aut (A)$
be an action of a locally compact group $G$ on a \ca~$A.$
\pause
We let $C_{\mathrm{c}} (G, A, \af)$ be the *-algebra of
\cfn s $a \colon G \to A$ with compact support,
with pointwise addition and scalar multiplication.
\pause
Using Haar measure in the integral,
we define multiplication by the
following ``twisted convolution'':
\[
(a b) (g) = \int_G a (h) \af_h (b ( h^{-1} g)) \, d h.
\]
\pause
Let $\Dt$ be the modular function of $G.$
We define the adjoint by
\[
a^* (g) = \Dt (g)^{-1} \af_g ( a (g^{-1})^*).
\]
\pause
We define a norm $\| \cdot \|_1$ on $C_{\mathrm{c}} (G, A, \af)$
by $\| a \|_1 = \int_G \| a (g) \| \, d g.$
\pause
One checks that $\| a b \|_1 \leq \| a \|_1 \| b \|_1$
and $\| a^* \|_1 = \| a \|_1.$
\pause
Then $L^1 (G, A, \af)$ is the Banach *-algebra obtained by
completing $C_{\mathrm{c}} (G, A, \af)$ in $\| \cdot \|_1.$
\end{dfn}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Twisted convolution (continued)}
\begin{exr}\label{Ex:CkConv}
Assuming suitable versions of Fubini's Theorem
for Banach space valued integrals,
\pause
check that that
multiplication in $C_{\mathrm{c}} (G, A, \af)$ is associative.
\pause
Further check
for $a, b \in C_{\mathrm{c}} (G, A, \af)$
that $\| a b \|_1 \leq \| a \|_1 \| b \|_1,$
\pause
that $(a b)^* = b^* a^*,$
\pause
and that $\| a^* \|_1 = \| a \|_1.$
\end{exr}
% \pause
\end{frame}
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\begin{frame}
\frametitle{When $G$ is discrete}
If $G$ is discrete, we choose Haar measure to be counting measure.
\pause
In this case,
$C_{\mathrm{c}} (G, A, \af)$ is, as a vector space,
the group ring $A [G],$
consisting of all finite formal linear combinations of elements in $G$
with coefficients in $A.$
\pause
The multiplication and adjoint are given by
\[
(a \cdot g) (b \cdot h) = (a [ g b g^{-1}]) \cdot (g h)
= (a \af_g (b)) \cdot (g h)
\andeqn
(a \cdot g)^* = \af_g^{-1} (a^*) \cdot g^{-1}
\]
for $a, b \in A$ and $g, h \in G,$
\pause
extended linearly.
\pause
This definition makes sense in the purely algebraic situation,
where it is called the {\emph{skew group ring}}.
\pause
\vspace{2ex}
We also often write $l^1 (G, A, \af)$ instead of
$L^1 (G, A, \af).$
% \pause
\end{frame}
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\begin{frame}
\frametitle{When $G$ is discrete (continued)}
Let $\af \colon G \to \Aut (A)$
be an action of a discrete group $G$ on a unital \ca~$A.$
\pause
In these notes, we will adopt the following fairly
commonly used notation.
For $g \in G,$
we let $u_g$ be the element of $C_{\mathrm{c}} (G, A, \af)$
which takes the value $1_A$ at $g$
and $0$ at the other elements of $G.$
\pause
We use the same notation for its image in $l^1 (G, A, \af)$
(above)
\pause
and in $C^* (G, A, \af)$
and $C^*_{\mathrm{r}} (G, A, \af)$ (defined below).
It is unitary,
and we call it the canonical unitary associated with~$g.$
\pause
\vspace{2ex}
In particular,
$l^1 (G, A, \af)$ is the set of all sums
$\sum_{g \in G} a_g u_g$ with $a_g \in A$
and $\sum_{g \in G} \| a_g \| < \infty.$
\pause
These sums converge in $l^1 (G, A, \af),$
and hence also in
$C^* (G, A, \af)$ and $C^*_{\mathrm{r}} (G, A, \af).$
\pause
A general element of $C^*_{\mathrm{r}} (G, A, \af)$
has such an expansion,
\pause
but unfortunately the series
one writes down generally does not converge.
See the discussion later.
