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\def\ox{\overline{x}}

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\title{
On non-commutative Minkowski spheres
}

\author{L. L. STACH\'O \ and \ W. WERNER }

\date{29/08/2011, Constanta}

\centerline{Work in progress}

\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{MOTIVATION}

$M := \big\{ (t,x) \in \RR\times\RR^3 : \ t^2 - \Vert x\Vert^2 =1,\  t\ge 1 \big\}$

\

$M =\big[$MINKOWSKI SPHERE in upper light cone$\big]$

\

$\RR^3 \equiv {\rm Mat}(1,3,\RR), \quad \Vert a\Vert = [a a^{{\rm T}} ]^{1/2}$

\

$$\phi(a) =\left[
\begin{matrix}
\phi_1(a)\\ \phi_2(a) \end{matrix} \right]  \qquad
\big( a\in\RR^3, \ \Vert a\Vert\ne 1 \big)$$

$$\phi_1(a) := \frac{1+\Vert a\Vert^2}{ 1-\Vert a\Vert^2} \ ,\quad
\phi_2(a) := 2 (1- \Vert a\Vert^2 a)^{-1} a  \ .$$

\

\bf Observation:\rm \  $\phi: {\rm Ball}(\RR^3) \leftrightarrow M$ \qquad\qquad by MAPLE (simplify)


}




\frame[<+->]{

\frametitle{MOTIVATION}

$$\phi^{-1}(t,x,y,z)=
\left\{
\begin{array}{ll}
\frac{r}{1 + \sqrt{1+r^2}} \big( x,y,z) \\
\ \ \phantom{{\int^k}}\hbox{if} \ \ (t,x,y,z)\ne(1,0,0,0)\\
& \phantom{x}\\
(0,0,0) \\
\ \ \phantom{{k^*}}\hbox{if} \ \ (t,x,y,z)=(1,0,0,0) \quad
\end{array}
\right.
$$

\

\

where \ \  $r^2 := x^2 + y^2 + z^2$


}



\frame[<+->]{

\frametitle{ALTERNATIVE VIEW}

\begin{itemize}

\item[]
${\rm Mat}(3,3,\RR)\times\RR^3$ instead of $\RR\times\RR^3$

\

\item[]
$M := \big\{ (t,x) \in \underbrace{{\rm Mat}(3,3,\RR)_+}_{pos.self-adj.op}\times\RR^3 : \
t^2 - \Vert x\Vert^2 =1,\  t\ge I \big\}$

\

\item[]

$$\phi(a) =\left[
\begin{matrix}
\phi_1(a)\\ \phi_2(a) \end{matrix} \right]  \qquad
\big( a\in\RR^3, \ \Vert a\Vert\ne 1 \big)$$


\begin{eqnarray*}
& &\phi_1(a) := \big( I+ a^{{\rm T}} a \big) \big( I-a^{{\rm T}} a \big)^{-1} \ ,\\
& & \phantom{x} \\
& &\phi_2(a) := 2a \big( I-a^{{\rm T}} a \big)^{-1} \ .
\end{eqnarray*}


\end{itemize}

}




\frame[<+->]{

\frametitle{ALTERNATIVE VIEW}

{\bf Observation}

\begin{eqnarray*}
\phi_1(a)^2 - & &\hskip-10mm \phi_2(a)^{{\rm T}} \phi_2(a) =\\
&=& \underbrace{( I - a^{{\rm T}} a)^{-1}}_{} \ (I + a^{{\rm T}}a)^2 \
\underbrace{(I - a^{{\rm T}} a)^{-1}}_{} -\\
& &\hskip-2mm
- \underbrace{(I - a^{{\rm T}} a)^{-1}}_{} \ \ (4 a^{{\rm T}} a) \ \ \ \underbrace{(I - a^{{\rm T}} a)^{-1}}_{} =\\
%&=& (I - a^{{\rm T}} a)^{-1} \big[ I + 2 a^{{\rm T}} a + (a^{{\rm T}} a)^2 - 4 a^{{\rm T}} a \big] (I - a^{{\rm T}} a)^{-1} =\\
%&=& (I - a^{{\rm T}} a)^{-1} \big[ I - 2 a^{{\rm T}} a + (a^{{\rm T}} a)^2 \big] (I - a^{{\rm T}} a)^{-1} =\\
%&=& (I - a^{{\rm T}} a)^{-1} \big[ I - 2 a^{{\rm T}} a + (a^{{\rm T}}a)^2 \big] (I - a^{{\rm T}} a)^{-1} =\\
&=& (I - a^{{\rm T}} a)^{-1} ( I - a^{{\rm T}} a )^2 (I - a^{{\rm T}}a)^{-1} =\\
&=& I \ .
\end{eqnarray*}

}



\frame[<+->]{

\frametitle{NON-COMMUTATIVE SCALARS}

Alternative case: \ $\langle a \vert a \rangle := a^{{\rm T}} a \in {\rm Mat}(3,3,\RR)$

\

{\bf Hilbert $C^*$-modules}

\

$E$ vector space, \ ${\cal A}$ $C^*$-algebra with $1$\ \ (over $\RR,\CC$)

\

\hskip 5mm
$\left\{
\genfrac{}{}{0pt}{}{\hbox{$\alpha x\in E$ LEFT}}{\hbox{\ $x\alpha\in E$ RIGHT}} \right\}$ \quad
${\cal A}$-module structure on $E$

\

$\langle x\vert y\rangle \in {\cal A}$ non-comm scalar prod.

\

$\langle y\vert x\rangle = \langle x\vert y\rangle^*$ ,
\quad
$\left\{
\genfrac{}{}{0pt}{}{\hbox{$\alpha \langle x\vert y\rangle = \langle \alpha x\vert y\rangle$}}
{\hbox{$\langle x\vert y\rangle \alpha = \langle x\alpha \vert y\rangle$}}
\right.$

\

\bf Example. \ \rm ${\rm Mat}(1,3,\RR) \sim {\cal L}(\RR^3,\RR)$ \quad
\vtop{
\hbox{Motivation$\to$left \ \ ${\cal A}={\rm Mat}{1,1,\RR}\equiv\RR$,}
\hbox{Alternative$\to$right \ \ ${\cal A}={\rm Mat}{3,3,\RR}$, \
${\rm Mat}(1,3,\RR)\hookleftarrow{\rm Mat}(3,3,\RR)$.}}

}



\frame[<+->]{

\frametitle{TRO SETTING}

{\bf T}ernary {\bf R}ing of {\bf O}perators

\

$H$ Hilbert space \qquad ${\cal T} \subset{\cal L}(H)$ \quad subspace preserving $xy^* z$

\

TRO = Hilbert $C^*$-module

\

\hskip10mm gen. constants: ${\cal A}_{left} =\{ x^*y: x,y\in{\cal T}\}^{C_1^* span}$ \quad ${\cal A} {\cal T}\subset {\cal T}$

\hskip36mm ${\cal A}_{right} =\{ xy^*: x,y\in{\cal T}\}^{C_1^* span}$ \quad ${\cal T}{\cal A} \subset {\cal T}$

\

\

\bf Remark. \rm Enveloping TRO: \ ${\cal L}(H)$.
\quad
IT SUFFICES
\begin{eqnarray*}
& &\phi(a) = \left[ \genfrac{}{}{0pt}{}{\phi_1(a)}{\phi_2(a)}\right] \qquad \big( a\in{\rm Ball}({\cal L}(H))\big)\\
& &\phi_1(a):= (1+a^*a)(1-a^*a)^{-1}, \ \ \phi_2(a):=2a(1-a^*a)^{-1}
\end{eqnarray*}

