

\documentclass{beamer}

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\font\Kis=cmr7

\font\Twelveopen=msbm10 %scaled\magstep1


\long\def\DEL#1{{ }}

\def\DDD{{\Delta}}
\def\SSS{{\openo S}}

\def\eps{\varepsilon}


\font\Goth=eufm10 scaled \magstep1

\def\RR{\mathbb{R}}
\def\KK{\mathbb{K}}
\def\MM{\mathbb{M}}
\def\DD{\mathbb{D}}
\def\NN{\mathbb{N}}
\def\PP{\mathbb{P}}
\def\FF{\mathbb{F}}
\def\TT{\mathbb{T}}
\def\ZZ{\mathbb{Z}}
\def\QQ{\mathbb{Q}}
%\def\OO{\mathbb{0}}
\def\OO{{\bf 0}}

\def\CC{\mathbb{C}}

\def\HHH#1{{\bf H}^{(#1)}}
\def\HHHx{{\bf H}}
\def\hhh{{\bf h}}

\def\XXX#1{{\bf X}^{(#1)}}
\def\XXXx{{\bf X}}
\def\xxx{{\bf x}}

\def\ot{\overline{t}}
\def\ox{\overline{x}}

\def\norm#1{\left\Vert #1 \right\Vert}

%\def\DD{\Delta}
%\def\DD{{\rm I\kern-1.6pt {\rm D}}}

%\def\idez#1{{\ideza}#1''}

\def\ideza{\setbox0=\hbox{\lower1.38ex\hbox{''}}\dp0=0pt\box0}


\title{
Strongly continuous one-parameter groups of
automorphisms of multilinear functionals
%Strongly continuous one-parameter groups of holomorphic unit ball
%automorphisms
}

\author{L.L. STACH\'O}

\date{21/09/2013, Ploie\c{s}ti}


\setbeamertemplate{background canvas}[vertical shading][bottom=red!10,top=blue!10]

\setbeamercovered{dynamic}



\usetheme{Madrid}



\begin{document}



%\section*{C\'\i moldal}



\frame{\titlepage} % a fenti \title es \author szovegek megjelenitese



% \section*{Tartalom}



% \frame{\tableofcontents} % a * nelkuli \section es \subsection kulcsszavak mellett megadott szovegeket egy oldalra kigyujti

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


\frame[<+->]{

\frametitle{PROBLEM SETTING}

{\bf J. Jamison - F. Botelho 2008}

\

$X_1,\ldots,X_N$ complex Banach spaces

\vskip3mm

$\mathcal{B} := \big\{ \hbox{bounded $N$-linear maps} \ X_1\times\cdots\times X_N\to\CC \big\}$

\vskip3mm

${\bf U}: \RR\to \hbox{{\Goth A}}:=
\big\{ \hbox{surjective linear isometries }\
\mathcal{B}\to\mathcal{B}\big\}$

\vskip1mm

\hskip10mm {\it strongly cont.} 1-parameter group of surj. lin. isom.

\vskip1mm

\hskip10mm ${\bf U}(t\!+\! h) \!=\! {\bf U}(t){\bf U}(h)$ \qquad $(t,h\!\in\!\RR)$


\hskip10mm $t\!\mapsto\! {\bf U}(t)\Phi$ continuous \ \quad $(\Phi\!\in\!{\cal B})$

\

\bf Conjecture:
$$\hbox{\it ${\bf U}(t) = \big[ \hskip-2mm
\underbrace{U_1^t \otimes\cdots\otimes U_N^t}_{\phi\mapsto \phi(U_1^t x_1,\ldots,U_N^t x_N)} \hskip-1mm\big]^*$
\hskip10mm
\vtop{
\hbox{$t\mapsto U_k^t$ \ \ str.cont. 1-par.grp. in}
\hbox{{\Goth A}$_k :=\{ \hbox{surj.lin.isom. $X_k\!\to\! X_k$}\}$}} }$$



}




\frame[<+->]{

\frametitle{NON-LINEAR ISSUE}


$\Big\{$Surj.lin.isom. of $X \Big\} =
\Big\{ F\in \underbrace{{\rm Aut}\big( {\rm Ball}(X)\big)}_{hol.aut.\ of\ unit\ ball} :\ F(0)=0 \Big\}$

\vskip5mm

$\Vert \phi\Vert =\hskip-17mm\sup\limits_{\hskip17mm
\Vert x_1\Vert =1,\ldots,\Vert x_N\Vert=1} \big\vert \phi(x_1,\ldots,x_N)\big\vert $
\qquad
in \ \ $\mathcal{B}$

\vskip5mm

\bf Question: \rm What about \it strongly cont. 1-par. groups \rm in ${\rm Aut}(\mathcal{B})$?

\vskip5mm
\bf Conjecture: \rm ${\rm Aut}(\mathcal{B})$ \ \ LINEAR \ \ for \ \ $N>2$ \ factors

\hskip35mm$\uparrow$

Stach\'o 1982: \ \ TRUE \ with \ HILBERT spaces of \ dim$>1$

\vskip5mm

Hilbert cases $N=1,2$: \quad JB$^*$-triples \ ${\bf H}$ \ resp. ${\mathcal L} \big( \HHH1,\HHH2\big)$

}



\frame[<+->]{

\frametitle{HILBERT CASE $N>2$; MAIN RESULT}


\begin{itemize}

\item[]
$\HHH1,\ldots,\HHH{N}$ \ Hilbert spaces

\vskip3mm

${\rm Aut}(\mathcal{B}) = \big\{ \big([U_1\otimes\cdots\otimes U_N] \circ I_\pi \big)^* : \
U_k \in\mathcal{U}(\HHH{k}), \ \pi \ \hbox{adm.index-perm.}\big\}$

\vskip3mm

${\bf U}: \RR\to {\rm Aut}(\mathcal{B})$ \quad
{\bf str*}.cont. 1-par.grp:

\hskip20mm
$t\mapsto\big[{\bf U}^t\phi\big]({\bf x}_1,\ldots,{\bf x}_N)$ cont.