% \pause
\end{frame}
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\begin{frame}
\frametitle{The integrated form of a covariant representation}
\begin{dfn}\label{D:IntForm}
Let $\af \colon G \to \Aut (A)$
be an action of a locally compact group $G$ on a \ca~$A,$
and let $(v, \pi)$ be a covariant representation
of $(G, A, \af)$ on a Hilbert space $H.$
\pause
Then the {\emph{integrated form}} of $(v, \pi)$ is the representation
$\sm \colon C_{\mathrm{c}} (G, A, \af) \to L (H)$
given by
\pause
\[
\sm (a) \xi = \int_G \pi (a (g)) v (g) \xi \, d g.
\]
\pause
(This representation is sometimes called
$v \times \pi$ or $\pi \times v.$)
\end{dfn}
\pause
\vspace{2ex}
One needs to be more careful with the integral here,
because $v$ is generally only strong operator \ct,
not norm \ct.
\pause
Nevertheless,
one gets $\| \sm (a) \| \leq \| a \|_1,$
so $\sm$ extends to a representation of $L^1 (G, A, \af).$
We use the same notation $\sm$ for this extension.
% \pause
\end{frame}
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\begin{frame}
\frametitle{The integrated form of a covariant representation
(continued)}
One needs to check that $\sm$ is a representation.
\pause
When $G$ is discrete and $A$ is unital,
the formula for $\sm$ comes down to $\sm (a u_g) = \pi (a) v (g)$
for $a \in A$ and $g \in G.$
\pause
Then
\begin{align*}
\sm (a u_g) \sm (b u_h)
& = \pi (a) v (g) \pi (b) v (g)^* v (g) v (h)
= \pi (a) \pi (\af_g (b)) v (g) v (h) \\
& = \pi (a \af_g (b)) v (g h)
= \sm \big( [a \af_g (b)] u_{g h} \big)
= \sm \big( (a u_g) (b u_h) \big).
\end{align*}
\pause
\vspace{2ex}
\begin{exr}\label{P:IntFIsRep}
Starting from this computation,
fill in the details of
the proof that the integrated form representation $\sm$
really is a nondegenerate representation
of $C_{\mathrm{c}} (G, A, \af).$
\end{exr}
% \pause
\end{frame}
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\begin{frame}
\frametitle{The integrated form of a covariant representation
(continued)}
\begin{thm}[Proposition~7.6.4 of Pedersen's book]\label{T:IntFBij}
Let $\af \colon G \to \Aut (A)$
be an action of a locally compact group $G$ on a \ca~$A.$
\pause
Then the integrated form construction defines a bijection
from the set of covariant representations
of $(G, A, \af)$ on a Hilbert space $H$
to the set of nondegenerate \ct\ representations
of $L^1 (G, A, \af)$ on the same Hilbert space.
\end{thm}
\pause
\vspace{2ex}
In particular,
since integrated form representations of $L^1 (G, A, \af)$
are necessarily contractive,
\pause
{\emph{all}} \ct\ representations of $L^1 (G, A, \af)$
are necessarily contractive.
% \pause
\end{frame}
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\begin{frame}
\frametitle{The integrated form
when $G$ is discrete}
If $G$ is discrete and $A$ is unital, then
there are homomorphic images of both $G$ and $A$
inside $C_{\mathrm{c}} (G, A, \af),$
\pause
given by $g \mapsto u_g$ and $a \mapsto a u_1,$
\pause
so it is clear how to get a covariant representation
of $(G, A, \af)$ from a nondegenerate representation
of $C_{\mathrm{c}} (G, A, \af).$
\pause
In general, one must use the multiplier algebra of
$L^1 (G, A, \af),$
which contains copies of $M (A)$ and $M (L^1 (G)).$
\pause
The point is that $M (L^1 (G))$
is the measure algebra of $G,$
and therefore contains the group elements as point masses.
\pause
\vspace{2ex}
\begin{exr}\label{P:IntFBij1}
Prove the theorem on the previous slide
when $G$ is discrete and $A$ is unital.
\end{exr}
\pause
\vspace{2ex}
For a small taste of the general case,
use approximate identities in $A$ to
generalize to the
case in which $A$ is not necessarily unital.
% \pause
\end{frame}
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\begin{frame}
\frametitle{The universal representation and the crossed product}
\begin{dfn}\label{D:CP}
Let $\af \colon G \to \Aut (A)$
be an action of a locally compact group $G$ on a \ca~$A.$
We define the {\emph{universal representation}}
$\sm$ of $L^1 (G, A, \af)$
\pause
to be the direct sum of all nondegenerate representations
of $L^1 (G, A, \af)$ on Hilbert spaces.