}



\frame[<+->]{

\frametitle{TRO SETTING}

\bf Proposition. \it $\phi: {\rm Ball}\big( {\cal L}(H)\big) \leftrightarrow M$ \ \ where
$$M:=\big\{ [t,x]\in{\cal L}(H)^2 : \ \underbrace{t\ge 0}_{pod.def},\ t^2 - x^* x =1 \big\} .$$
moreover \qquad
$\phi^{-1} \left[ t,x \right] = x(1+t)^{-1}$

\

\

\bf Corollary. \it
If $\big( E,{\cal A}, \langle . \vert . \rangle\big)$ is a right $[$left$]$ Hilbert $C^*$-module,

\

\hskip5mm $\phi_1(a):=(1+\langle a\vert a\rangle) (1-\langle a\vert a\rangle)^{-1}$,

\hskip5mm $\phi_2(a):=2a(1-\langle a\vert a\rangle)^{-1}$ $\big[$resp. $2(1-\langle a\vert a\rangle)^{-1}a \big]$

\

then \quad $[ \phi_1,\phi_2 ]: {\rm Ball}(E) \leftrightarrow
\big\{ [t,x]\in{\cal A}_0\times E: \ t^2-\langle x\vert x\rangle =1\big\}$

}




\frame[<+->]{

\frametitle{GEOMETRIC ISSUE (motivation)}

\bf Example. \rm $H:=\CC$, \quad
$\phi(a)=\left[\frac{1+\vert a\vert^2}{1+\vert a\vert^2}, \frac{2a}{1+\vert a\vert^2}\right]$ \ $\Longrightarrow$
$$\big[ \phi_1^\prime(a)u \big]^2 - \big\vert \phi_2^\prime(a)u \big\vert^2 =
-4 \frac{\vert u\vert^2}{[1 -\vert a\vert^2\big]^2} \ .$$
Riemann(-Poincar\'e-Bergman) metric on $B:={\rm Ball}(\CC)$ is given by
$$\langle u\vert u \rangle_a^B :=
\vert g_a^\prime(0)^{-1} u \vert^2 =
\frac{\vert u\vert^2}{\big[ 1-\vert a\vert^2 \big]^2}
\qquad (a\in B,\ u\in\CC)$$
$g_a(z):=(z+a)/(1-\overline{a} z)$ \ M\"obius transformation.

\

\bf Observation. $\Psi:=\phi^{-1}$, $\Longrightarrow$
$$\widetilde{t}^2 - \vert \widetilde{x} \vert^2 =
-4 \Big\langle \big[\Psi^\prime(t,x)\big](\widetilde{t},\widetilde{x}) \Big\vert
\big[\Psi^\prime(t,x)\big](\widetilde{t}, \widetilde{x}) \Big\rangle_{\Psi(t,x)}^B .$$


}



\frame[<+->]{

\frametitle{GEOMETRIC ISSUE (TRO setting)}

$\phi: B \!:=\!{\rm Ball}\big({\cal L}(H)\big) \!\leftrightarrow\! M \!:=\!\{ (t,x):\, t\!\ge\! 0, \ t^2 \!-\! x^*x \!=\!1\}$,
\quad $\Psi:=\phi^{-1}$

$g_a(z):= (1-aa^*)^{-1/2}(z+a)(1-a^*z)^{-1}(1-a^*a)^{-1/2}$ \quad M\"obius trf.
\begin{eqnarray*}
\langle u \vert v\rangle_a^B &:=&
\big[[g_a^\prime(0)]^{-1} u\big]^* \big[ [g_a^\prime(0)]^{-1} v\big] =\\
&=& (1-a^*a)^{-1/2} u^* (1-aa^*)^{-1} v (1-a^*a)^{-1/2} .
\end{eqnarray*}
\bf Theorem. \it For \ $(\ot,\ox)\in T_{(t,x)} M$ \ we have \
$\ot t + t\ot = \ox^* x + x^* \ox$ \ and \rm

$\langle (\ot,\ox) \vert (\ot,\ox) \rangle_{(t,x)}^M :=
-4 \Big\langle \big[ \Psi^\prime(t,x)\big](\ot,\ox) \Big\vert \Psi^\prime(t,x)\big](\ot,\ox) \Big\rangle_{\Psi(t,x)}^B =$

$(I+t)^{1/2} \Big\{
2\big[ \ot^2 - \ox^* \ox\big] - \Big[ \ox^* x(I+t)^{-1} x^* \ox -
\ot x^* \ox - \ox^* x \ot + \ot (t-I) \ot \Big]
\Big\} (I+t)^{1/2}$
%For \ $(t,x)\in M,\ (\ot,\ox)\in T_{(t,x)} M$ \ we have
%$$\phi^{\#} \Big\langle (\ot,\ox)\Big\vert (\ot,\ox) \Big\rangle_{(t,x)} \! :=\! -4
%\Big\langle [\Psi^\prime(t,x)](\ot,\ox)\Big\vert
%[\Psi^\prime(t,x)](\ot,\ox) \Big\rangle_{\Psi(t,x)}^B$$

\

{\bf Problem}. \rm Find "smart" {\it indefinite} inner products \
$\langle .\vert . \rangle_{(t,x)}$ \
on \
${\cal L}(H)^2$

\hskip18mm extending \ \
$\phi^{\#} \langle . \vert . \rangle_{(t,x)}$ \ \ from \ \ $T_{t,x} M$

}



\frame[<+->]{

\frametitle{GEOMETRIC ISSUE (further motivation)}

\bf Remark. \rm
In the 1-dimensional case $H=\CC$,
we have
$$\phi^\# g_b \left[ \genfrac{}{}{0pt}{}{t}{x}\right] =
\phi\circ g_b \circ \phi^{-1} \left[ \genfrac{}{}{0pt}{}{t}{x}\right] =
\left[{
\genfrac{}{}{0pt}{}{\frac{1+\vert b\vert^2}{1-\vert b\vert^2} t +
\frac{2b}{1-\vert b\vert^2} x}{\frac{2b}{1-\vert b\vert^2} t +
\frac{1+\vert b\vert^2}{1-\vert b\vert^2} x} } \right] ;
\leqno{1)}$$
2) \ $g_{{\rm tanh}(a)}$ is the exponential of
the vector field
$[a-\overline{a}u^2] \frac{\partial}{\partial u}$.
Hence
$$\phi^\# g_{{\rm tanh}(a)} \left[ \genfrac{}{}{0pt}{}{t}{x}\right] =
\left[\genfrac{}{}{0pt}{}{ {\rm cosh}(2a) t + {\rm sinh}(2a) x }{ {\rm sinh}(2a) t + {\rm cosh}(2a) x }
\right]
\qquad{\rm for} \quad
\left[ \genfrac{}{}{0pt}{}{t}{x}\right] \in M \ .$$
In particular
$\phi^\# g_b$ is the restriction of a
linear mapping of $\CC^2$ to $M$.