\vskip3mm
\item[]
\bf Lemma. \it \quad
${\bf U}(t) = \big[ U_{1,t} \otimes\cdots\otimes U_{N,t} \big]^* \qquad (t\in\RR)$

\vskip5mm

\item[]
\bf THEOREM. \it
There are possibly {\bf unbounded self-adjoint}
$A_k : {\rm dom}(A_k) \to \HHH{k}$
(defined on dense linear submanifolds)
with
$${\bf U}(t) \!=\! \Big( \big[ \exp(itA_1) \big] \otimes\cdots\otimes \big[ \exp(itA_N) \big] \Big)^*
\qquad (t\in\RR) .$$

\vskip5mm

%\item[]
\bf Corollary. \it
If \ ${\bf W}: \RR\to
{\rm Aut}\big( \mathcal{L}(\HHH{1}\! ,\HHH{2})\big)$
is a
{\bf str*.cont. lin.} 1-par.grp then \
${\bf W}(t)X \!=\! \exp(t A_1) X \exp(t A_2)$ \
\rm

\end{itemize}

}




\frame[<+->]{

\frametitle{BACKGROUND OF TECH. DIFFICULTIES}

$\TT :=\{ \kappa:\ \vert \kappa\vert=1\}$

\vskip5mm


{\bf Ambiguous representation} \ \ for unitary ops
\vskip5mm
\begin{center}
$U_1\otimes\cdots\otimes U_N = V_1\otimes\cdots\otimes V_N$
\break
$\Updownarrow$
\break
$\displaystyle{V_k =\kappa_k U_k, \ \ \kappa_k\in\TT \quad (k=1,\ldots,N)\qquad
\prod_{k=1}^N \kappa_k =1 }$
\end{center}




}



\frame[<+->]{

%\frametitle{GEOMETRY OF \ $\PP(\HHH1)^*\otimes\cdots\otimes\PP(\HHH{N})^*$}
\frametitle{ELEMENTARY FUNCTIONALS}

\begin{itemize}

\item[]
$\hhh_1^* \otimes\cdots\otimes \hhh_N^* : \
({\bf x}_1,\ldots,{\bf x}_N) \mapsto \prod_{k=1}^N \hhh_k^*({\bf x}_k) =
\prod_{k=1}^N \langle {\bf x}_k \vert \hhh_k\rangle $
\DEL{
$$[L_1\otimes\cdots\otimes L_N]
\hhh_1^* \otimes\cdots\otimes \hhh_N^*  =
[L_1^* \hhh_1]^* \otimes\cdots\otimes [L_N^* \hhh_N]^* .$$
}%ENDDEL

\DEL{\item[]
\bf Remark. \rm For unit vectors \  ${\bf g},\hhh \in \HHHx$,
\vskip3mm
$\phantom{xxxxx}
{\rm dist}(\TT{\bf g} ,\TT\hhh)
\vtop{
\hbox{$\displaystyle{=
{\rm dist} ({\bf g},\TT\hhh) =
\min_{\vert\kappa\vert=1}
\big[ 2 - 2\, {\rm Re} \langle {\bf g} \vert \kappa \hhh \rangle \big]^{1/2} =}$}
\hbox{$=
\sqrt{2} \, \big[ 1 - \vert \langle {\bf g} \vert \hhh \rangle \vert \big]^{1/2} .$
}}$
}%ENDDEL

\vskip5mm

\item[]
\bf Lemma. \it
$\Phi := {\bf g}_1^*\otimes\cdots\otimes {\bf g}_N^*$,
$\Psi:=\hhh_1^*\otimes\cdots\otimes \hhh_N^*$ \quad
(unit vectors)
$$\big\Vert \Phi - \Psi \big\Vert \le \eps \ \ \Longrightarrow \ \
\underbrace{{\rm dist} (\TT{\bf g}_k, \TT \hhh_k)}_{
\min_{\kappa,\mu\in\TT}
\Vert \kappa {\bf g}_k - \mu \hhh_k\Vert}
\le 2^{N-1} \eps$$
\rm

\vskip5mm

\item[] $\PP(\HHHx):= \{ \TT {\bf g}:\ {\bf g}\in \partial{\rm Ball}(\HHHx) \}
%\quad
%{\rm dist}(\TT{\bf g} ,\TT\hhh) := \min_{\kappa,\lambda \!\in\!\TT} \vert \kappa {\bf g} \!-\! \lambda \hhh \vert
$


\vskip5mm

%\item[]
\bf Lemma. \it
${\bf F}: \RR\to \PP(\HHHx)$
cont. \ \ $\Longrightarrow$ $\exists\ t\mapsto \hhh_t \in \partial {\rm Ball}(\HHHx)$ cont.
$${\bf F}(t) =\TT \hhh_t \qquad (t\in\RR)$$


\end{itemize}

}



\frame[<+->]{

\frametitle{ADJUSTED STRONG* CONTINUITY}

\begin{itemize}

\item[]
\bf Lemma$^*$. \it \ \
$\Psi(t) = \hhh_{1,t}^* \otimes\cdots\otimes \hhh_{N,t}^*$ \quad (with unit vectors)
\vskip3mm
$\Psi:\RR\to{\cal B}$ \ continuous, \ $\Longrightarrow$ \qquad
$\exists\, \kappa_1,\ldots,\kappa_N:\RR\to\TT$
$$\prod_{k=1}^N \kappa_k(t)\equiv 1 \ , \qquad
t\mapsto \kappa_k(t)\hhh_{k,t} \ \ \hbox{norm-continuous}$$
\rm

%\vskip3mm \bf Proof. \rm Assembling continuous local sections.

\vskip5mm


\item[]
\bf Proposition. \it $t\mapsto \big[ \overbrace{U_{1,t}}^{unit.}\otimes\cdots\otimes \overbrace{U_{N,t}}^{unit.} \big]^*$
\ \  str*.cont. $\RR\to\hbox{{\Goth A}}$,
\quad $\Longrightarrow$
\vskip3mm
$\exists\, \kappa_1,\ldots,\kappa_N:\RR\to\TT$ \qquad
$t\mapsto \kappa_k(t) U_{k,t}$ \ str.cont. \ $(k=1,\ldots,N)$
\rm

\end{itemize}

}



\frame[<+->]{

\frametitle{EXTENSION: MORE BANACH SPACE GEOMETRY}

\begin{itemize}

\item[]
$\XXX{1},\ldots,\XXX{N}$ {\bf uniformly smooth} Banach spaces

\vskip3mm
\bf Recall: \rm
$\XXXx$ is uniformly convex (rotond):

\hskip20mm$\inf\Big\{ 1 - \norm{{\bf x}+{\bf y}}/2: \ {\bf x},{\bf y}\in\partial{\rm Ball}(\XXXx) \Big\}>0$

\vskip3mm
\hskip14mm $\XXXx$ is uniformly smooth:
\vskip-2mm
\hskip20mm$\lim\limits_{\tau\searrow 0} {1\over \tau} \sup
\Big\{ \norm{{\bf x}+{\bf y}}/2 + \norm{{\bf x}-{\bf y}}/2 -1 : \
\norm{{\bf x}}=1, \ \norm{{\bf y}} \le \tau \Big\} =0 $

\vskip6mm

\item[]
1)$\XXXx$ unif.sm. $\iff$
$\XXXx^*$ unif.convex,

\hskip3mm
$\XXXx$ unif.convex $\iff$
$\XXXx^*$ unif.sm

%2) In a unif.conv. space weak convergence +
%convergence of norms implies norm (strong) convergence.
2*) $\XXXx$ unif.convex: \ weak converg. + conv of norms $\Rightarrow$ norm conv.

3) Unif.conv. and unif.sm. spaces are reflexive.