\pause
Then we define the {\emph{\cp}} $C^* (G, A, \af)$
to be the norm closure of $\sm (L^1 (G, A, \af)).$
\end{dfn}
\pause
% \vspace{0.5ex}
One could of course equally well use
the norm closure of $\sm (C_{\mathrm{c}} (G, A, \af)).$
\pause
\vspace{0.5ex}
There is a minor set theoretic detail:
the collection of all nondegenerate representations
of $L^1 (G, A, \af)$ is not a set.
\pause
There are several standard ways to deal with this problem,
but in these notes we will ignore the issue.
\pause
% \vspace{0.5ex}
\begin{exr}\label{P:FixSet}
Give a set theoretically correct definition of the \cp.
\end{exr}
\pause
% \vspace{0.5ex}
The important point is to preserve the universal
property below.
% \pause
\end{frame}
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\begin{frame}
\frametitle{The universal representation and the crossed product
(continued)}
It follows that every covariant representation of $(G, A, \af)$
gives a representation of $C^* (G, A, \af).$
\pause
(Take the integrated form,
and restrict elements of $C^* (G, A, \af)$ to the
appropriate summand in the direct sum in the definition above.)
\pause
The \cp\ is,
essentially by construction,
the universal \ca\ for covariant representations of $(G, A, \af),$
\pause
in the same sense that if $G$ is a locally compact group,
then $C^* (G)$ is
the universal \ca\ for unitary representations of~$G.$
\pause
\vspace{2ex}
There are many notations in use for crossed products,
including:
\begin{itemize}
\item
$C^* (G, A, \af)$ and $C^*_{\mathrm{r}} (G, A, \af).$
\item
$C^* (A, G, \af)$ and $C^*_{\mathrm{r}} (A, G, \af).$
\item
$A \rtimes_{\af} G$ and $A \rtimes_{\af, {\mathrm{r}}} G$
(used in Williams' book).
\item
$A \times_{\af} G$ and $A \times_{\af, {\mathrm{r}}} G$
(used in Davidson's book).
\item
$G \times_{\af} A$ and $G \times_{\af, {\mathrm{r}}} A$
(used in Pedersen's book).
\end{itemize}
% \pause
\end{frame}
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\begin{frame}
\frametitle{The universal representation and the crossed product
when $G$ is discrete}
\begin{thm}\label{L:UnivCP}
Let $\af \colon G \to \Aut (A)$
be an action of a discrete group $G$ on a unital \ca~$A.$
\pause
Then $C^* (G, A, \af)$ is the universal \ca\ generated by
a unital copy of $A$
(that is, the identity of $A$ is supposed to be the identity of the
generated \ca)
and
\pause
unitaries $u_g,$ for $g \in G,$
\pause
subject to the relations $u_g u_h = u_{g h}$ for $g, h \in G$
and $u_g a u_g^* = \af_g (a)$ for $a \in A$ and $g \in G.$
\end{thm}
\pause
\vspace{2ex}
\begin{cor}\label{C:UnivCPZ}
Let $A$ be a \uca, and let $\af \in \Aut (A).$
Then the \cp\ $C^* (\Z, A, \af)$ is the universal \ca\ generated by
a copy of $A$ and a unitary $u,$
subject to the relations
$u a u^* = \af (a)$ for $a \in A.$
\end{cor}
% \pause
\end{frame}
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\begin{frame}
\frametitle{The universal representation and the crossed product
when $G$ is discrete (continued)}
\begin{exr}\label{P:PfUniv}
Based on the discussion above,
write down a careful proof of the theorem.
\end{exr}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Regular covariant representations}
So far, it is not clear that there are any covariant
representations.
\pause
% \vspace{2ex}
\begin{dfn}[7.7.1 of Pedersen's book]\label{D:RegRpn}
Let $\af \colon G \to \Aut (A)$
be an action of a locally compact group $G$ on a \ca~$A.$
Let $\pi_0 \colon A \to L (H_0)$ be a representation.
\pause
We define the {\emph{regular covariant representation}}
$(v, \pi)$ of $(G, A, \af)$ on the Hilbert space $H = L^2 (G, H_0)$
of $L^2$ functions from $G$ to $H_0$
as follows.
\pause
For $g, h \in G,$ set
\[
(v (g) \xi) (h) = \xi (g^{-1} h).
\]
\pause
For $a \in A$ and $g \in G,$ set
\[
(\pi (a) \xi) (h) = \pi_0 (\af_{h^{-1}} (a)) ( \xi (h)).