}


\frame[<+->]{

\frametitle{GEOMETRIC ISSUE (TRO setting)}

\bf Notation. \rm \ \
$A_a(u) := \big[ a-ua^*u\big]\frac{\partial}{\partial u}$ \ \
$\big( a\in {\cal L}(H)\big)$

\

For fixed
$a\in{\cal L}(H)$ and $u\in{\rm Ball}{\cal L}(H)$,

$u_\tau := [\exp(\tau A_a)]u$ $(\tau\!\in\!\RR)$ \ \
satisfies \ \
$\frac{d}{d\tau} u_\tau \!=\! A_a(u_\tau)$, $u_0 \!=\! u$.

\

Well-known: \
$b=[\exp(A_a)]0 \ \Rightarrow \ g_b(u)=[\exp(A_a)]u$ \
$(\Vert u\Vert \!<\! 1)$.

\

\bf Proposition. \it
Let $a\in{\cal L}(H)$ and $\left[ \genfrac{}{}{0pt}{}{t}{x}\right]\in M$.
Then
$$\phi^\# A_a \left[ \genfrac{}{}{0pt}{}{t}{x}\right] =
\left[
\genfrac{}{}{0pt}{}{a^*x+x^*a}{a(1+t)+x(1+t)^{-1}x^*a}
\right] \ .$$


}


\frame[<+->]{

\frametitle{BEYOND TRO: Self-adjoint operators}

${\cal A}={\cal A}(H) \!=\! \{ a\!\in\! {\cal H}:\, a^* \!=\! a\}$ \ \ REAL JORDAN ALGEBRA in ${\cal L}(H)$
$$\phi^\# A_a \left[ \genfrac{}{}{0pt}{}{t}{x}\right] =
\left[ \genfrac{}{}{0pt}{}{a x + x a}{a t + t a} \right] =
\left[ \begin{matrix} 0 &L_a +R_a \\ L_a+R_a &0 \end{matrix} \right]
\left[ \genfrac{}{}{0pt}{}{t}{x}\right]$$
%where \ \
%$L_a : {\cal L}(H) \ni z \mapsto az$, \ $R_a : {\cal L}(H) \ni z \mapsto za$.
\bf Theorem. \it
$a\in{\cal A} \ \Rightarrow$ \ $1)$
$g_{{\rm tanh}(a)}: {\rm Ball}({\cal A}) \leftrightarrow {\rm Ball}({\cal A})$,

\

$2) \ \phi: {\rm Ball}({\cal A}) \!\leftrightarrow\!
M\cap {\cal A}^2 = \big\{ \big[ \genfrac{}{}{0pt}{}{t}{x}\big] \in{\cal A}_+\times{\cal A}: \ t^2 \!-\! x^2\!=\! 1 \big\}$

\

$3) \ \phi^\# g_{{\rm tanh}(a)} \! =\!
\phi\circ [\exp A_a] \circ \phi^{-1} \left[ \genfrac{}{}{0pt}{}{t}{x}\right] \!=\!
\left[
\genfrac{}{}{0pt}{}{ {\rm cosh}(L_a+R_a) t + {\rm sinh}(L_a+R_a) x }
{ {\rm sinh}(L_a+R_a) t + {\rm cosh}(L_a+R_a) x }
\right]$
%For the terms
%${\rm cosh}(L_a+R_a) z$, ${\rm sinh}(L_a+R_a) z$
%$(z=t,x)$
%here we have

\

\begin{eqnarray*}
& &{\rm cosh}(L_a+R_a) z =
\frac{1}{2} [\exp (a)] z [\exp(a)] +
\frac{1}{2} [\exp (-a)] z [\exp(-a)] \ , \\
& &{\rm cosh}(L_a+R_a) z =
\frac{1}{2} [\exp (a)] z [\exp(a)] -
\frac{1}{2} [\exp (-a)] z [\exp(-a)] \ .
\end{eqnarray*}

}



\frame[<+->]{

\frametitle{BEYOND TRO: JB*-triples}

\bf Observation. \rm \ $\phi_1(a) = 1+2 a^* \phi_2(a)$ \quad in TRO case

\


$\big( E,\Vert .\Vert,\{ \ldots \} \big)$ \ JB$^*$-triple: \ \
$\{ \ldots\}:E^3\to E$, \ $D(a,b):x\mapsto\{ abx\}$

\

1) $\{ xyz\}$ \ symmetric bilinear in $x,z$, \ antilinear in $y$

2) Jordan-identity for \ $\{ \ldots\}$

3) $\Vert \{ aaa \} \Vert = \Vert a\Vert^3$

4) $\big\Vert \exp\big( \zeta D(a,a)\big)\big\Vert \le 1$ \ if \ ${\rm Re}(\zeta) \le 0$

\

Canonical JB$^*$-product on ${\cal L}(H)$: \ $\{ xyz \}:= \frac{1}{2} xy^*z + \frac{1}{2} zy^*x$

\

$\phi_2(a):= 2\big[ 1 - D(a,a) \big]^{-1} a$, \ \
$\phi_1(a) := 1 + \underbrace{D\big( \phi_2(a),a\big)}_{= D\big( a,\phi_2(a)\big)} \in{\cal L}(E)$

\

\bf Proposition. \it $\phi^{-1} \left[ \genfrac{}{}{0pt}{}{t}{x} \right] = (1+t)^{-1} x$ \ for \
$\left[ \genfrac{}{}{0pt}{}{t}{x} \right] \in {\rm range}(\phi)$


}

\end{document}



Find a convenient indefinite inner product \
$\langle (\ot_1,\ox_1), (\ot_2,\ox_2) \rangle_{(t,x)}$ \
on
${\cal L}(H)\times {\cal L}(H)$
such that \quad
$\langle (\ot,\ox), (\ot,\ox) \rangle^M_{(t,x)} :=
\langle (\ot,\ox), (\ot,\ox) \rangle_{(t,x)}$
for $(t,x)\in M$ and $(\ot,\ox)\in T_{(t,x)} M$ \quad
with the inner product
$$\langle (\ot,\ox), (\ot,\ox) \rangle^M_{(t,x)} \! :=\! -4
\big\langle [\Psi^\prime(t,x)](\ot,\ox),
[\Psi^\prime(t,x)](\ot,\ox) \big\rangle_{\Psi(t,x)}^B
\ \
{\rm for} \ (t,x)\in M, \ (\ot,\ox) \in T_{(t,x)} M$$
with the natural Riemannian inner product on
the unit ball $B := \{ a: \ a^*a<I\}$,
$$\eqalign{
\langle u,v\rangle_a^B &:=
\big[[g_a^\prime(0)]^{-1} u\big]^* \big[ [g_a^\prime(0)]^{-1} v\big] =\cr
&= (I-a^*a)^{-1/2} u^* (I-aa^*)^{-1} v (I-a^*a)^{-1/2} .\cr}$$


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


$$\eqalign{
\phi_1^\prime(a)x &=
(a^*x + x^*a)(I-a^*a)^{-1} + (I+a^*a)(I-a^*a)^{-1} (a^*x + x^*a) (I-a^*a)^{-1} =\cr
&= (I-a^*a)^{-1} \big[ (I-a^*a) (a^*x + x^*a) +
(I+a^*a) (a^*x + x^*a) \big] =\cr
&= (I-a^*a)^{-1} (a^*x + x^*a) (I-a^*a)^{-1} \ .\cr
\phi_2^\prime(a)x &=
 2x (I-a^*a)^{-1} + 2 \underbrace{a (I-a^*a)^{-1}}_{(I-aa^*)^{-1} a}
(a^*x + x^*a) (I-a^*a)^{-1} = \cr
&= 2 (I-aa^*)^{-1}\big[ (I-a^*a)x + a(a^*x + x^*a) \big] (I-a^*a)^{-1} =\cr
&= 2 (I-aa^*)^{-1} [ x + ax^* a ] (1- a^*a)^{-1} \ \cr}$$