\end{itemize}

}


\frame[<+->]{

\frametitle{EXTENSION: MORE BANACH SPACE GEOMETRY}

\begin{itemize}

\vskip3mm
\item[]
\bf Lemma. \it Let $\phi^{(k)},\psi^{(k)}\in \partial {\rm Ball}[\XXX{k}]^*$. \ \
T.f.a.e.:

$(1)$ $\phi^{(1)}\otimes\cdots\otimes\phi^{(N)} = \psi^{(1)}\otimes\cdots\otimes\psi^{(N)}$,
%if and only if

$(2)$ $\exists\ \kappa_1,\ldots,\kappa_N\in\TT$ \ \ with \ \
$\prod_k \kappa_k=1$,  $\psi^{(k)}=\kappa_k\phi_k$. %$(k=1,\ldots,N)$.

\vskip6mm
\item[]
\bf Lemma. \it Let $\phi^{(k)}_j\in \partial {\rm Ball} [\XXX{k}]^*$ be nets such that
$$\phi^{(k)}_j\otimes\cdots\otimes\phi^{(N)}_j \to
\phi^{(k)}\otimes\cdots\otimes\phi^{(N)} \quad {\rm pointwise}.$$
Then \ \
$\exists \ \big[ \kappa_j^{(k)} \big]$ in $\TT$ \ with \
$\norm{\kappa_j^{(k)}\phi_j^{(k)} - \phi^{(k)}} \to 0 \quad (k\!=\! 1,\ldots,N).$


\end{itemize}

}


\frame[<+->]{

\frametitle{BANACH EXTENSIONS (continued)}

$\XXX{1},\ldots\XXX{N}$ unif.smooth Banach spaces

\vskip3mm
\bf Proposition. \it Let
$t(\in\RR) \mapsto \phi^{(k)}_t\otimes\cdots\otimes\phi^{(N)}_t(\xxx_1,\ldots,\xxx_N) $
cont.

for any fixed
$\xxx^{(1)},\ldots,\xxx^{(N)}$ \ where \
$\phi_t^{(k)} \in \partial {\rm Ball}[\XXX{k}]^*$.

Then \ $\exists$ functions \ $t(\in\RR)\mapsto \kappa_t^{(k)}\in\TT$ \
such that
$$\prod_{k=1}^N \kappa_t^{(k)} =1 \quad (t\in\RR), \qquad
t\mapsto \kappa_t^{(k)} \phi_t^{(k)} \ \hbox{\rm is norm-cont.} $$

\bf Corollary. \it If $U_{k,t} \in{\cal L}(\XXX{k})$ $(t\in\RR,\ 1\le k\le N)$ are {\bf isometries} and
$t\mapsto \big[ U_{1,t}\otimes\cdots\otimes U_{N,t} \big]^*$ is {\bf str*}.cont. \ then
 \ $\exists$ functions \ $t(\in\RR)\mapsto \kappa_t^{(k)}\in\TT$ \
such that each $t\mapsto U_{k,t}$ is {\bf str.}cont. (pointwise cont.).


}



\frame[<+->]{

\frametitle{SEPARATE COMMUTATIVITY (smooth Banach setting)}

\begin{itemize}

\item[]
$t\mapsto {\bf U}(t) := \big[ U_{1,t} \otimes\cdots\otimes U_{N,t} \big]^*$
str.cont. 1-par.grp
\vskip1mm
Proposition $\Longrightarrow$
without loss of generality:
\begin{eqnarray*}
&1)&\ \ t\mapsto U_{k,t} \in \mathcal{U}(\XXX{k}) \ \
\hbox{strongly continuous} ;\\
&2)&\ \ U_{k,0}={\rm Id}, \quad U_{k,-t}=U_{k,t}^* \quad (t\in \RR).
\end{eqnarray*}


\item[]
\bf Lemma. \it
$\{ U_{k,t}:\ t\in\RR\}$ \ $(k=1,\ldots,N)$ \
are Abelian.

\vskip1mm
\hskip17mm
\rm
[ A priory only \ \ $U_{k,t} U_{k,s} \in \TT U_{k,s} U_{k,t}$ ! ]

\DEL{
\bf Proof. \rm
Induction on $n=1,2,\ldots$ $\Rightarrow$ \ \
$\exists$ constants $\kappa_k^{(n)}\in\TT$ \ with

$\phantom{xxxxx}
U_{k,nt} = \kappa_k^{(n)} [U_{k,t}]^n
\quad (1\le k \le N), \qquad \prod_{k=1}^N \kappa_k^{(n)} =1$
}%ENDDEL


\vskip9mm

\item[]
\bf Theorem. \it
$\exists$ $\kappa_1,\ldots,\kappa_N:\RR\!\to\!\TT$, \ \
$\exists$
%not necessarily str.cont.
$\underline{\hbox{1-par.grp-s}}$ \
$t\mapsto U_k^t$
% \in {\cal U}(\HHH{k})$
$$U_{k,t} = \kappa_k(t) U_k^t , \quad \kappa_k(0)=1  \qquad (t\in\RR, \ 1\le k\le N).$$
\rm

\end{itemize}


}



\frame[<+->]{

\frametitle{LOCALIZATION (a Hilbert space argument)}


\begin{itemize}

\item[]
$${\bf U}(t) = \Big(
\big[ \kappa_1(t) U_1^t\big]\otimes\cdots\otimes\big[ \kappa_N(t) U_N^t\big] \Big)^*$$
%\big[ \kappa_1(t) U_1^t\big]^*\otimes\cdots\otimes\big[ \kappa_N(t) U_N^t\big]^* \Big)^*$$

\vskip1mm

\item[]
$k$ fixed, \quad $\XXXx \equiv \HHHx \!:=\! \HHH{k}$
$$\hbox{$t\!\mapsto\! U^t := U_k^t \!\in\!{\cal U}( \HHHx )$ {\bf non.cont.} 1-pgr., \
$\kappa=\kappa_n:\RR\!\to\!\TT$}$$

%\item[]
\vskip-8mm
$$\hbox{$t\!\mapsto\! \kappa(t) U^t$ str.cont. \ 1-prg. \hskip-12mm ---------},
\hskip5mm
\kappa(-t) \!=\!\overline{\kappa(t)} , \quad \kappa(0)\!=\! 1 .
$$

\vskip3mm
\item[]
%$\exists\, \{ {\bf x}_\alpha:\ \alpha\in\mathcal{A} \}$ \quad orthogonal family
\bf Cyclic decomp: \rm \quad
$\bf {\bf H} = \oplus_{\alpha\in\mathcal{A}} \HHHx_\alpha, \qquad
\HHHx_\alpha := \overline{{\rm Span}} \{ U^t {\bf x}_\alpha :\ t\in \RR\}$

\vskip3mm

\item[]
\bf Main idea:\ \rm It suffices to see
$$\kappa(t) U^t \vert {\HHHx}_{\alpha} = \chi_\alpha(t) \widetilde{U}_\alpha^t$$
$\exists$ $\chi_\alpha \in\mathcal{C}(\RR,\TT)$,\quad
$t\mapsto \widetilde{U}_\alpha^t$ \
str.cont. 1-prg.
in \ ${\cal U}( {\HHHx}_{\alpha})$