\]
\pause
The integrated form of $\sm$
will be called a regular representation
\pause
of
any of $C_{\mathrm{c}} (G, A, \af),$
$L^1 (G, A, \af),$
$C^* (G, A, \af),$
and (when defined)
$C^*_{\mathrm{r}} (G, A, \af).$
\end{dfn}
% \pause
\end{frame}
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\begin{frame}
\frametitle{The Hilbert space of the regular covariant representation}
The easy way to construct $L^2 (G, H_0)$ is to take it to be the
completion of $C_{\mathrm{c}} (G, H_0)$ in the norm
coming from the scalar product
\[
\langle \xi, \et \rangle
= \int_G \langle \xi (g), \et (g) \rangle \, d g.
\]
% \pause
\end{frame}
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\begin{frame}
\frametitle{Reduced crossed products}
\begin{exr}\label{P:RegIsRep}
Suppose that $G$ is discrete.
Prove that
a regular representation really is a covariant representation.
\end{exr}
\pause
\vspace{2ex}
If $A = \C,$ $H_0 = \C,$
and $\pi_0$ is the obvious representation of $A$ on $H_0,$
then the regular representation is
the usual left regular representation of~$G.$
\pause
\vspace{2ex}
\begin{dfn}\label{D:RedCP}
Let $\af \colon G \to \Aut (A)$
be an action of a locally compact group $G$ on a \ca~$A.$
\pause
Let $\ld \colon L^1 (G, A, \af) \to L (H)$
be the direct sum of all regular
representations of $L^1 (G, A, \af).$
\pause
We define the {\emph{reduced crossed product}}
$C^*_{\mathrm{r}} (G, A, \af)$
to be the norm closure of $\ld (L^1 (G, A, \af)).$
\end{dfn}
\pause
\vspace{2ex}
As with \cp s, in these notes we ignore the set theoretic difficulty.
% \pause
\end{frame}
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\begin{frame}
\frametitle{The relationship between reduced
and full crossed products}
Implicit in the definition of $C^*_{\mathrm{r}} (G, A, \af)$
is a representation of $L^1 (G, A, \af),$
\pause
hence of $C^* (G, A, \af).$
\pause
Thus, there is a \hm\ %
$C^* (G, A, \af) \to C^*_{\mathrm{r}} (G, A, \af).$
By construction, it has dense range,
and is therefore surjective.
\pause
Moreover, by construction,
any regular representation of $L^1 (G, A, \af)$
extends to a representation of $C_{\mathrm{r}}^* (G, A, \af).$
\pause
\vspace{2ex}
\begin{thm}[Theorem~7.7.7 of Pedersen's book]\label{T:CPAmen}
Let $\af \colon G \to \Aut (A)$
be an action of a locally compact group $G$ on a \ca~$A.$
\pause
If $G$ is amenable,
then $C^* (G, A, \af) \to C^*_{\mathrm{r}} (G, A, \af)$
is an isomorphism.
\end{thm}
\pause
\vspace{2ex}
The converse is true for $A = \C$:
if $C^* (G) \to C^*_{\mathrm{r}} (G)$
is an isomorphism,
then $G$ is amenable.
\pause
But it is not true in general.
For example, if $G$ acts on itself by translation,
\pause
then $C^* (G, \, C_0 (G)) \to C^*_{\mathrm{r}} (G, \, C_0 (G))$
is an isomorphism for every $G.$
\pause
(We will do this below for a discrete group.)
% \pause
\end{frame}
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\begin{frame}
\frametitle{The crossed product is not too small}
\begin{thm}\label{T:InjOnL1}
Let $\af \colon G \to \Aut (A)$
be an action of a locally compact group $G$ on a \ca~$A.$
\pause
Then $C_{\mathrm{c}} (G, A, \af) \to C^*_{\mathrm{r}} (G, A, \af)$
is injective.
\end{thm}
\pause
\vspace{2ex}
We will prove this below in the case of a discrete group.
The proof of the general case can be found in Lemma~2.26
of the book of Williams.
\pause
It is, I believe, true that
$L^1 (G, A, \af) \to C^*_{\mathrm{r}} (G, A, \af)$ is injective,
and this can probably be proved by working a little harder
in the proof of Lemma~2.26 of the book of Williams,
but I have not carried out the details and I do not know a reference.
% \pause
\end{frame}
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\begin{frame}
\frametitle{When $G$ is discrete: integrated form of a regular
representation}
We specialize to the case of discrete~$G.$
\pause
The main tool is the structure of regular representations.