\noindent
Thus we have

$$\eqalign{
&{\rm range}(\phi) = M :=
\{ (t,x) \in {\cal L}(H)\times {\cal L}(H):
\ t=t^* \ge I, \
t^2 - x^*x =I\} \ ,\cr
&\phi : B:= \{ a:\ a^*a< I\} \leftrightarrow
M:=\{ (t,x):\ t=t^* \ge I, \ t^2 - x^*x =I\} , \cr
&\Psi := \phi^{-1} : (t,x) \mapsto x(I+t)^{-1} \cr}$$

\noindent
{\bf Aim.} Find a convenient indefinite inner product \
$\langle (\ot_1,\ox_1), (\ot_2,\ox_2) \rangle_{(t,x)}$ \
on
${\cal L}(H)\times {\cal L}(H)$
such that \quad
$\langle (\ot,\ox), (\ot,\ox) \rangle^M_{(t,x)} :=
\langle (\ot,\ox), (\ot,\ox) \rangle_{(t,x)}$
for $(t,x)\in M$ and $(\ot,\ox)\in T_{(t,x)} M$ \quad
with the inner product
$$\langle (\ot,\ox), (\ot,\ox) \rangle^M_{(t,x)} \! :=\! -4
\big\langle [\Psi^\prime(t,x)](\ot,\ox),
[\Psi^\prime(t,x)](\ot,\ox) \big\rangle_{\Psi(t,x)}^B
\ \
{\rm for} \ (t,x)\in M, \ (\ot,\ox) \in T_{(t,x)} M$$
with the natural Riemannian inner product on
the unit ball $B := \{ a: \ a^*a<I\}$,
$$\eqalign{
\langle u,v\rangle_a^B &:=
\big[[g_a^\prime(0)]^{-1} u\big]^* \big[ [g_a^\prime(0)]^{-1} v\big] =\cr
&= (I-a^*a)^{-1/2} u^* (I-aa^*)^{-1} v (I-a^*a)^{-1/2} .\cr}$$






\frame[<+->]{


\frametitle{PROBLEM SETTING}


$H$ Hilbert space

\

$E:={\cal L}(H)$

\

$M:= \big\{ \big[{t\atop x}\big]\in E_+\times E:\
t^2- x^* x =1, \ t\ge 1\big\}$

\

$\phi:{\rm Ball}(E) \leftrightarrow M$


$$\phi(u) := \left[
{{(1+a^*a)(1-a^*a)^{-1}}
\atop
{2a(1-a^*a)^{-1}}} \right]
\quad{\rm with}\quad
\phi^{-1} \left[ {t\atop x}\right] =
x/(1+t) \ .$$

}



\frame[<+->]{

\frametitle{}

}




\end{document}

\frame[<+->]{
\frametitle{TOOLS OF PROOF}

1) Ternary rings of operators (TRO)

$$\vtop{
\hbox{Hilbert C*-modules \ $\simeq$ \ TRO}
\hbox{\hskip 21mm $\langle x \vert y \rangle z \ \simeq \ x y^* z$ }} $$

\

2) Potapov's  transformations (in TRO)

$$M_a (x) := (1-aa^*)^{-1/2}
\underbrace{(x+a)(1-a^*x)^{-1}}_{\hbox{M\"obius}}
(1-a^*a)^{-1/2} $$

\

Operator calculation for $M_a^{\#} \big[ dY(X) \big] M_{-a}^{\#}$ \ at $a$

}



\frame[<+->]{

\frametitle{JORDAN STRUCTURE OF BOUNDED SYMMETRIC DOMAINS}

$Z$ complex Banach space, \quad $0 \in D \subset Z$ bded symm domain

\

1983 Kaup:
\vtop{
\hbox{Renorm $Z$ with $\gamma_0^D$ \ (infinitesimal Carath\'eodory norm)}
\hbox{$D \sim {\rm Ball}(Z)$ \hskip 11mm (biholomorphic eqivalence)}}

\

{\bf JB*-triple} \quad  $(Z,\{\ldots\})$

\begin{itemize}

\item[] $\{ xyz \}$ \ symm bilin. in  $x,z$, \ conj-lin. in $y$

\item[] $\{ ab \{ xyz \}\} =
\{\{ abx\} yz\} - \{ x \{bay\} z\} + \{ xy \{ abz\}\}$
\quad (Jordan)

\item[] $\Vert \{ xxx \} \Vert = \Vert x \Vert^3$ \hskip 62mm (C*)

\item[] $D(a) := [ x \mapsto \{ aax \} ] \in {\rm Her} (Z,\Vert . \Vert)$

\end{itemize}

}


\frame[<+->]{

\frametitle{BASIC INFINITESIMAL JB*-TRANSLATIONS}

$S_a := [\hbox{Symmetry of ${\rm Ball}(Z)$ at $a$}]$

$X_a(x) := \lim_{\tau\to 0+}\big(  S_{\tau a/2} S_0 (x) \big)$

$$X_a = [a - \{ xax \}] {\partial\over\partial x} \quad
\hbox{{\bf complete} vector field in} \ {\rm Ball}(Z)$$


$$\Phi_a := \exp ( X_a )
\in {\rm Aut}({\rm Ball}(Z))  \quad \hbox{FLOW}$$

$${d\over d\tau} \Phi_{\tau a} (x) = X_a \big( \Phi_{\tau a} (x) \big) $$

\

$$\hbox{Example:
\vtop{
\hbox{$Z:=\CC$, $\{ xay \} := x\overline{a} y$,  }
\hbox{Let $a\in[-1,1]$, \ \  $X_a(x) = a- ax^2$ }
\hbox{}
\hbox{$\Phi_{\tau a}(x) =
\displaystyle{ x+\tanh \tau a \over 1-x \tanh \tau a } $ }}
}$$

}



\frame[<+->]{

\frametitle{JB*-M\"OBIUS TRANSFORMATIONS}

$c(a) := \Phi_a(0)$

$c: Z \leftrightarrow {\rm Ball}(Z)$ \ \ bianalytic


\

{\bf DEF.} \ \ $M_{\Phi(a)} := \Phi_a$ \qquad
[Example: $Z=\CC$ $\Rightarrow$ $M_p(x) ={{x+p}\over {1-\overline{p} x}}$]

$$\Psi \in{\rm Aut}(Z) \Rightarrow
\exists!\ p\in {\rm Ball}(Z),\ U\in{\rm Unitary}(Z) \quad
\Psi = M_p \circ U$$

\

Kaup 1983:

$$M_p (x) = p + B(p)^{1/2} [1 - D(x,p) ]^{-1} x $$

\

$B(p) := 1 - 2 D(p) + Q(p)^2$ \quad BERGMAN op.

$D(x,p): y\mapsto \{ xpy \}$ \ polarized der. \quad

$Q(p): x\mapsto \{ pxp\}$ \ quadratic repr.

}


\frame[<+->]{

\frametitle{INVARIANT CONNECTION ON A JB*-BALL}

{\bf THEOREM.} $(Z,\{\ldots\})$ complex JB*-triple $\Longrightarrow$

There is a unique ${\rm Aut}$-invariant connection on ${\rm Ball}(Z)$.

