% $Header: /cvsroot/latex-beamer/latex-beamer/solutions/generic-talks/generic-ornate-15min-45min.en.tex,v 1.5 2007/01/28 20:48:23 tantau Exp $

\documentclass[xcolor=pdftex,dvipsnames,table]{beamer}



\def\tr {{\rm tr}}
\def\FA{{\rm A}}
\def\FB{{\rm B}}
\def\Ex {{\mathbb{E}}}
\def\bbH {{\mathbb{H}}}
\def\span {{\rm span}}
\def\cH {{\cal H}}
\def\cF {{\cal F}}
\def\cU {{\cal U}}
\def\SO {{\rm SO}}
\def\sch {{\rm Sch}}
\def\SU {{\rm SU}}
\def\dom {{\rm dom}}
\def\Real {{\mathbb R}}
\newcommand{\ignore}[1]{}

% This file is a solution template for:

% - Giving a talk on some subject.
% - The talk is between 15min and% - Style is ornate.



% Copyright 2004 by Till Tantau <tantau@users.sourceforge.net>.
%
% In principle, this file can be redistributed and/or modified under
% the terms of the GNU Public License, version 2.
%
% However, this file is supposed to be a template to be modified
% for your own needs. For this reason, if you use this file as a
% template and not specifically distribute it as part of a another
% package/program, I grant the extra permission to freely copy and
% modify this file as you see fit and even to delete this copyright
% notice.


\mode<presentation>
{
  \usetheme{Darmstadt}
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  \setbeamercovered{transparent}
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}

\usepackage{umoline}

\usepackage{graphics}

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\usepackage[english]{babel}
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%\usepackage{pgf,pgfarrows,pgfnodes,pgfautomata,pgfheaps}

\usepackage[latin1]{inputenc}
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\usepackage{times}
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\title[Here You Can Type the Title of Your Presentation.] % (optional, use only with long paper titles)
{Here You Can Type the Title of Your Presentation}

%\subtitle{} % (optional)

\author[Name of Author] % (optional, use only with lots of authors)
{Author~Name\\
\scriptsize{joint work with S. Author}}
% - Use the \inst{?} command only if the authors have different
%   affiliation.
\institute[Universities of Tehran] % (optional, but mostly needed)
{
  Department of Mathematics and Statistics\\
  University of Tehran}

\date{\scriptsize{IMC44, University of Ferdowsi, August 27-30, 2013, Mashhad, Iran}}

\begin{document}

%\setbeamercovered{invisible}



\begin{frame}
  \titlepage
\end{frame}


\section[Recall]{Recall}
\subsection{Notations and background}


\begin{frame}{}
\begin{definition}
Here define your definition
\end{definition}

$\lambda:$ left regular representation.\\
$\Sigma:$ equiv. classes of unitary rep. \\

\begin{definition}
\begin{itemize}
\item  Topological algebra.
%
\item Banach algebra.
%
\item  Normed Algebra
\end{itemize}
\end{definition}
%
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{}
\begin{block}{Question}
For which groups $G$, $L^{1}(G)$ is  amenable?
\end{block}
%
%\onslide<1->{$G_e$ connected component of identity.}
\begin{block}<1->{Conjecture}
$G$ is amenable.
\end{block}

\begin{block}<1->{Known results}
\begin{itemize}
\item {\textcolor{blue}{[Forrest-Runde]}} $G$ is Abelian.
%
\item {\textcolor{blue}{[Forrest-Samei-Spronk]}}  $G$ \alert{compact} gp  and  $\FA(G)$ w. amen. $\Rightarrow$ $G_e$ abelian.
\end{itemize}
\end{block}

\begin{theorem}<2->[Choi-G.]
$L^{1}(ax+b)$ is not w. amen.
\end{theorem}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[$ax+b$]{$ax+b$}
\subsection{What is $ax+b$ group?}

\begin{frame}{An important non-compact group}

\begin{definition}
\begin{itemize}
\item $G=\left\{
\left(
\begin{array}{ccc}
  a & b  \\
 0  &  1  \\
\end{array}
\right):\  a\in\Real_+^*, b\in \Real\right\}$.
%
\item $\Real^*_+:=(\Real_+,*)$ with  Haar measure $t^{-1}dt$.\\
$\Real_+^*$ acts on $\Real$ by mult.

