\magnification=\magstep1

\def\eps{\varepsilon}
\def\frac#1#2{{#1\over#2}}
\def\mathcal#1{{\cal #1}}
\def\artanh{\mathop{\hbox{\rm artanh}}}

\def\CC{{\mathchoice
{\rm C\mkern-8mu\vrule height1.45ex depth-.05ex
width.05em\mkern9mu\kern-.05em}
{\rm C\mkern-8mu\vrule height1.45ex depth-.05ex
width.05em\mkern9mu\kern-.05em}
{\rm C\mkern-8mu\vrule height1.ex depth-.07ex
width.035em\mkern9mu\kern-.035em}
{\rm C\mkern-8mu\vrule height.65ex depth-.1ex
width.025em\mkern8mu\kern-.025em}}}


\def\RR{{\rm I\kern-1.6pt {\rm R}}}
\def\OO{{\rm \hbox{$\vert$}\kern-3.6pt {\rm O}}}
\def\MM{{\rm I\kern-1.6pt {\rm M}}}
\def\TT{{\rm T}\kern-4.8pt {\rm T} \kern2pt}

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


\centerline{\bf JORDAN MANIFOLDS}

\bigskip
\bigskip
\noindent
\bf Notations
\bigskip
\rm

\item{(N1)}
$(E,\{\ldots\})$ \quad real Banach $J^*$-triple with norm $\Vert .\Vert$.
\hfill\break
Complex triples are also included in a natural manner.

\medskip
\item{(N2)}
$M$ \quad Banach real ${\cal C}^\infty$-manifold modeled over $E$.
\hfill\break
$M$ is, as usual, complex, iff there is an (almost) complex structure $J$ on $TM$.
\hfill\break
If we emphasize that $M$ is a {\it complex} manifold we mean by that (as usually) that
\hfill\break
its coordinate transitions are {\it holomorphic}.

\medskip
\item{(N3)}
$\overline{{\cal A}} :=\{$complete atlas of $M\} =$
\hfil\break
$=\{$local \hbox{Fr\'echet}-diffeomorphisms from open subsets of $M$ to $E\}$
\hfill\break
$X: {\rm dom}(X) \leftrightarrow {\rm ran}(X)$ for $X\in\overline{{\cal A}}$.

\medskip
\item{(N4)}
${\cal A} :=\{ \Phi_p :\ p\in M\}$ \quad
atlas on $M$. For each point $p\in M$,
\hfill\break
$\Phi_p : U_p \to E$ where $U_p\ \hbox{open}\subset M$ and
$\Phi_p(p)=0$.

\medskip
\item{(N5)}
${\cal G} := \{ \Phi_p\circ \Phi_q^{-1} \}$ \quad coordinate transition maps.
\hfill\break
${\cal G}^* := \{$finite composition maps from ${\cal G}\}$.

\medskip
\item{(N6)}
${\cal P} := \big\{ \{ \ldots \}_p: \ p\in M\big\}$ \quad local triple products in $M$,
\hfill\break
$\{ \ldots \}_p : {T_p M}^3\to T_p M$ $(p\in M)$\quad trilinear operations.
\hfill\break
I.e. a tensor field of type (3,1).

\medskip
\item{(N7)} Given a local coordinate $\Phi\in\overline{\cal A}$,
\hfill\break
$\Gamma_\Phi : {\rm ran}(\Phi) \to {\cal L}(E\otimes E, E)$ Christoffel symbol.
\hfill\break
Transformation rule for \ $\Psi:= \psi\circ \Phi$:
\hfill\break
$\Gamma_\Psi\big(\psi(z)\big) \big( \psi^\prime(z)u, \psi^\prime(z) v\big) =
\psi^{\prime\prime} (z) (u,v)  + \psi^\prime(z) \Gamma_\Phi(z) (u,v)$ \ $(z\in{\rm ran}(\Phi);\ u,v\in E)$.
This transformation law is used in oder to well-define the concept of an affine connection:

\medskip
\item{(N8)} Given a local coordinate $\Phi\in\overline{\cal A}$,
\hfill\break
$R_\Phi : {\rm ran}(\Phi) \to {\cal L}(E\otimes E, {\cal L}(E))$ curvature tensor:
\hfill\break
$R_\Psi(z) (u,v)w = $
\hfill\break
$= \Gamma_\Phi(z) \big( \Gamma_\Phi(z)(u,w),v \big) +
\Gamma_\Phi(z) \big( \Gamma_\Phi(z)(v,w),u \big) -
\big[ \Gamma_\Phi^\prime(z) v \big] (u,w) +
\big[ \Gamma_\Phi^\prime(z) u \big] (v,w)$.
\hfill\break
Transformation rule (as in N7):
\hfill\break
$R_\Psi \big( \psi(z)\big)   \big( \psi^\prime(z)u, \psi^\prime(z) v \big) \psi^\prime(z) w =
\psi^\prime(z) R_\Phi (z) ( u,v ) w$.

\bigskip
\bigskip
\noindent
\bf Definition. \rm
\medskip
\item{(D1)}
$(M,\overline{{\cal A}},{\cal P})$ is a {\it Jordan manifold} if
\hfill\break
the mapping $(p,u,v,w)\mapsto \{ uvw\}_p$ is continuously differentiable.
\hfill\break
Submanifolds can be defined.

\medskip
\item{(D2)} Given two Jordan manifolds
$(M,\overline{{\cal A}},{\cal P})$ and $(N,\overline{{\cal B}},{\cal Q})$,
\hfill\break
a continuously differentiable map $S:M\to N$ is a {\it Jordan morphism} if
\hfill\break
$\forall\ p\in M \quad \forall\ u,v,w\in T_p(M) \qquad
S^\prime(p) \{ uvw \}_p = \{ [S^\prime(p)u] [S^\prime(p)v] [S^\prime(p)w] \}_{S(p)}$.
\hfill\break
Isomorphism and automorphisms can be defined as usually.
\hfill\break
${\rm Aut}(M) := \{ S : M\leftrightarrow M$ with $S, S^{-1}$ Jordan morphisms$\}$.

\medskip
\item{(D3)} Given
$(M,\overline{{\cal A}},{\cal P})$ and $(N,\overline{{\cal B}},{\cal Q})$
with $U\ \hbox{open}\subset M$ and $V\ \hbox{open}\subset N$,
\hfill\break
a continuously differentiable map $S:U\leftrightarrow V$ is a {\it local Jordan automorphism} if
\hfill\break
$S$ is a Jordan isomorphism between $U$ and $V$ as Jordan submanifolds.
\hfill\break
${\rm Aut}_{{\rm loc}} (M) := \{$local Jordan automorphism in $M\}$.
\hfill\break
Notice that $S_1\circ S_2 \in {\rm Aut}_{{\rm loc}} (M)$ whenever
$S_1,S_2 \in {\rm Aut}_{{\rm loc}} (M)$ with ${\rm ran}(S_1) \cap {\rm dom}(S_2) \ne\emptyset$.

\bigskip
\bigskip
\noindent
\bf Convention. \rm
Henceforth $(M,\overline{{\cal A}},{\cal P})$ stands for a Jordan manifold.

\bigskip
\bigskip
\noindent
\bf Example. \rm (E0) Riemann spaces can be regarded as Jordan manifolds:
\hfill\break
If $\langle .\vert .\rangle_p$ is the inner product on $T_p M$ then
we take
\hfill\break
$\{ uvw\}_p :=
{1\over 2} \langle u\vert v\rangle_p w + {1\over 2} \langle w\vert v\rangle_p u$.


\medskip
\noindent
\bf Example. \rm
The triple products in ${\cal P}$ need not be isomorphic to each other:
$$\hbox{$M:=\CC$ \ \ with \ \ $\{ uvw\}_p := {\rm Re}(p) u\overline{v} w$.}
\leqno{\rm (E1)}$$

\noindent
\bf Conjecture. \rm (HARD) \it

If we assume all the $\{\ldots\}_p$ to be
$JB^*$-triple products then they are isomorphic.