Its Christoffel symbol is

$$\Gamma(p) (X,Y) =
2 B(p)^{1/2} \big\{ [ B(p)^{-1/2} X(p) ] p [ B(p)^{-1/2} Y(p) ]  \big\}$$

\

Interpretation: \ $\nabla_X Y = dX(Y) - \Gamma(X,Y)$

If $X,Y$ are represented by $f,g: G \to Z$
in a local coordinate $\xi: G \to {\rm Ball}(Z)$ \
then $\nabla_X Y$ is represented by
$$v\mapsto
\vtop{
\hbox{$f^\prime(v) g(v) -$}
\hbox{$[\xi^{-1}]^\prime \big( \xi(v) \big)
\Big[
\Gamma\big( \xi(v) \big) \Big( \xi^\prime(v) f(v) , \xi^\prime(v) g(v) \Big)
\Big]$}}$$

}



\frame[<+->]{

\frametitle{INGREDIENTS OF PROOF}

${\rm Ball}(Z)$ symmetric $\Longrightarrow$ at most one Aut-invariant connection

\

$\Gamma(0) \equiv 0$

\

Trasformation rule: \ \ $M_p:0\mapsto p$, \ $[M_p]^{-1} = M_{-p}$,
$$
\Gamma(p) \big(x,y \big)
\vtop{
\hbox{$ = M_p''(0)(M_{-p}'(0)x,M_{-p}'(0)y)$}
\hbox{$ = 2B(p)^{1/2} \{ [M_{-p}'(0)x] p [M_{-p}'(0)y] \}$,}
\hbox{$ = 2B(p)^{1/2}\{ [B(p)^{-1/2}x] a [B(p)^{-1/2}y] \}$}} $$

$$M_p^\prime(x) = B(p)^{1/2} B(x,-p)^{-1}$$

$B(x,b) := 1 - 2 D(x,b) + Q(x)Q(b)$ \ \ polarized Bergman op.

}


\frame[<+->]{

\frametitle{GEODESICS OF $\nabla$}

$\gamma : I \to {\rm Ball}(Z)$ is $\nabla$-geodesic:
$$\ddot\gamma (t) +
2 B\big( \gamma(t)\big)^{1/2}
D\Big(
B\big( \gamma(t)\big)^{-1/2} \dot\gamma(t) , \gamma(t) \Big)
B\big( \gamma(t)\big)^{-1/2} \dot\gamma(t) = 0  $$

$\gamma_{p,v} :$ \ \ $\nabla$-geodesic with
$\gamma_{p,v}(0)=p$, \ \ $\dot\gamma(0)=v$

\

\

{\bf Theorem.} $\gamma_{p,v} (t) =
M_p \big( T {\rm artanh} ( t \ {\rm id}_\Omega ) \big)$ \

\

where

\

$T: {\rm Re} {\cal C}_0(\Omega) \to Z$ \ Gelfand repr.

$T ({\rm id}_{\Omega}) = M_p^\prime(0)^{-1} v$, \ \
$T (\varphi\psi\chi) = \{ (T \varphi) (T\psi) (T\chi)\}$

}


\frame[<+->]{

\frametitle{LOCAL M\"OBIUS TRF IN REAL JORDAN TRIPLES}

$E$ real Banach space, $\{ \ldots \} :E^3 \to E$

\

$(E,\{ \ldots \})$ {\bf Jordan triple}: \ \
$\{ \ldots \}$ trilinear \ + \ Jordan identity

\

$D(a,b)x:=\{ abx \}$, \
$Q(a,b)x:=\{ axb\}$, \
$B(a,b) := 1 -2 D(a,b) + Q(a)Q(b)$

\

$M_p(x) := p + B(p)^{1/2} [ 1 + D(x,p) ]^{-1} x$,

\

$\Phi_v := \exp\big( X_v \big)$ \quad flow of \ \
$X_v := [v-\{ xvx\}] \partial/\partial x$

\

\

{\bf Lemma.} $\exists\, \eps > 0$ \ \
$v\mapsto \Phi_v(0)$  bianal. on $\eps {\rm Ball}(E)$;

\

$\forall\, x,v \in \eps {\rm Ball}(E)$
\qquad
$M_{\Phi_v(0)}(x) = \Phi_v(x)$



}



\frame[<+->]{

\frametitle{LOCAL M\"OBIUS TRF IN REAL JORDAN TRIPLES}

${\rm Der}(E,\{\ldots\}) :=
\Big\{ \ell \!\in \! {\cal L}(E):\ \ell\{ xyz\} \equiv
\{ (\ell x)yz\} \! + \!  \{ x(\ell y)z \} \! + \! \{ xy(\ell z)\} \Big\}$

\

$L_\ell := [\ell x] {\partial \over \partial x}$ \ \
$(\ell\in {\rm Der}(E,\{\ldots\}))$

\

Example: \ $D(a,b)-D(b,a)\in {\rm Der}(E,\{\ldots\})$

\

\

{\bf Definition.} $\Phi:U\to E$ is a {\bf local M\"obius transformation} on $E$
if

$U$ is a $0$ nbh in $E$, \ \
$\Phi$ is bianalytic
of the form \ \ $M_p \circ \exp(L_m) \vert U$

\

\

{\bf Proposition.}

1) $\big\{ X_v + L_\ell :\ v\in E,\ \ell\in {\rm Der}(E,\{\ldots\}) \big\}$
Banach-Lie algebra

$\phantom{11)}$ with \
$\big( [f(x) {\partial\over \partial x} , g(x) {\partial\over \partial x} ] :=
f^\prime (x) g(x) - g^\prime (x) f(x))  {\partial\over \partial x}$.

\

2) $\exp(X_v+L_\ell)$ \ \ is a local M\"obius trf \ \
for small \ $v,\ell$.

}



\frame[<+->]{

\frametitle{JORDAN-M\"OBIUS MANIFOLDS}

{\bf Definition.} \ \
$({\cal M},{\cal A})$ is a {\bf Jordan-M\"obius manifold} on $E$ if

\

${\cal A} \supset \{ Y_p :\ p\in {\cal M}\}$ \ with \
$Y_p: U_p\to {\cal M}$ \ \   such that

\

$Y_p(0) =p$, \ \ $U_p \subset E$ open 0-nbh;

each
chart transition  \ $Y_p^{-1}\circ Y_q$  \ is a
local M\"obius trf \ if \
$q\in Y_p (U_p)$

\

\

{\bf Example.} 1) ${\cal M}:={\rm Ball}(Z)$ of complex JB*-triple, \ \
$Y_p := M_p : {\cal M}\leftrightarrow {\cal M}$

\

2) ${\cal M}:= \varrho {\rm Ball}(E)$, \ \
$Y_p := M_ p \vert U_p$ \ \
where \ \
$\varrho$ is so small that

the sections of
$(p,x)\mapsto M_p(x)$ \ are bianalytic on $[\varrho {\rm Ball}(E)]^2$;

$U_p$ is so small that \ $M_p(U_p) \subset \varrho {\rm Ball}(E)$

{\bf E.g.} $E:=\{$real lower triag $n\times n$ T\"oplitz matrices$\}$
\quad $\{ xyz\}:=xyz$, $\varrho=1$


}



\frame[<+->]{

\frametitle{MAIN RESULTS ON JORDAN-M\"OBIUS MANIFOLDS}

$({\cal M},{\cal A})$ Jordan-M\"obius manifold on $(E,\{\ldots\})$,

$Y_p : U_p\to {\cal M}$ \ $(p\in {\cal M})$ \ as before

\


{\bf Theorem.}
There is a unique connection $\nabla_X Y$ on ${\cal M}$

which is invariant under all local transformations of ${\cal M}$

whose repesentations in the charts $Y_q$ $(q\in {\cal M})$
are local M\"obius trf.