\DEL{
\item[]
\begin{eqnarray*}
& & {\cal A} \ := \ \big\{ \hbox{the comm. $C^*$-algebra
gen. by $\{ U^t:\ t\in\RR \} \big\}$} ,\\
& & {\bf T} \ :  \ {\cal C}(\Omega)\leftrightarrow{\cal A} \quad
\hbox{the Gelfand representation of ${\cal A}$},\\
& & u^t \ := \ {\bf T}^{-1} U^t \qquad (t\in\RR).
\end{eqnarray*}


\item[]
\vskip-3mm
$$
\HHHx = {\scriptstyle{\bigoplus}}_{j\in J} \ {\HHHx}_{{\bf x}_j}
%= \overline{{\rm Span}} \{ \kappa(t)U^t {\bf x} :\ t\in \RR\}
 , \qquad
{\HHHx}_{{\bf x}} := \overline{{\rm Span}} \{ A {\bf x} :\ A\in {\cal A}\}$$

\vskip2mm

with any maximal orthogonal family
$\big\{ {\HHHx}_{{\bf x}_j} : \, j\!\in\! J\big\}$
% such that
%${\HHHx}_{{\bf x}_j} \!\perp\! {\HHHx}_{{\bf x}_k}$ \ $(j\!\ne\! k\!\in\! J)$
%guaranteed by the
($\exists$ byZorn Lemma).
}%ENDDEL


\end{itemize}

}



\frame[<+->]{

\frametitle{LOCAL FUNCTION REPRESENTATION (Hilbert spaces)}

\begin{itemize}

\item[]
Henceforth \ \ $\alpha$ \ \ {\bf fixed}
\vskip2mm
%$t\mapsto U^t\vert \HHHx_\alpha \in\mathcal{U}(\HHHx_\alpha)$ \qquad
%1-prg. in $\mathcal{U}(\HHHx_\alpha)$
\bf Gelfand repr. \rm \quad
${\bf T}: \mathcal{C}(\underbrace{\Omega}_{comp.}) \leftrightarrow
\overline{{\rm Span}}_{t\in\RR} U^t$, \quad $\underbrace{u^t}_{\Omega\to\TT} \mapsto U^t$

\DEL{
\vskip3mm

\item[] ${\bf x}_\alpha$ \ {\bf cyclic} vector: \
$\overline{{\rm Span}}\{ U^t{\bf x}_\alpha: t\in\RR\} =\HHHx_\alpha$
}%ENDDEL

\vskip3mm

\item[]
$\sim$\bf Halmos's idea \rm \quad $\HHHx_\alpha$ cyclic $\Rightarrow$ \quad
%For the Gelfand-Neumark repr.
%$${\bf T}_0 : \overline{{\rm Span}}\{ U^t\vert\HHHx_\alpha:\ t\in\RR\}
%\leftrightarrow \mathcal{C}(\Omega)$$
%\hskip28mm $u^t := {\bf T}_0 (U^t\vert\HHHx_\alpha) \in \mathcal{C}(\Omega,\TT)$

%\vskip3mm

$\exists$
prob. Radon $(\Omega,\underbrace{\mu}_{=\mu_\alpha})$, \
${\bf T} \hookrightarrow {\bf T}_\alpha : L^2(\Omega,\mu) \leftrightarrow \HHHx_\alpha$ isometry \


$$\Big\langle\, U^t {\bf T}_\alpha f \, \Big\vert \ {\bf T}_\alpha g \,\Big\rangle =
\int_{\omega\in\Omega} \!\!\!\!
u^t (\omega) f(\omega) \overline{g(\omega)} \ \mu(d\omega)$$


\item[]
\vskip3mm
\bf We have to prove: \quad \it
$\exists\
\chi\in\mathcal{C}(\Omega,\TT)$ , \ \
$t\mapsto\widetilde{u}^t$ 1-prg. in $\mathcal{C}(\Omega,\TT)$

$$\hbox{$\kappa(t) u^t =
\chi(t) \widetilde{u}^t$ \  $(t\in\RR)$ \ and \
$t\mapsto{\bf M}_{\widetilde{u}^t}$ is str.cont.}$$
\rm

\end{itemize}

}




\frame[<+->]{

\frametitle{PROBABILISTIC ARGUMENTS}

\begin{itemize}

\item[]
\bf Notations, assumptions \rm

\vskip3mm

$(\omega,\mu)$ prob. Radon measure, \ $\Omega$ compact

\vskip1mm

${\bf M}_a: L^2(\Omega,\mu) \ni f \mapsto af$ \ \ $(a\in L^\infty(\Omega,\mu))$ \quad mult.op.

\vskip1mm

$u^t\in\mathcal{C}(\Omega,\TT), \quad u^{t+s} = u^t u^s$

\vskip1mm

$\kappa:\RR\to\TT$ , \quad \ $\underline{\hbox{$t\mapsto \kappa(t) {\bf M}_{u^t}$ str.cont.}}$

\item[]
\vskip8mm
\bf Lemma. \it
$\{ a_1,a_2,\ldots \}\ \hbox{bounded}\subset L^\infty(\Omega,\mu)$, $\Longrightarrow$

\vskip3mm

${\bf M}_{a_1}, {\bf M}_{a_2},\ldots \to 0$ strongly
$\iff$
$a_n \to 0$ {\bf stochastically} (wrt. $\mu$):
$$\lim_{n\to\infty} \mu\{ \omega\in\Omega:\ \vert a_n(\omega) \vert > \eps\} =0
\qquad (\eps>0)$$
\rm


\end{itemize}


}


\frame[<+->]{

\frametitle{PROBABILISTIC ARGUMENTS}

\begin{itemize}

\item \ \
$\displaystyle{
\Delta(h) := \sup_{\vert t\vert \le h} \
\int_{\omega_1,\omega_2\in \Omega} \!\!\!\!
\vert u^t(\omega_1)-u^t(\omega_2)\vert^2 \ \mu(d\omega_1)\mu(d\omega_2)
}$

\DEL{
\begin{eqnarray*}
&\Delta\!\!\!\!\!\!& (\delta) := \sup_{\vert t\vert \le \delta} \
\int_{\omega_1,\omega_2\in \Omega} \!\!\!\!
\vert u^t(\omega_1)-u^t(\omega_2)\vert^2 \ \mu(d\omega_1)\mu(d\omega_2) \ ,\\
&\Omega\!\!\!\!\!\!& _{t,r}^{(2)} :=
\big\{ (\omega_1,\omega_2)\in\Omega^2 :\
d\big( u^t(\omega_1),u^t(\omega_2) \big) < r \big\} \ .
\end{eqnarray*}
}%ENDDEL