When $G$ is discrete, we can write $L^2 (G, H_0)$
as a Hilbert space direct sum $\bigoplus_{g \in G} H_0,$
\pause
and elements of it can be thought of as families
$( \xi_g)_{g \in G}.$
\pause
The following formula for the integrated form of a regular
representation is just a calculation.
\pause
\begin{lem}\label{L:StructRR}
Let $\af \colon G \to \Aut (A)$
be an action of a discrete group $G$ on a \ca~$A.$
\pause
Let $\pi_0 \colon A \to L (H_0)$ be a representation,
and let
$\sm \colon C^*_{\mathrm{r}} (G, A, \af) \to L (H) = L (L^2 (G, H_0))$
be the associated regular representation.
\pause
Let $a = \sum_{g \in G} a_g u_g \in C^*_{\mathrm{r}} (G, A, \af),$
with $a_g = 0$ for all but finitely many~$g.$
\pause
For $\xi \in H$ and $h \in G,$ we then have
\[
( \sm (a) \xi) (h)
= \sum_{g \in G} \pi_0 (\af_h^{-1} (a_g)) \big( \xi (g^{-1} h ) \big).
\]
\end{lem}
% \pause
\end{frame}
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\begin{frame}
\frametitle{When $G$ is discrete: integrated form of a regular
representation (continued)}
In particular, picking off coordinates in $L^2 (G, H_0)$ gives:
\pause
\vspace{2ex}
\begin{cor}\label{C:StructRR}
Let the hypotheses be as in the Lemma,
and let $a = \sum_{g \in G} a_g u_g \in C^*_{\mathrm{r}} (G, A, \af).$
\pause
For $g \in G,$
let $s_g \in L (H_0, H)$ be the isometry which sends
$\et \in H_0$ to the function $\xi \in L^2 (G, H_0)$
given by
\[
\xi (h) = \left\{ \begin{array}{ll}
\et & \hspace{3em} h = g \\
0 & \hspace{3em} h \neq g.
\end{array} \right.
\]
\pause
Then
\[
s_h^* \sm (a) s_k = \pi_0 \big( \af_h^{-1} ( a_{h k^{-1} }) \big)
\]
for all $h, k \in G.$
\end{cor}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Comparing norms}
Let $\af \colon G \to \Aut (A)$
be an action of a discrete group $G$ on a \ca~$A.$
Define norms on $C_{\mathrm{c}} (G, A, \af)$ as follows:
\pause
\begin{itemize}
\item
$\| \cdot \|_{\I}$
is the supremum norm.
\pause
\item
$\| \cdot \|_{1}$
is the $l^1$ norm.
\pause
\item
$\| \cdot \|$
is the restriction of the \ca\ norm on $C^* (G, A, \af).$
\pause
\item
$\| \cdot \|_{\mathrm{r}}$
is the restriction of the \ca\ norm on $C^*_{\mathrm{r}} (G, A, \af).$
\end{itemize}
\pause
\vspace{2ex}
\begin{lem}\label{L:FGpCP}
For every $a \in C_{\mathrm{c}} (G, A, \af),$
we have
$\| a \|_{\I} \leq \| a \|_{\mathrm{r}} \leq \| a \| \leq \| a \|_1.$
\end{lem}
% \pause
\end{frame}
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\begin{frame}
\frametitle{Comparing norms: the proof}
The middle of this inequality follows from the definitions.
\pause
The last part follows from the
observation above
that all \ct\ representations of $L^1 (G, A, \af)$
are norm reducing.
\pause
Here is a direct proof:
for $a = \sum_{g \in G} a_g u_g \in C_{\mathrm{c}} (G, A, \af),$
with all but finitely many of the $a_g$ equal to zero, we have
\pause
\[
\left\| \ssum{g \in G} a_g u_g \right\|
\leq \ssum{g \in G} \| a_g \| \cdot \| u_g \|
= \ssum{g \in G} \| a_g \|
= \left\| \ssum{g \in G} a_g u_g \right\|_1.
\]
\pause
% \vspace{2ex}
We prove the first part of this inequality.
Let $a = \sum_{g \in G} a_g u_g,$
with all but finitely many of the $a_g$ equal to zero,
and let $g \in G.$
\pause
Let $\pi_0 \colon A \to L (H_0)$
be an injective nondegenerate representation.
\pause
With the notation of the previous corollary,
we have
\pause
\[
\| a_g \|
= \| \pi_0 (a_g) \|
= \| s_g^* \sm (a) s_1 \|
\leq \| \sm (a) \|
\leq \| a \|_{\mathrm{r}}.