\

{\bf Definition.}
$({\cal M},{\cal A})$ is a {\bf uniform} Jordan-M\"obius manifold if

$\exists\ \varrho > 0 \qquad U_p \supset \varrho{\rm Ball}(E)$.

\

{\bf Proposition.} Uniform Jordan-M\"obius manifolds are {\it symmetric}.

\

{\bf Question.} Is it possible to embed any JM-manifold into a
{\it uniform} one?


}



\frame[<+->]{

\frametitle{MORE EXAMPLES}

3)  [Chu-Isidro-Kaup-Stach\'o]

\hskip 5mm
$({\cal H} , \langle .\vert .\rangle)$ complex Hilbert space, \ \
$\{ xyz\} :=
{1\over 2} \langle x\vert y\rangle z + {1\over 2} \langle z\vert y\rangle x$

\hskip 5mm
$E := e_0^\perp= \{ z\in{\cal H}: \ \langle z\vert e_0\rangle=0 \}$,
 \ \ $\{\ldots\}$ from ${\cal H}$, \ \ $e_0$ unit vector

\hskip 5mm
${\cal S}:=$[unit sphere], \quad $e^\sim := \zeta e:\ \vert\zeta\vert=1 \}$

\hskip 5mm
$R_e=[$unitary op. on ${\cal H}$ with $R_e(e)=e_0 ]$
$$\vtop{
\hbox{
${\cal M}:= \{ e^\sim: \ e\in {\cal S}\}$ ,}
\hbox{$Y_{e^\sim} (x) := [R_e (e_0 + x)/\Vert e_0+x\Vert]^\sim \quad
(e^\sim \in {\cal M},\ x\in E)$}}$$

4) Kaup's complex symmetric hermitian Banach manifolds are JM

$p_0\in {\cal M}$ base point, \ $S_p=[$symm at $p]$, \ $E:=T_{p_0}{\cal M}$

$X_v(p) :=
[\tau\mapsto S_{q(\tau)} \circ S_{p_0}(p)]^{{\cal A}} \in T_p{\cal M}$
\quad $(q(.)^{{\cal A}} = v \in E)$

$\{ xvy \} = {\partial^3 \over \partial x \partial\overline{v}\partial x}
[X_x,[X_v,X_y]] \in E$

$$Y_p : [\exp(v-\{ zvz\}){\partial\over\partial z}]0 \mapsto [\exp X_v]p$$



}



\frame[<+->]{

\frametitle{JORDAN MANIFOLDS}

{\bf Definition.} $({\cal M},{\cal A},{\cal P})$ is a {\bf Jordan manifold} if

\begin{itemize}

\item[] $({\cal M},{\cal A})$ \ \
connected real ${\cal C}^\infty$-Banach-manifold

\item[] ${\cal P}= \big\{ \{\ldots\}_p : \ p\in{\cal M}\big\}$, \quad
$\{\ldots\}_p$\ real J-triple product on $T_p{\cal M}$

\item[] $(p,x,y,z)\mapsto  \{ xyz\}_p$ \ \ smooth on
${\cal M}\times [T{\cal M}]^3$

\end{itemize}

\

\hskip 5mm
{\bf Morphisms} between Jordan manifolds:

\hskip 15mm
smooth mappings whose derivatives are

\hskip 15mm
triple product homomorphisms

\hskip 15mm
on the respective tangent spaces.

\

\hskip 5mm
{\bf Local morphisms}: \ \ morphisms between submanifolds

}



\frame[<+->]{


\frametitle{EXAMPLES OF JORDAN MANIFOLDS}

1) Riemann spaces can be regarded as Jordan manifolds:

\hskip 5mm
$\langle .\vert .\rangle_p$ inner product on $T_p M$ \ $\longrightarrow$ \
$\{ uvw\}_p :=
{1\over 2} \langle u\vert v\rangle_p w +
{1\over 2} \langle w\vert v\rangle_p u$

\

\

2) Jordan-M\"obius manifolds are Jordan manifolds:

\hskip 5mm
$\{ uvw \}_p =Y_p^\prime(0) \Big\{
\big[ [Y_p^\prime(0)]^{-1}u \big]
\big[ [Y_p^\prime(0)]^{-1}v \big]
\big[ [Y_p^\prime(0)]^{-1}w \big] \Big\}$

\

\

3) In general,
the triple products in ${\cal P}$ need not be isomorphic:

\hskip 5mm
$M:=\RR$, \ $\{ uvw\}_p := p u v w$

\hskip 5mm
$\{ \ldots \}_p \sim \{ \ldots \}_q$ \ \ iff  \ \ ${\rm sgn}(p)={\rm sgn}(q)$


}



\frame[<+->]{

\frametitle{ISOMORPHISM OF LOCAL TRIPLE PRODUCTES}

{\bf Theorem.} If the members of ${\cal P}$ are finite dimensional
JB*-triple products then they are isomorphic.

\

{\bf Proof.} With convergent subsequences of {\bf grids} in tangent spaces.

\

\

{\bf Conjecture.}
The condition ${\rm dim}({\cal M})< \infty$ can be dropped

}


\frame[<+->]{

\frametitle{HOMOGENEOUS JORDAN MANIFOLDS}

{\bf Definition.} $({\cal M},{\cal A},{\cal P})$ is {\bf homogeneous}:

\hskip15mm
$\forall\ p,q\in {\cal M} \quad \exists\ \Psi:{\cal M} \leftrightarrow {\cal M}$
Jordan automorphism \ \ $\Psi(p) = q$

\

\hskip5mm
{\bf Locally homogeneous}: \ \
$\forall\ p,q\in {\cal M} \quad \exists\ {\cal U}$ $p$-nbh \ \ $\exists\ {\cal V}$ $q$-nbh

\hskip25mm
$\exists\ \Theta:{\cal U} \leftrightarrow {\cal V}$
Jordan automorphism \ \ $\Theta(p) = q$

\

\

{\bf Example.} $(1,\infty)$ as Jordan-M\"obius manifold with \ \
$Y_p(x):={x+p \over 1+ xp}$

\hskip18mm only locally homogeneous

}



\frame[<+->]{

\frametitle{COMPLEX HOMOGENEOUS JORDAN MANIFOLDS}

Kaup 1977 $\longrightarrow$

\hskip10mm
$({\cal M},{\cal A},{\cal P})$
complex symm. hom. Jordan manifold

\

\hskip10mm
all $\{ \ldots\}_p$ are JB*-triples \
$\Longrightarrow$ \ \
$({\cal M},{\cal A},{\cal P})$ {\bf hermitian}

\hskip60mm
$\sim$  \ canonical manifold of a

\hskip66mm
complex hermitian J*-triple

\

\

{\bf Example.} Equivalence classes of tripotents in a JB*-triple
$\big( Z,\{\ldots\} \big)$

\hskip18mm
$\big( T_{{e^\sim}} {\cal M}, \{ \ldots \}_{{e^\sim}} \big) \ \sim \
\big( Z_{1/2}(e),\{ \ldots\}\vert_{Z_{1/2}(e)} \big)$


\

\

{\bf Problem.} Can symmetry be dropped?