\vskip7mm

\item[]
\bf Proposition. \it
\DEL{$\displaystyle{
\lim_{h\to 0} \ \int_{\omega_1,\omega_2\in\Omega}
\big\vert u^h(\omega_1) \!-\! u^h(\omega_2) \big\vert^2 \
\mu(d\omega_1) \mu(d\omega_2) = 0 }$
}%ENDDEL
\quad $\Delta(h)\searrow 0$ \ as \ $h\searrow 0$

\vskip3mm
\bf Proof. \rm Use the fact:

\vskip5mm

\quad ${\bf M}_{\kappa(h)u^h}\buildrel{str}\over\to{\bf M}_{u^0}$ \ i.e. \quad
$\lim\limits_{h\to 0} \ \mu\{ \omega:\ \vert \kappa(h) u^h(\omega) - 1 \vert > \eps\} =0$

\item[]
\begin{eqnarray*}
\Rightarrow\hskip17mm&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\big\vert&
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
u^h(\omega_1)-u^h(\omega_2) \big\vert =
\big\vert \kappa(h)\big( u^h(\omega_1)-u^h(\omega_2)\big) \big\vert \le \\
&\le& \big\vert \kappa(h) u^h(\omega_1)- 1 \big\vert+
\big\vert \kappa(h)u^h(\omega_2) -1 \big\vert .
\end{eqnarray*}



%\item[]
\vskip-8mm
$$\Rightarrow\quad
\lim_{h\to 0} \mu\otimes\mu \big\{ (\omega_1, \omega_2)\in\Omega^2 :\
\big\vert u^h(\omega_1) - u^h(\omega_2) \big\vert  >\eps\big\}=0$$


\end{itemize}

}



\frame[<+->]{

\frametitle{PROBABILISTIC ARGUMENTS}

\begin{itemize}


\item[]
Arc length distance on $\TT$: \quad
$d(\kappa_1,\kappa_2) :=
2 \, {\rm arcsin}\, {\vert \kappa_1 - \kappa_2\vert \over 2}$

\vskip3mm

\item
\quad
$\Omega_{t,r}^{(2)} :=
\big\{ (\omega_1,\omega_2)\in\Omega^2 :\
d\big( u^t(\omega_1),u^t(\omega_2) \big) < r \big\}$

\DEL{
\item
$u^{mnt} = [u^{nt}]^m$ $(t\in\RR)$, $\Rightarrow$ \quad
$\Omega_{t/(mn),\pi/(mn)}^{(2)} \subset
\Omega_{t/n,\pi/n}^{(2)}$
}


\item[]
\vskip-2mm
\begin{eqnarray*}
& &(\omega_1,\omega_2)\in\Omega^{(2)}_{t/n,\pi/n} \Longrightarrow \\
& &\log_* \big[ u^{kt/n}(\omega_1)/u^{kt/n}(\omega_2)\big] =
{k\over n} \log_* \big[ u^t(\omega_1)/u^t(\omega_2)\big]
\quad (\vert k\vert\le n) ,\\
\DEL{
& &u^{kt/n}(\omega_1)/u^{kt/n}(\omega_2) =
\exp\Big( {k\over n} \log_* \big[ u^t(\omega_1)/u^t(\omega_2)\big]
\Big) \quad (k\in\ZZ) .
}%ENDDEL
\end{eqnarray*}

\item[]
\vskip-5mm
$$\Omega_{t,\pi}^{(2)} \supset \Omega_{t/2,\pi/2}^{(2)}
\supset \Omega_{t/3!,\pi/3!}^{(2)} \supset \cdots$$

\vskip3mm

\item
\bf Proposition. \it
For any $t\in\RR$ we have \ \
$$\displaystyle{\mu\otimes\mu \Big(
\bigcap\limits_{n=1}^\infty \Omega_{t/n!, \pi/n!}^{(2)} \Big) \ge
1- \underbrace{\Delta(\vert t\vert) } }_{\hskip-22mm\sup\limits_{\vert \tau\vert \le \vert t\vert} \
\int\limits_{\omega_1,\omega_2\in \Omega}
\vert u^\tau(\omega_1)-u^\tau(\omega_2)\vert^2 \ \mu(d\omega_1)\mu(d\omega_2)}$$
\rm

\end{itemize}

}



\DEL{
\frame[<+->]{

\frametitle{PROOF}

\begin{itemize}

\item[] Use the following facts:

\begin{eqnarray*}
\Delta(\vert t\vert) \!\!\! &\ge& \!\!\! {1\over n} \sum_{k=1}^n
\int _{\omega_1,\omega_2\in\Omega}
\vert u^{kt/n}(\omega_1) - u^{kt/n}(\omega_2) \vert^2 \
\mu(d\omega_1)\mu(d\omega_2) \\
&=& \!\!\! \frac{1}{n} \sum_{k=1}^n
\int _{\omega_1,\omega_2\in\Omega} \hskip-6mm
4 \sin^2 {1\over 2} d\big( u^{kt/n}(\omega_1) , u^{kt/n}(\omega_2) \big) \
\mu(d\omega_1)\mu(d\omega_2)
\DEL{
\\
&=& 4 \int _{\omega_1,\omega_2\in\Omega}
\phi_n \Big(\frac{1}{2} d\big( u^{t/n}(\omega_1) , u^{t/n}(\omega_2) \big) \Big) \
\mu(d\omega_1)\mu(d\omega_2) \ .
}%ENDDEL
\end{eqnarray*}

\item[]
\begin{eqnarray*}
\frac{1}{n} \sum_{k=1}^n \sin^2 k x
\!\!\!\! &\ge&\!\!\!  \frac{4}{\pi^2} \, \frac{(n+1)(2n+1)}{6} x^2 \ge
\frac{4 n^2 }{ 3\pi^2} \, \sin^2 x \quad {\rm for} \ \ 0\le x\le \frac{\pi}{2n} \\
&\ge&\!\!\!  \frac{1}{4} \qquad {\rm for} \ \ \frac{\pi}{2n} \le x \le \frac{\pi}{2} .
\end{eqnarray*}
\end{itemize}


}
}%ENDDEL



\frame[<+->]{

\frametitle{CONCLUSIONS WITH PROBABILITY}

\begin{itemize}

\item[]
\bf Corollary. \it
\DEL{
$\exists$ \ $\alpha:\Omega^2 \to\RR$ \quad
$\mu\otimes\mu$-measurable function with
}%ENDDEL
$\alpha(\omega_1,\omega_2):={1\over i/\ell} \log_* \big[ u^{1/\ell}(\omega_1)/ u^{1/\ell}(\omega_2)\big]$
for $(\omega_1,\omega_2)\in D_\ell$ \
well-def. $\mu\otimes\mu$ a.e.
$$\frac{u^q(\omega_1)}{u^q(\omega_2)} = \exp\big( i\, q \, \alpha(\omega_1,\omega_2) \big)
\quad (q \!\in\! \QQ) \qquad \forall_{\mu\otimes\mu} \! (\omega_1,\omega_2) \!\in\!\Omega^2$$
%for \ $\mu\otimes\mu$-almost every \ $(\omega_1,\omega_2)\in\Omega^2$.