\]
This completes the proof.
% \pause
\end{frame}
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\begin{frame}
\frametitle{$A$ is a subalgebra of the reduced crossed product}
The lemma implies that
the map $a \mapsto a u_1,$
from $A$ to $C^*_{\mathrm{r}} (G, A, \af),$
is injective.
\pause
We routinely identify $A$
with its image in $C^*_{\mathrm{r}} (G, A, \af)$ under this map,
thus treating it as a subalgebra of $C^*_{\mathrm{r}} (G, A, \af).$
\pause
\vspace{2ex}
Of course, we can do the same with the full \cp\ $C^* (G, A, \af).$
% \pause
\end{frame}
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\begin{frame}
\frametitle{For finite groups, no completion is needed}
\begin{cor}\label{C:FGpCP}
Let $\af \colon G \to \Aut (A)$
be an action of a finite group $G$ on a \ca~$A.$
\pause
Then the maps
$C_{\mathrm{c}} (G, A, \af)
\to C^* (G, A, \af) \to C^*_{\mathrm{r}} (G, A, \af)$
are bijective.
\end{cor}
\pause
\vspace{2ex}
\begin{proof}
When $G$ is finite,
$\| \cdot \|_1$ (the $l^1$ norm)
is equivalent to $\| \cdot \|_{\I}$ (the supremum norm),
and is complete in both.
\pause
The lemma now implies that both C*~norms are
equivalent to these norms,
so $C_{\mathrm{c}} (G, A, \af)$ is complete in both C*~norms.
\end{proof}
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\end{frame}
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\begin{frame}
\frametitle{Coefficients in reduced crossed products}
When $G$ is discrete but not finite,
things are much more complicated.
We can get started:
\pause
\begin{prp}\label{P:CondExpt}
Let $\af \colon G \to \Aut (A)$
be an action of a discrete group $G$ on a \ca~$A.$
\pause
Then for each $g \in G,$
there is a linear map $E_g \colon C^*_{\mathrm{r}} (G, A, \af) \to A$
with $\| E_g \| \leq 1$
\pause
such that if
$a = \sum_{g \in G} a_g u_g \in C_{\mathrm{c}} (G, A, \af),$
then $E_g (a) = a_g.$
\pause
\vspace{2ex}
Moreover, with $s_g$ as above,
we have
$s_h^* \sm (a) s_k = \pi_0 \big( \af_h^{-1} (E_{h k^{-1}} (a)) \big)$
for all $h, k \in G.$
\end{prp}
\pause
\begin{proof}
The first part is immediate from the inequality
$\| a \|_{\I} \leq \| a \|_{\mathrm{r}}$ above.
\pause
\vspace{2ex}
The last statement follows by continuity from
``picking off coordinates'' in the regular representation.
\end{proof}
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\end{frame}
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\begin{frame}
\frametitle{Coefficients in reduced crossed products: Discussion}
Thus, for any $a \in C^*_{\mathrm{r}} (G, A, \af),$
\pause
and therefore also for $a \in C^* (G, A, \af),$
\pause
it makes sense to talk about its coefficients $a_g.$
\pause
The first point is that if
$C^* (G, A, \af) \neq C^*_{\mathrm{r}} (G, A, \af)$
\pause
(which can happen if $G$ is not amenable,
but not if $G$ is amenable),
\pause
the coefficients $(a_g)_{g \in G}$ do not even uniquely determine
the element~$a.$
\pause
This is why we are only considering reduced \cp s here.
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\begin{frame}
\frametitle{Coefficients in reduced crossed products: Properties}
Here are the good things about coefficients.
\pause
\vspace{2ex}
\begin{prp}\label{P:Faithful}
Let $\af \colon G \to \Aut (A)$
be an action of a discrete group $G$ on a \ca~$A.$
\pause
Let the maps $E_g \colon C^*_{\mathrm{r}} (G, A, \af) \to A$
be as in the previous proposition.
Then:
\pause
\begin{enumerate}
\item\label{P:Faithful:G}
If $a \in C^*_{\mathrm{r}} (G, A, \af)$
and $E_g (a) = 0$ for all $g \in G,$
then $a = 0.$
\pause
\item\label{P:Faithful:Inj}
If $\pi_0 \colon A \to L (H_0)$ is a nondegenerate representation
such that $\bigoplus_{g \in G} \pi_0 \circ \af_g$ is injective,
then the regular representation $\sm$ of $C^*_{\mathrm{r}} (G, A, \af)$
associated to $\pi_0$ is injective.