}


\frame[<+->]{

\frametitle{REAL SYMMETRIC NON-JORDAN-M\"OBIUS JORDAN MANIFOLD}

Corach-Porta-Recht 1993

$A$ C*-algebra with $1$, \quad $E:=\{ a\in A:\ a=a^*\}$,
\quad $\{ xyz \} := {1\over 2} xyz + {1\over 2} zyx$

${\cal M} := \{ a\in A :\ a > 0\} = \{ g^*g : \ g$ inv.$\}$

$L_g: {\cal M} \leftrightarrow {\cal M}, \ x\mapsto g^* x g$ \ linear

$S: p\mapsto p^{-1}$ \ \ symmetry on ${\cal M}$

$S_{g^*g} := L_g \circ S \circ L_{g^{-1}} : p \mapsto g^* g p^{-1} g^*g$

\

$\{ uvu\}_{g^*g} : \vtop{
\hbox{$= L_g \{ (L^{-1}_g u) (L^{-1}_g u) (L^{-1}_g u) \} =$}
\hbox{$= u (g^* g)^{-1} v (g^*g)^{-1} u$ \qquad well-def.}}$

\

$L_g,S_p$ \ ${\cal P}$-automorphisms,

$X_v:= {d\over d\tau} \big\vert_{\tau=0+} S_{\exp(\tau v/2)} \circ S_1$, \qquad
$\exp(X_v) p = \exp(v/2)p\exp(v/2)$

\

{\bf Proposition.} In the chart
$Y: \big[\exp (v-\{ xvx\}) {\partial\over\partial x}\big] 0
\mapsto
[\exp X_v] 1$
the transformations $\exp(X_w)$ $(w\in E)$
are not in general $\{ \ldots \}$-M\"obius type






}

\end{document}

\frame[<+->]{

\frametitle{DISJOINTIFICATION \ OF \ $\PP$}

$\displaystyle{\SSS := \bigcup_{e\in M} \{ e\} \times Z_1(e) \ =
\big\{ (e,x):\ e\in M, \ x\in J_e \big\} \ \subset Z\times Z}$

\

$Z^-(e) := Z_{1/2}(e) \oplus Z_1^-(e)$, \quad $Z^+(e) := Z_0(e)\oplus Z^+(e)$, \quad
$P^\pm(e): Z\to Z^\pm(e)$

\

{\bf THM.} \ $\SSS$ \ real anal. dir. subm. of $Z^2$, \quad ${\rm T}_{(e,x)} \SSS = Z_1^-(e) \times Z_1(e)$

\

For \ $(e,x)\in\SSS$ \ and \ $(v,y)\in Z^-(e)\times Z$ \ \ t.f.a.e.

1) $(v,y)\in {\rm T}_{(e,x)} \SSS$, \quad
2) $P_0(e)y=0$ and $P_{1/2}(e)y = 2 \{ P_{1/2} v, e, x\}$,

3) $y = \{ v,e,x\} + \{ e,v,y \} + \{ e,e,y\}$

\


{\bf Proof.} \ Investigate the map

$\Phi_{(e,x)}(w,z) := $

$\Big( \
[ \exp K( e ,P^-(e)w)] [ e+ P^+(e)w ] \ , \
[\exp K(e, P^-(e)w) ] (x+z) \
\Big)$

}


\frame[<+->]{

\frametitle{TANGENT VECT0R FIELDS TO \ ${\Twelveopen S}\hskip-2.5mm{\Twelveopen S}$}

{\bf COR.} \ \
${\rm T}_{(e,x)} \SSS \supset
\Big\{ \Big( P_{1/2}(e) D(a,b)e, \ D(a,b)x \Big): \ (a,b)\in Z\!\times\! Z \Big\}$

\

If \ \ $V,Y$ \ \ smooth v.fields on \ $Z$ \ and \ $V$ tangent to $M$. \ \ T.f.a.e.

\

$\ $ 1) $\forall\ (e,x)\in\SSS \quad [\exp tY]x \in Z_1\big( [\exp tV]e \big)$ \qquad
for \ $\vert t\vert$ \ small

\

$\ $ 2) $Y(x) = \{ \{ V(e),e,x \} + \{ e,V(e), x\} + \{ e,e,Y(x)\}$, \qquad $(e,x)\in\SSS$

\

If $V$ complete in $M$  \ and \ $Y$ complete in $Z$ \quad $\Rightarrow$ \ 1)+2) is equivalent to

\

$\ $ 3) $[\exp(tY)] Z_1(e) = Z_1\big( [\exp tV]e \big)$, \quad $(e,x)\in\SSS,\ t\in\RR$


}


\frame[<+->]{

\frametitle{COORDINATES VIA  \ $(ST)$}

$C_u^{(e)} : M \ni f \mapsto P_{1/2}(f) D(u,e) f$ \quad $e\in M,\ u\in Z_{1/2}(e))$

\

\

{\bf REM.} \ Solution of $(ST)$: \ $t\mapsto [\exp tC_u^{(e)}]e \in M$

%$\{ s(w),s(w),w \} = \{ s(w),w,e(w)\}$, \quad $J_w = J_{s(w)} = Z_1\big( s(w)\big)$

\

\

{\bf DEF.} \ \ $Y_e(u) := [\exp C_u^{(e)}]e$, \qquad $e\in M,\ u\in Z_{1/2}(e)$

\

\

{\bf THM.} \ $C_u^{(e)}$ \ \ equivar. complete v.field on $M$,

\

\hskip14mm  $\exists\ e\mapsto W_e$ $0$-nbh. in $Z_{1/2}(e)$ \quad $Y_e \vert W_e$ real-bianal.


$$Y_e(0) = e, \quad [\exp D(u,e)] J_e = J_{Y_e(u)} \qquad\qquad (u\in W_e)$$


}



\frame[<+->]{

\frametitle{COMPLEX STRUCTURES}

$\widetilde{Y}^{(e)} (u) := \Big[ \big[\exp C_u^{(e)}\big]e \Big]^\sim =
Y_e(u)^\sim {\phantom{\Big]^{\int}}}^{\kern-10mm{{\rm equiv.class}\atop\downarrow}} \ \ \in\MM$

\

\

$\overline{Y}^{(e)} (u) := Z_{1/2} \Big[ \big[\exp C_u^{(e)}\big]e \Big] =
J_{Y_e(u)} \hskip6mm \in \PP$

\

$\widehat{Y}^{(e)} (u) := i D\big( \big[ \exp C_u^{(e)} \big] e \big)
\hskip24mm \in\DD$

\

{\bf THM.} \ $\big\{ \widetilde{Y}^{(e)} :\ e\in M\big\}$, \
$\big\{ \overline{Y}^{(e)} :\ e\in M\big\}$, \
$\big\{ \widehat{Y}^{(e)} :\ e\in M\big\}$

\

\hskip14mm holomorphic atlases for \ $\MM,\PP,\DD$

$$\vtop{
\hbox{\hskip5mm$Z_{1/2}(e)$}
\hbox{\hskip5mm$\swarrow \ \downarrow \ \searrow$}
\hbox{$\DD\leftrightarrow \MM  \leftrightarrow\PP$}}
\qquad
\vtop{
\hbox{\hskip23.5mm$u$}
\hbox{\hskip14mm$\swarrow \hskip6mm \downarrow \hskip6mm \searrow$}
\hbox{$\widehat{Y}^{(e)}(u)\leftrightarrow \widetilde{Y}^{(e)}(u)  \leftrightarrow \overline{Y}^{(e)}(u)$}}
$$


}



\frame[<+->]{

\frametitle{HOLOMORPHIC MAPS ON \ $\MM$, $\PP$}

{\bf THM.} \ ${\openo U}$ open $\subset \MM$, \
$U:= %\bigcup{\openo U} :=
\{ e\in M:\ \exists\ \widetilde{f}\in \MM \ \ e\in\widetilde{f}\}$,

$\Phi:{\openo U}\to B$ map to compl. Banach space .