%\vskip5mm

\DEL{
\item[]
\bf Proof. \rm
$t_m:=\frac{1}{m}$, \quad
$S_m := \bigcap\limits_{n=1}^\infty \Omega^{(2)}_{t_m/n!, \pi/n!}$, \quad
$D_\ell := S_\ell \setminus
\bigcup_{m=1}^\ell S_m$

\item[]
%$D_i\cap D_j=\emptyset$ $(i\ne j)$; \quad
$$\mu\otimes\mu\big( \bigcup_{\ell=1}^m D_\ell \big) \ge
\mu\otimes\mu\big( S_m \big) \ge 1 - \Delta(t_m) \nearrow 1
\quad (m\to\infty)$$

\item[]
$(\omega_1,\omega_2) \!\in\! D_\ell$, \quad
$q =\frac{k}{\ell n!} = \frac{k}{n!}t_\ell$ \quad
$\big[\Rightarrow (\omega_1,\omega_2) \in \Omega^{(2)}_{t_\ell/n!,\pi/n!}\big]$
\begin{eqnarray*}
\DEL{
& &(\omega_1,\omega_2) \in D_\ell
\Longrightarrow \ \
(\omega_1,\omega_2) \in \Omega^{(2)}_{t_\ell/n!,\pi/n!}
\Longrightarrow \ \\
}%ENDDEL
& &{ u^q(\omega_1) \over u^q(\omega_2) } =
{ u^{(k/n!)t_\ell}(\omega_1) \over u^{(k/n!)t_\ell}(\omega_2) } =
\exp \Big( {k\over n!} \log_* \big[
u^{t_\ell}(\omega_1) / u^{t_\ell}(\omega_2) \big] \Big) \\
\DEL{
& &\hskip12mm
=  \exp \Big( {q\over t_\ell} \log_* \big[
u^{t_\ell}(\omega_1) / u^{t_\ell}(\omega_2) \big] \Big)
.
}%ENDDEL
\end{eqnarray*}

\item[]
$$\alpha(\omega_1,\omega_2) :=
{1\over i\, t_\ell} \log_* \big[
u^{t_\ell}(\omega_1) / u^{t_\ell}(\omega_2) \big]
\qquad \big( (\omega_1,\omega_2) \in D_\ell )$$
}%ENDDEL

\vskip5mm

\item[]
\bf Theorem. \it
$t\mapsto u^t\in\mathcal{C}(\Omega)$ 1-prg., \ $t\mapsto \kappa(t) {\bf M}_{u^t}$ str.cont.
$\Longrightarrow$

\vskip2mm

$\exists\ \chi\in\mathcal{C}(\RR,\TT)$ \quad $\exists \widetilde{\alpha}:\Omega\to \RR$ \ \
$\mu$-mesurable
$$\kappa(t) u^t(\omega) =
\chi(t) \exp\big( it \widetilde{\alpha}(\omega)\big) \quad (t\in\RR) \qquad
\forall_\mu \, \omega \!\in\!\Omega$$



\end{itemize}


}



\frame[<+->]{

\frametitle{FINISH OF THE PROOF (for Hilbert case)}

\begin{itemize}


\item[]
\bf Cor. $\Longrightarrow$


\rm
$\forall_\mu\, \omega_0 \quad
\mu\big\{ \omega : \
u^q(\omega) = u^q(\omega_0) e^{ iq \alpha(\omega,\omega_0)}
\ \ (q\!\in\!\QQ) \big\} =1$


\item[]
Fix $\omega_0$. \ \ Then \ $\exists! \, \chi_0: \QQ \to \TT$ \ with
\vskip-3mm
$$\chi_0(q) \exp\big( iq \alpha(\omega,\omega_0) \big) =
\kappa(q) u^q(\omega) \ \
(q\!\!\in\!\QQ) \qquad \forall_\mu\, \omega\in\Omega$$

\item Define: \quad $\widetilde{\alpha}(\omega):= \alpha(\omega,\omega_0)$

\vskip3mm

\item[]
$t\!\mapsto\! u^t e^{-it \widetilde{\alpha}(\omega)}$ \ {\bf stoch.}cont.
\quad
$\big[ t_n\!\to\! t$ $\Rightarrow$
$e^{it_n \widetilde{\alpha}(\omega)} \!\to\!
e^{it \widetilde{\alpha}(\omega)}$
$\forall_\mu \, \omega \big]$

$\kappa(t) u^t e^{-iq\widetilde{\alpha}}$ $(t\!\in\!\RR)$ are constant
$\mu$-alm. everywhere \quad[$\Leftarrow$def. $\widetilde{\alpha}$]

\vskip2mm

\item
$q\mapsto \chi_0(q) := \big[ \mu\hbox{-a.e. value of
$\kappa(q) u^q \exp( -iq\widetilde{\alpha})$} \big]$

\hskip10mm admits a cont.ext. \ $\chi:\RR\to\TT$ \ from $\QQ$


\end{itemize}


}


\DEL{

\frame[<+->]{

\frametitle{FINISH OF THE PROOF}

\begin{itemize}

\item[]
\bf Obtained: \it
If  \ $t\mapsto U_k\in\mathcal{U}(\HHHx)$ 1-prg. with cyclic vector,

$\kappa:\RR\to\TT$ \ and \ $t\mapsto \kappa(t) U^t$ str.cont., \ then

$\exists\, \chi\in\mathcal{C}(\RR,\TT)$ \
\hskip5mm $\exists\, A: {\bf D}\to \HHHx$ \ unb.selfadj.op.
$$\kappa(t) U^t = \chi(t) \exp(it A) \qquad (t\in\RR)$$

\item[]
\vskip2mm
\bf Application: \rm ${\bf U}(t) =  {\bf B}^t{\bf C}^t $ \ where \
$\big[$wrt. $\HHH{k} \!=\! {\scriptstyle{\bigoplus\limits_{j\in J_k}}} \ {\HHHx}_j^{(k)} \big]$
\vskip-2mm
$${\bf B}^t :=
\widetilde{U}_1^t \otimes\cdots\otimes \widetilde{U}_N^t  , \quad
{\bf C}^t := M_t^{(1)} \otimes\cdots\otimes M_t^{(N)}$$
\vskip-2mm
$$M_t^{(k)} :=
{\scriptstyle{\bigoplus\limits_{j\in J_k}}} \
\chi_j^{(k)}(t) {\rm Id}_{{\HHHx}_j^{(k)}} ,
\quad
\widetilde{U}_k^t :=
{\scriptstyle{\bigoplus\limits_{j\in J_k}}} \ \exp \big( it A_j^{(k)} \big)$$
with suitable families $[\chi_j^{(k)} : \ j\in J_k]$ of cont. functions
$\RR\!\to\!\TT$, respectively not necessarily bounded self-adjoint operators
$$A_j^{(k)} : {\bf D}_j^{(k)}\to{\HHHx}_j^{(k)} , \qquad
{\bf D}_j^{(k)} \ \hbox{dense linear submanifold}\subset {\HHHx}_j^{(k)}$$