\item $G\simeq\Real\rtimes \Real_+^*=\{(b,a):b\in \Real, a\in \Real_+^*\}$.
\end{itemize}
\end{definition}

\begin{block}{Properties of $G=ax+b$}
\begin{itemize}
\item[] Left Haar measure: $d\mu(a,b)=a^{-2} da\ db$.
%
\item[] Modular function:  $\Delta(a,b)=\frac{1}{a}$.
%
\end{itemize}
\end{block}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{Representation theory of $ax+b$}
\begin{block}{$\infty$-dim irred rep}
Mackey machine for induced rep:
%
$$\pi_{\pm}:G\rightarrow {\cal U}(L^2(\Real^*_+,dt/t)),$$
\vspace{-0.5cm}
\begin{itemize}
 \item  $\pi_{+}(b,a)\xi(t) = e^{- 2\pi i bt}\xi(at),$
%
 \item $\pi_{-}(b,a)\xi(t) = e^{+2\pi i bt}\xi(at).$
\end{itemize}
\end{block}

\begin{block}<2->{coefficient functions}
\begin{itemize}
\item $(\xi *_{\pi_+} \eta )(b,a) = \int_0^\infty e^{-2\pi i bt} \xi(at)\overline{\eta(t)}\, t^{-1}dt, $
 %
\item $(\xi *_{\pi_-} \eta )(b,a) = \int_0^\infty e^{2\pi i bt} \xi(at)\overline{\eta(t)}\, t^{-1}dt.$
\end{itemize}
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}{}
$\cH$= $L^2(\Real_+^*,dt/t)$.\\
$\cF$: Fourier transform on ${\mathbb R}$.\\
\begin{itemize}
\item $K:\cH\rightarrow \cH, K\xi(t)=t\xi(t)$.
\item $\imath:L^2(\Real_+^*)\rightarrow L^2(\Real)$.
\item $_a\xi(t)=\xi(at)$.
\end{itemize}



\begin{block}<2->{coefficient functions}
\begin{itemize}
\item $(\xi *_{\pi_+} \eta )(b,a) =\cF(\imath K^{-1}(_a\xi_1\overline{\eta_1}))(\chi_b), $
 %
\item $(\xi *_{\pi_-} \eta )(b,a) = \cF(\imath K^{-1}(_a\xi_1\overline{\eta_1}))(\chi_{-b}).$
%
\end{itemize}
\end{block}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{$ax+b$ is ``nice''.}
\begin{frame}{}
\begin{definition}
Irred rep $\pi$ is \alert{square integrable} if $\exists$ $0\neq \xi*_\pi\eta\in L^2(G)$.
\end{definition}

{\bf Note.} $\pi$ sq. integ. iff $\pi<\lambda$.
\begin{block}{Orthogonality rel in $ax+b$}
$\eta_1,\eta_2,\xi_1,\xi_2\in C_c^2(\Real_+^*)$. Then
% Then $\xi_1*_{\pi_\pm}\eta_1$ and $\xi_2*_{\pi_\pm}\eta_2$ belong to $L^2(G)$, and
%

$\langle \xi_1*_{\pi_+}\eta_1,\xi_2*_{\pi_+}\eta_2\rangle_{L^2(G)}  =  \langle\eta_2,\eta_1\rangle_{\cH}\langle K^{-\frac{1}{2}}\xi_1,K^{-\frac{1}{2}}\xi_2\rangle_{\cH}.$\\