T.f.a.e. $\phantom{ \Big]^{\int} }$

\

\hskip5mm 1) \ $\Phi$ is holomorphic

\

\hskip5mm 2) $\forall\ e \in U \ \quad \exists\ V \hbox{open}\subset M \ \quad \exists\ \phi:V\to B$ holomorphic \
s.\hskip-0.3mm th.

\

\hskip 20mm $\phi(f) = \Phi(f^\sim) \qquad \hbox{whenever} \quad f \in U\cap V$

\

\

{\bf REM.} \ $\Psi: [$nbh. of $J_e] \to B$ \ is holomorphic $\iff$

\

\hskip13.6mm $\Phi\big( \underbrace{J\cap M}_{\in \MM} \big) := \Psi(J)$ \ \ is holomorphic


}



\end{document}


\frame[<+->]{

\frametitle{NUMERICAL TREATMENT}

\begin{itemize}

\item[]
It follows
$$a_{8+n} =
{\rm Re}\!\!\! \!\!\!\!\!\!\! \sum_{k=\in\{1,2,3,5\}} \!\!\!\!\!\!\!\!
\rho_k^n \alpha_k, \quad
b_{8+n} =
{\rm Re}\!\!\!  \!\!\!\!\!\!\! \sum_{k=\in\{1,2,3,5\}} \!\!\!\!\!\!\!\!
\rho_k^n \beta_k, \quad
c_{8+n} =
{\rm Re}\!\!\! \!\!\!\!\!\!\!  \sum_{k=\in\{1,2,3,5\}} \!\!\!\!\!\!\!\!
\rho_k^n \gamma_k$$

\item[]
\begin{eqnarray*}
& &\alpha_1 \approx \ \ 0.0001015421, \quad
\alpha_2 \approx 3375.0749184657,\\
& &\beta_1 \approx   -0.0000538719,\quad
\beta_2 \approx 3552.5887804916,\\
& &\gamma_1\approx   -0.0000206576\quad
\gamma_2\approx 4039.314501374481984826,
\end{eqnarray*}

\item[]
etc.

\item[]
\begin{eqnarray*}
& &\alpha_5 \approx    2\cdot(-0.0380308729 +  1.4388617102 i) ,\\
& &\beta_5 \approx \ \  2\cdot(2.6921768571   -1.4860158672 i) ,\\
& &\gamma_5\approx \ \  2\cdot(2.34766856938 +  0.9004836949 i) .
\end{eqnarray*}

\end{itemize}

}


\frame[<+->]{

\frametitle{GENERATOR FUNCTIONS}

\begin{itemize}

\item[]
$v_n=\sum_{p=1}^K v_p A_p$, \qquad $n>K$

$\phantom{xxxxx}$


\item[]
$g(t) := \sum_{n=1}^\infty t^n v_n$ \qquad\qquad scalar$\to$vector

$\phantom{xxxxx}$

\item[] Classical Fibonacci: \quad $f_n=f_{n-1}+f_{n-2}$, $f_0=0$, $f_1=1$

\item[]
$\phantom{xxxxx}
g(t) = \sum_{n=1}^\infty t^n f_n$ \quad scalar$\to$scalar

\item[]
$\phantom{xxxxx}
g(t)-t g(t)-t^2g(t) = \sum_{n=2}^\infty (\underbrace{f_n-f_{n-1}-f_{n-2}}_0)t^n + f_0 + (f_1-f_0)t$

\item[]
$\phantom{xxxxx}
g(t) = t/(1-t-t^2) = \alpha_1 (t-\rho_1)^{-1} + \alpha_2 (t-\rho_2)^{-1}$

\item[]
$\phantom{xxxxx}
g(t) = \sum_{n=0}^\infty \big( \underbrace{\beta_1 \rho_1^n + \beta_1 \rho_1^n}_{f_n} \big) t^n$


\end{itemize}

}


\frame[<+->]{

\frametitle{GENERATOR FUNCTION VIA PHASE SPACE}

\begin{itemize}

\item[]
$g(t):= v_1 + t v_2 + t^2 v_3 + \cdots =
\sum_{n=0}^\infty t^n v_{n+1}$

\item[]
\begin{eqnarray*}
g(t)\!\!\!\! &-&\!\!\!\! \Big[ t g(t)A_1 +\cdots + t^K g(t) A_K +
\sum_{n=K+1}^\infty t^n b \Big] =\\
%&=& \sum_{n=0}^\infty t^n v_{n+1} -
%\sum_{n=1}^\infty t^n v_n A_1 -
%\sum_{n=2}^\infty \! t^n v_{n-1} A_2 - \cdots -
%\sum_{n=K}^\infty \! t^n v_{n+1-K} A_K - \!\!\!\!
%\sum_{n=K}^\infty \!\! t^n b =\\
%&=& v_1 + \sum_{n=1}^{K-1} t^n \big[ v_{n+1} -
%( v_n A_1 + v_{n-1} A_2 +\cdots+ v_1 A_n)\big] +\\
%& &\hskip5mm + \!\!
%\sum_{n=K}^\infty t^n
%\big[ v_{n+1} -
%(v_n A_1 + v_{n-1} A_2 + \cdots + v_{n+1-K} A_K +b)\big] =\\
&=& v_1 + \sum_{n=1}^{K-1} t^n \Big[ v_{n+1} - \sum_{r=1}^n v_{n+1-r} A_r \Big]
\end{eqnarray*}

\item[]
$$g(t) =  \Big[ {t^K \over 1 - t} b + \sum_{n=0}^{K-1} t^n b_n \Big]
\Big[ I - \sum_{p=1}^{K-1} t^p A_p \big]^{-1}$$

\item[]
$\phantom{xxxxxxxxxxxxxxxxxx}
b_n := v_{n+1} - \!\!\! \sum_{1\le r\le n} v_{n-r} A_r$


\end{itemize}

}

\def\eps{\varepsilon}

\frame[<+->]{

\frametitle{THEOREM 2}

\it
\begin{itemize}

\item[]
$f:(-\eps,\eps)\to\CC^d$ \qquad rational function

\item[]
$f(t) = u(t) U(t)^{-1}$ \qquad
$u:\RR\to \CC^d$, \ $U:\RR\to\CC^{d\times d}$

\item[]
$\Longrightarrow$

$\phantom{xxxxx}$

$f =\big[$gen. funct. of
$v_1,v_2,\ldots\in\CC^d$ with affine recursion prop.$\big]$

\item[]
$n\ge K$, \quad $\Rightarrow$ \ \
Taylor coefficients of \ $f$

$\phantom{xxxxx}$

$n!^{-1} d^n / dt^n \vert_{t=0} f(t) =$

$\phantom{xxxxx}$

\item[]
$=v^{(0)} +
\sum_{q=1}^N
\sum_{p=0}^{\min\{ n-K,m_q\}}
\left( {n-K \atop p} \right) \lambda_q^n v^{(p,q)}$.

\end{itemize}

}


\end{document}