\pause
\item\label{P:Faithful:1}
If $a \in C^*_{\mathrm{r}} (G, A, \af)$
and $E_1 (a^* a) = 0,$ then $a = 0.$
\end{enumerate}
\end{prp}
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\begin{frame}
\frametitle{Proof of the properties of coefficients}
(\ref{P:Faithful:G}):
Let $\pi_0 \colon A \to L (H_0)$ be a representation,
and let the notation be as above.
\pause
If $a \in C^*_{\mathrm{r}} (G, A, \af)$ satisfies $E_g (a) = 0$
for all $g \in G,$
then $s_h^* \sm (a) s_k = 0$ for all $h, k \in G,$
\pause
whence $\sm (a) = 0.$
\pause
Since $\pi_0$ is arbitrary, it follows that $a = 0.$
% This proves~(\ref{P:Faithful:G}).
\pause
\vspace{0.5ex}
(\ref{P:Faithful:Inj}):
Suppose $a \in C^*_{\mathrm{r}} (G, A, \af)$ and $\sm (a) = 0.$
\pause
Fix $l \in G.$
\pause
Taking $h = g^{-1}$ and $k = l^{-1} g^{-1}$ in
the previous proposition,
\pause
we get $(\pi_0 \circ \af_g) (E_l (a)) = 0$
for all $g \in G.$
\pause
So $E_l (a) = 0.$
\pause
This is true for all $l \in G,$
so $a = 0.$
\pause
\vspace{0.5ex}
(\ref{P:Faithful:1}):
As before,
let $a = \sum_{g \in G} a_g u_g \in C_{\mathrm{c}} (G, A, \af).$
\pause
Then
$a^* a = \sum_{g, h \in G} u_g^* a_g^* a_h u_h,$
so
\pause
\[
E_1 (a^* a) = \sum_{g \in G} u_g^* a_g^* a_g u_g
= \sum_{g \in G} \af_g^{-1} \big( E_g (a)^* E_g (a) \big).
\]
\pause
In particular, for each fixed $g,$ we have
$E_1 (a^* a) \geq \af_g^{-1} \big( E_g (a)^* E_g (a) \big).$
\pause
By continuity, this inequality holds
for all $a \in C^*_{\mathrm{r}} (G, A, \af).$
\pause
Thus, if $E_1 (a^* a) = 0,$
then $E_g (a)^* E_g (a) = 0$ for all $g,$
\pause
so $a = 0$ by Part~(\ref{P:Faithful:G}).
\pause
This completes the proof.
% \pause
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\begin{frame}
\frametitle{Injective representations of $A$ always give injective
regular representations of the reduced crossed product}
It is true for general \lcg s,
not just discrete groups,
that the regular representation of $C^*_{\mathrm{r}} (G, A, \af)$
associated to an injective representation of $A$ is injective.
See Theorem~7.7.5 of Pedersen's book.
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\begin{frame}
\frametitle{The conditional expectation}
The map $E_1$
used in Part~(\ref{P:Faithful:1}) of the previous proposition
is an example of what is called a
{\emph{conditional expectation}}
(from $C^*_{\mathrm{r}} (G, A, \af)$ to $A$)
\pause
that is, it has the properties given in the following exercise.
\pause
Part~(\ref{P:Faithful:1}) of the previous proposition
asserts that this conditional expectation is faithful.
\pause
\begin{exr}\label{Pb:CondExpt}
Let $\af \colon G \to \Aut (A)$
be an action of a discrete group $G$ on a \ca~$A.$
\pause
Let $E = E_1 \colon C^*_{\mathrm{r}} (G, A, \af) \to A$
be as above.