$ \langle \xi_1*_{\pi_-}\eta_1,\xi_2*_{\pi_-}\eta_2\rangle_{L^2(G)}  =  \langle\eta_2,\eta_1\rangle_{\cH}\langle K^{-\frac{1}{2}}\xi_1,K^{-\frac{1}{2}}\xi_2\rangle_{\cH}.$
%
$
\langle \xi_1*_{\pi_+}\eta_1,\xi_2*_{\pi_-}\eta_2\rangle_{L^2(G)}  =  0.$
\end{block}

\begin{block}{Corollary}
$\pi_+$ and $\pi_-$ are sq. inter.
\end{block}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{Square integrable representations}
$G:$ LCG.\\
$\Delta_G$: modular function.
%
\begin{theorem}[Dufllo-Moore Orthog. rel. for sq. integ.]
$\pi$: irred. sq. integ. rep.\\
  Then $\exists !$ densely defined $K$ ``formal degree'' on ${\cH}_\pi$
s.t.
\begin{itemize}
%\item $\pi(g)K\pi(g)^{-1}=\Delta_G(g)^{-1}K$ for every $g\in G$.
%
\item $\xi*_\pi\eta\in L^2(G)$ iff $\xi\in\dom K^{-\frac{1}{2}}$.
%
\item For $\eta,\eta'\in {\cH}_\pi$ and $\xi,\xi'\in \dom (K^{-\frac{1}{2}})$, we have
%
$\langle \xi*_\pi\eta,\xi'*_\pi\eta'\rangle_{L^2(G)}=\langle\eta',\eta\rangle_{{\cH}_\pi}\langle K^{-\frac{1}{2}}\xi,K^{-\frac{1}{2}}\xi'\rangle_{{\cH}_\pi}.$
\end{itemize}
%
\end{theorem}

{\bf Note.}
 $\pi$, $\sigma$ be nonequiv.  sq. integ.  rep. Then,
%
$$\langle\xi*_\pi\eta,\xi'*_{\sigma}\eta'\rangle_{L^2(G)}=0.$$


\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{frame}{}
\begin{block}{Proposition}
$\FA(G)=\FA_{\pi_+}\oplus_1 \FA_{\pi_-}$.
\end{block}
%
\begin{block}{Important property for $\FA_{\pi_\pm}$}
$\|\xi*_{\pi_\pm}\eta\|_{\FA}=\|\xi\|\|\eta\|.$
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Reduced Heisenberg}
\begin{frame}{}
\begin{definition}
${\mathbb H}_r:=\{(p,q,e^{2\pi i \theta}):p,q\in \Real,\theta\in[0,1)\}$, with gp mult.
%
$(p,q,e^{2\pi i \theta})\cdot(p',q',e^{2\pi i \theta'})=(p+p',q+q',e^{2\pi i (\theta+\theta')}e^{\pi i (pq'-qp')}).$
\end{definition}

\begin{block}{Representations}
For $n\in{\mathbb Z}\setminus\{0\}$,  $\sch_n:{\mathbb H}_r\rightarrow \cU(L^2(\Real))$ is defined by
$$\sch_n(p,q,e^{2\pi i \theta})\xi(x)=e^{2\pi i nq(-x+\frac{p}{2})}e^{2\pi i n\theta}\xi(-p+x).$$
\end{block}

\begin{block}{Proposition}
\begin{itemize}
\item
$\langle \xi_1*_{\pi_n}\eta_1,\xi_2*_{\pi_n}\eta_2\rangle_{L^2(G)}=\frac{1}{n}\langle\eta_2,\eta_1\rangle_{\cH}\langle \xi_1,\xi_2\rangle_{\cH}.$
%
\item $\langle \xi_1*_{\pi_n}\eta_1,\xi_2*_{\pi_m}\eta_2\rangle_{L^2(G)}=0, \ \mbox{ whenever } n\neq m.$
%
\item $\FA(\bbH_r)= \oplus_{n\neq 0}\FA_{\sch}(\bbH_r) \oplus \FA(\bbH_r:{\mathbb T}).$
\end{itemize}
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}{}
\begin{block}{Question}
Is $\FA(G)$ weakly amenable when $G$ is the motion group?
\end{block}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}