\end{itemize}


}



\frame[<+->]{

\frametitle{FINISH OF THE PROOF}

\begin{itemize}

\item[]
$t\mapsto {\bf B}^t$, $t\mapsto {\bf C}^t$ \ str.cont. \qquad [products of bded str.cont. functions]
\vskip3mm
${\bf C}^{t+h} \!=\! {\bf U}(t+h) {\bf B}^{-t-h} \!=\!
%{\bf U}(t) {\bf U}(h) {\bf B}^{-t} {\bf B}^{-h} \!=\!
{\bf U}(t) {\bf B}^{-t} {\bf U}(h) {\bf B}^{-h} \!=\!
{\bf C}^t {\bf C}^h $

\item[]
\vskip5mm
\bf Lemma. \it
$\exists$ real constants
$a_j^{(k)}$ $(j\in J_k;\ k=1,\ldots,N)$ \ \ with
$${\bf C}^t =
\Big[ {\scriptstyle{\bigoplus_{j\in J_1}}}
\exp\big({i a_j^{(1)} t}\big) \, {\rm id}_{{\HHHx}_j^{(1)}} \Big]
\!\!\otimes\cdots\otimes
\Big[ {\scriptstyle{\bigoplus_{j\in J_N}}}
\exp\big({i a_j^{(N)} t}\big) \, {\rm id}_{{\HHHx}_j^{(N)}} \Big]$$

\item[]
\bf Conclusion. \it With \
$\widehat{A}_j^{(k)} \exp \big( it A_j^{(k)} \big):= A_j^{(k)} + a_j^{(k)} {\rm id}_{{\HHHx}_j^{(k)}}$, \
$$M_t^{(k)} \widetilde{U}_1^t \chi(t) = {\scriptstyle{\bigoplus\limits_{j\in J_k}}} \
\exp\big( it \widehat{A}_j^{(k)} \big)$$
implying
$${\bf U}(t) = \bigotimes_{k=1}^N \Big[ \oplus_{j\in J_k}
\exp\big( it \widehat{A}_j^{(k)} \big)\Big] \quad (t\in\RR).
\quad {\rm Qu.e.d.}$$


\end{itemize}

}

}%ENDDEL



\frame[<+->]{

\frametitle{INGREDIENTS OF PROOF IN HILBERT CASE}


\begin{itemize}
\vskip3mm
\item[1)] Reducibility of surjective isometries by a
topological direct sum of sparable subspaces.

\vskip3mm
\item[2)] Surjective linear isometries are scalar type operators.

\vskip3mm
\item[3)] Strong convergence of scalar type operators on separable subspaces
can be treated with representations by multiplications of random variables
with stochastically convergent functions.

\end{itemize}

}



\frame[<+->]{

\frametitle{PROBLEMS CONCERNING GENERALIZATIONS}

\bf Remark. \rm The proof works if
\it

\vskip3mm
\hskip7mm each \ $\XXX{k} =\big[ \HHH{k} \ \ \hbox{with an equivalent smooth norm}\big]$,

\vskip1mm
\hskip7mm {\bf +} the surjective linear isometries on each $\XXX{k}$ are of scalar type.


\vskip5mm
\bf Conjecture. \it If \ $\XXXx=\big[$Hilbert space with an equivalent norm$\big]$ then
any surjective linear isometry of $\XXXx$ is unitary with respect some scalar product.
\rm
\vskip1mm(In finite dimensions this is TRUE)

\vskip5mm
\bf Problem. \it Let $\XXX{1},\ldots,\XXX{N}$ be Banach spaces and
let $U$ be a surj. lin. isometry of $\mathcal{B}(\XXX{1},\ldots,\XXX{N})$.
Under which geometric hypothesis can we write it in the form
$U=\big[ U_1\otimes\cdots\otimes U_N \big]^*$ ?



}




\frame[<+->]{

\frametitle{CARTAN FACTORS}

Cartan factors: Types 1--6

\bigskip

\it Every $JB^*$-triple embeds into an $\ell^\infty$-product of Cartan factors
\rm

\bigskip

Type 1: ${\cal L}(\HHH1,\HHH2)$

Types 2,3: ${\cal L}^{({\rm T},\pm)}(\HHHx) =\{ L\in{\cal L}(\HHHx) :\ L=\pm L^{{\rm T}} \}$

Type 4: Spin factor on $\HHHx$, \quad $\{ {\bf x} {\bf a}{\bf y} \}:=
\langle {\bf x} \vert {\bf a}\rangle {\bf y} + \langle {\bf y} \vert {\bf a}\rangle {\bf x} -
\langle {\bf x} \vert \overline{{\bf y}}\rangle \overline{{\bf a}}$

Type 5: $\OO^{1\times 2}$ \quad\ \ $1\!\times\! 2$ complex octonion matrices (dim=16)

Type 6: ${\cal H}_3(\OO)$ \quad $3\!\times\! 3$ complex Hermitian octonion matrices (dim=27)

\bigskip\bigskip
$\hhh\mapsto\overline{\hhh}$ \ \ conjugation: \
$\overline{{\bf e}_\alpha}={\bf e}_\alpha$, $\overline{i{\bf e}_\alpha}=-i{\bf e}_\alpha$
for a fixed ON basis

$L\mapsto L^{{\rm T}}$ transposition (wrt. conjugation \ $\overline{ \cdot \hskip-4.5mm \phantom{Mk}}$):
$L^{{\rm T}} \hhh = \overline{L^* \overline{\hhh}}$

}


\frame[<+->]{

\frametitle{STR. CONT. 1PRGs of LIN. AUTs on FACTORS}

Factors of type $> 3$ are not interesting \qquad (str. cont. wrt. Hilbert norm)

\bigskip
Type 1: ${\cal L}(\HHH1,\HHH2) \simeq \{$2-lin. funct. $\HHH1\times\HHH2\to\CC\}$ $\longrightarrow$ Case $N=2$.

\bigskip
Types $k=2,3$:

\hskip 10mm
$\hbox{{\Goth A}}_k := \big\{$surjective lin. isom. of \ ${\cal L}^{({\rm T},(-1)^k)}(\HHHx) \ \big\}$

$$\hbox{ ${\cal L}^{({\rm T},(-1)^k)}(\HHHx) \simeq \Big\{ \
{\hbox{symm.} \atop \hbox{antisymm.}}$ \ \ 2-lin. $\HHHx\times\HHHx\to\CC$ funct. $\Big\}$ }$$


$${\bf U} \in \hbox{{\Goth A}}_2 \ \iff \ \exists\, U \in{\cal U}(\HHHx) \qquad
\underline{ {\bf U}: L\mapsto ULU^{{\rm T}} }$$

\bigskip
}



\frame[<+->]{

\frametitle{TYPES 2,3}


\bf Theorem. \it Let \ $k=2,3$, \quad
${\bf U}_k:\RR\to \hbox{{\Goth A}}_k$ \ str. cont. 1prg. \ \
Then
$$\exists\, A \ \hbox{\rm unbded. self-adj. $\HHHx$-op.} \qquad
{\bf U}(t) =  \big[ L \mapsto \exp(it A) L \exp(it
\underbrace{A^{{\rm T}}}_{\overline{A}} ) \big] $$
\bf Proof. \rm
${\bf U}_k : \underbrace{U_t}_{unit.} \otimes U_t^{{\rm T}} \big\vert F_k$

$\exists \sigma: \RR\to \{-1,1\} \qquad t\mapsto \sigma(t) U(t)$ str.cont.