Prove that $E$ has the following properties:
\pause
\begin{enumerate}
\item\label{P:CondExpt:Idem}
$E (E (b)) = E (b)$ for all $b \in C^*_{\mathrm{r}} (G, A, \af).$
\pause
\item\label{P:CondExpt:Pos}
If $b \geq 0$ then $E (b) \geq 0.$
\pause
\item\label{P:CondExpt:Norm}
$\| E (b) \| \leq \| b \|$ for all $b \in C^*_{\mathrm{r}} (G, A, \af).$
\pause
\item\label{P:CondExpt:Mod}
If $a \in A$ and $b \in C^*_{\mathrm{r}} (G, A, \af),$
then $E (a b) = a E (b)$ and $E (b a) = E (b) a.$
\end{enumerate}
\end{exr}
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\begin{frame}
\frametitle{The limits of coefficients}
Unfortunately,
in general $\sum_{g \in G} a_g u_g$
does not converge in $C^*_{\mathrm{r}} (G, A, \af),$
and
it is very difficult to tell exactly which families of coefficients
correspond to elements of $C^*_{\mathrm{r}} (G, A, \af).$
\pause
In fact, the situation is intractable even for the case
of the trivial action of $\Z$ on $\C.$
In this case,
$l^1 (\Z, A, \af) = l^1 (\Z).$
The crossed product is the group \ca\ $C^* (\Z),$
which can be identified with $C (S^1).$
% (We write $C^* (\Z)$ because, $\Z$ being amenable,
% we have $C^*_{\mathrm{r}} (\Z) = C^* (\Z).$)
The map $l^1 (\Z) \to C (S^1)$ is given by Fourier series:
the sequence $a = (a_n)_{n \in \N}$ goes to the function
$\zt \mapsto \sum_{n \in \Z} a_n \zt^n.$
\pause
(This looks more familiar
when expressed in terms of $2 \pi$-periodic functions on $\R$:
it is $t \mapsto \sum_{n \in \Z} a_n e^{i n t}.$)
\pause
Failure of convergence of
$\sum_{n \in \Z} a_n u_n$ corresponds to the fact that
the Fourier series of a \cfn\ need not converge uniformly.
\pause
Identifying the coefficient sequences which correspond to elements
of the crossed product corresponds to giving a criterion for
exactly when a sequence $(a_n)_{n \in \N}$ of complex numbers
is the sequence of Fourier coefficients of some \cfn\ on $S^1,$
\pause
a problem for which I know of no satisfactory solution.
% \pause
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\begin{frame}
\frametitle{The limits of coefficients (continued)}
Let's pursue this a little farther.
The regular representation of $\Z$ on $l^2 (\Z)$
gives an injective map $\ld \colon C^* (\Z) \to L (l^2 (\Z)).$
\pause
Let $\dt_n \in l^2 (\Z)$ be the function
% $\dt_n (n) = 1$
% and $\dt_n (k) = 0$ for $k \neq n.$
\[
\dt_n (k) = \left\{ \begin{array}{ll}
1 & \hspace{3em} k = n \\
0 & \hspace{3em} k \neq n.
\end{array} \right.
\]
\pause
Then the Fourier coefficient $a_n$ is recovered as
$a_n = \langle \ld (a) \dt_0, \, \dt_n \rangle.$
That is, $\ld (a) \dt_0 \in l^2 (\Z)$ is given by
$\ld (a) \dt_0 = \sum_{n \in \Z} a_n \dt_n.$
\pause
Thus, the sequence of Fourier coefficients of a \cfn\ is
always in $l^2 (\Z).$
(Of course, we already know this,
but the calculation here can be applied to more general \cp s.)
\pause
Unfortunately, this fact is essentially useless for the
study of the group \ca.
Not only is the Fourier series of a \cfn\ always in $l^2 (\Z),$
but the Fourier series of a function in $L^{\infty} (S^1),$
which is the group von Neumann algebra of $\Z,$
is also always in $l^2 (\Z).$
\pause
One will get essentially no useful information from a
criterion which can't even exclude any elements of $L^{\infty} (S^1).$
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\begin{frame}
\frametitle{The limits of coefficients (continued)}
Even if one understands completely what all the elements
of $C^*_{\mathrm{r}} (G)$ are,
and even if the action is trivial,
understanding the elements of the reduced crossed product requires that
one understand all the elements of the completed
tensor product $C^*_{\mathrm{r}} (G) \otimes_{\mathrm{min}} A.$
\pause
As far as I know, this problem is also in general intractable.
\pause
\vspace{2ex}
There is just one bright spot:
although we will not prove it here,
there is an analog for general \cp s by $\Z$ of the fact
that the Cesaro means of the Fourier series of a \cfn\ always
converge uniformly to the function.
See Theorem~8.2.2 of Davidson's book.
\pause
\vspace{2ex}
The discussion above is meant to point out
the difficulties in dealing with crossed products by infinite groups.
\pause
Despite all this, for some problems, finite groups are harder.
\pause
Computing the K-theory of a crossed product by $\Z / 2 \Z$
is harder than computing the K-theory of a crossed product by any
of $\Z,$ $\R,$ or even a (nonabelian) free group!
% \pause
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\end{document}