SEP. CONT. + LOC. + PRB. $\longrightarrow$ Theorem.

}



\frame[<+->]{

\frametitle{GENERAL JB*-TRIPLES}

\begin{itemize}

\item[]
$E$ JB*-triple \quad $E \hookrightarrow {\rm Atomic} \big( E^{**} \big) =
\bigoplus_{k\in\mathcal{K}} \underbrace{F_k}_{Cartan}$
\vskip-2mm
${\bf U}: \RR\to {\rm Aut}(E)$ str. cont, 1prg.

$\Pi_k: E\to F_k$ canonical proj.

\vskip5mm

\item[]
$t\mapsto {\bf U}(t)^{**} \circ \Pi_k$
$\underbrace{\hbox{str. cont.}}_{???}$ 1prg. $\RR\to{\rm Aut}(F_k)$


\vskip2mm
\item[]
If YES , Gelfand-Neumark description for str.cont. 1prg of ${\rm Aut}(E)$.

\DEL{
$t\mapsto {\bf U}(t)^{**} \big\vert \Pi_k E$ str.cont. 1prg. $\RR\to {\rm Aut}(\Pi_k E)$

$\Pi_k(E)$ is a weak*-dense JB*subtriple of $F_k$.
}%ENDDEL
\vskip5mm

\item[]
\bf Problems. {\rm (1)} \it $E$ w*-dense JB*-subtriple in $F$ (Cartan factor),
$t\mapsto {\bf U}(t)$ str.cont. 1prg. $\RR\to{\rm Aut}(E)$ $\Longrightarrow^?$
$t\mapsto {\bf U}^{w*}(t)$ str.cont.?

{\rm (2)} \it Category description of w*-dense JB*-subtriples of Cartan factors

\vskip2mm
\item[]
\bf Conjecture. [$\sim$\rm 2003 Isidro-Stacho] $E$ w*-dense JB*-subtrp in factor Type 1
$\iff$ $\exists E_0\subset E \ w*-dense \ {\rm TRO}$ (ternary ring of operators)


\end{itemize}

}




\frame[<+->]{

\frametitle{PROBLEMS IN NON-LINEAR CASE}

\begin{itemize}

\item[]
$N<3$: \qquad $\HHHx,\mathcal{L}(\HHH1,\HHH2)$ \ \ JB*-triples

\vskip3mm

\item[]
$E$ \ JB*-triple \qquad $B:={\rm Ball}(E)$

\vskip3mm

$t\mapsto \Phi^t \in{\rm Aut}(B)$ \quad str.cont 1-prg: \qquad
$t\mapsto\Phi^t(x)$ norm.cont. $\forall\, x$

\vskip3mm

\item[]
$\Phi^t = M_{\Phi^t(0)} \circ \hskip-10mm \underbrace{U_t}_{\hskip7mm{\rm LIN}\in{\rm Aut(B)}}$
\hskip10mm
$M_a(x)=a+\hskip-7mm{\underbrace{\beta(a)}_{I-2D(a)+Q(a)^2}}^{\hskip-7mm 1/2} [I-D(x,a)]^{-1} x$

\vskip3mm

\item[] \bf Lemma. \it $t\mapsto\Phi^t$ str.cont. $\iff$ \ \
$t\mapsto \Phi^t(0)$ cont., $t\mapsto U_t$ str.cont.
\rm

\vskip5mm
\item[]
\bf Question: \it
$\Phi^t \buildrel{?}\over= \exp \big[ t\big( a \!-\! \{ aa^*x\} \!+\! Ax\big)\partial/\partial x\big]$ \quad
$A$ unbded.$\{...\}$-der.



\end{itemize}

}



\frame[<+->]{

\frametitle{EXAMPLE IN HILBERT SPACE}

\begin{itemize}

\item[]
%$E=\HHHx =L^2(\TT) \equiv L^2({\rm mod}_{2\pi} \RR)$ \quad periodic funct.
$E:=L^2(\RR)$ \quad
%
%\vskip3mm
%
$S^t: f\!\mapsto\! f(x \!+\! t)$ \quad shifts \hskip8mm INF.GEN: \ $A:=\frac{d}{dx}$

\vskip3mm

\item[]
Consider: \ \ $\Phi^t = \exp \Big[ t\big( a \!-\! \{ aa^*f\} \!+\! Af\big)\frac{\partial}{\partial f}\Big]$
%\quad $\displaystyle{a \!=\!\! \sum_{k=-n}^n \!\! \alpha_k e^{ikx}}$
\quad $\Phi^t(0): x\mapsto f(t,x)$

%\vskip3mm
\item[]
%\bf Ex.: $a:=\cos x$, \quad $\Phi^t(0): x\mapsto f(t,x)$
$$\frac{\partial f}{\partial t} = a - \!
\int_{-\pi}^\pi  \!\! f(t,\xi) \overline{a(\xi)} \, d\xi \cdot f + \frac{\partial f}{\partial x}
\qquad f(0,x)\equiv 0$$

\item[]
$y\!:=\! x\!+\! t$, $u\!:=\! t$, $f(t,x)\!=\! g(y,u)$, $b(u,y)\!=\! a(y\!-\! u)$ \ $\Rightarrow$

$$\frac{\partial g}{\partial u} = b(\underbrace{x-u}_{retard.}) - \!
\int_{-\pi}^\pi  \!\! g(u,\eta) \overline{b(\eta)} \, d\eta \cdot g , \qquad
g(0,y)\equiv 0$$


\end{itemize}


}



\bf Reference

\rm L.L. STACH\'O,

\it
On strongly continuous one-parameter groups of

automorphisms of multilinear functionals,

\rm
J. of Math. Anal. and Appl., {\bf 363/2}, 419-431 (2010)

\vskip10mm
\bf Acknowledgement

\rm
Supported by the project

\it TAMOP-4.2.2.A-11/1/KONV-2012-0060  -

%Creating the Center of Excellence at the University of Szeged
%Impulzuslézerek alkalmazása az anyagtudományban és a biofotonikában
Impulse Lasers for Use in Materials Science and Biophotonics

\rm
of the European Union and co-financed by the European Social Fund.

%\vskip20mm

%\bf MANY THANKS FOR YOUR ATTENTION




\end{document}