\bigskip
\bigskip
\noindent
\bf Lemma. \it If for every couple of points $p,q\in M$ there are neighborhoods
$U,V\subset M$
along with a diffeomorphism
$S:U \leftrightarrow V$
such that
$q=S(p)$ and
$S^\prime(p)$ is an isomorphism between $\{\ldots\}_p$ and $\{\ldots\}_q$
then there is an atlas ${\cal A} \subset \overline{{\cal A}}$ with the properties {\rm (N4)}.

\bigskip
\noindent
\bf Proof. \rm
Fix any $o\in M$ and a local coordinate $X\in \overline{\cal A}$
with $X(o)=0$.
For any $p\in M$ choose a local diffeomorhism
$S_p$ with $S_p(o)=p$ and the properties in (D2).
We redefine the triple product $\{\ldots\}$ on $Z$
as the image of $\{\ldots\}_o$ by $X^\prime(0)$ and we set
$X_p := X\circ S_p^{-1}$ $(p\in M)$.


\bigskip
\noindent
\bf Definition. \rm

\item{(D4)} A Jordan manifold $M$ is {\it homogeneous} if
\hfill\break
${\rm Aut}(M)$ is trasitive on it:
$\forall\ p,q\in M \quad \exists\ S\in{\rm Aut}(M) \qquad S(p)=q$.

\medskip
\item{(D5)} A Jordan manifold is {\it locally homogeneous} if
\hfill\break
${\rm Aut}_{{\rm loc}}(M)$ is trasitive on it.

\bigskip
\bigskip
\noindent
\bf Examples. \rm

\item{(E2)} [Corach-Porta-Recht].
\hfill\break
$M:={\bf A}^+$ the positive invertible elements
of a unital $C^*$-algebra ${\bf A}$.
\hfill\break
$E:= {\bf A}$ with $\{ xyz\} := {1\over 2} xy^*z + {1\over 2} zy^*y$ as usually.
\hfill\break
$U_a := M$ and $X_a(b) := b-a$  for all $a,b\in M$.
\hfill\break
$\{ uvw\}_a = \{ uvw\}$ \quad $(a\in M; \ u,v,w \in T_a M \equiv E)$.
\hfill\break
\it Then, for any invertible $g\in{\bf A}$ we have
$L_g := [a\mapsto (g^*)^{-1} a g^{-1}] \in {\rm Aut}(M)$.
\hfill\break
In particular $M$ is homogeneous because $\{ L_g(1):\ e\in M\} = \{ g^{-2}: e\in M\} =M$.
\rm

\medskip
\item{(E3)} [Chu-Isidro-Kaup-Stach\'o]. A case with equivalence classes of tripotents.
\hfill\break
${\bf H}$ complex Hilbert space with inner product $\langle .\vert .\rangle$. \
${\bf S}:=$[unit sphere].
\hfill\break
Triple product
$\{ xyz\} := {1\over 2} \langle x\vert y\rangle z + {1\over 2} \langle z\vert y\rangle x$
\ $(x,y,z\in {\bf H})$.
\hfill\break
$E := e_0^\perp= \{ z\in{\bf H}: \ \langle z\vert e_0\rangle=0$ with a fixed $e_0\in{\bf S}$.
\hfill\break
$M:= \{ \TT e: \ e\in{\bf H}, \ \langle e\vert e\rangle=1\}$ with
$\TT := \{ \zeta\in\CC:\ \vert\zeta\vert=1\}$.
\hfill\break
For $e\in {\bf S}$, let us write $e^\sim := \TT e$.
\hfill\break
$T_{e^\sim} M \equiv e^\perp = \{ z\in {\bf H}: \ \langle z\vert e\rangle=0\}$.
\hfill\break
$\{ uvw\}_{e^\sim} := - \{ uvw\} \in T_{e^\sim} M$ for $u,v,w\in T_{e^\sim} M$.
\hfill\break
Given $e\in{\bf S}$, let ${\bf R}_e$ be a rotation of ${\bf H}$ with ${\bf R}_e(e)=e_0$.
\hfill\break
$\Phi_{e^\sim}^{-1}: e\perp z\mapsto [\CC {\bf R}_e(e+z)]\cap {\bf S}$ \ \ $(e\in {\bf S})$.
\hfill\break
\it Then ${\cal X} := \{ \Phi_{e^\sim} : \ e^\sim \in M\} = \{ \Phi_{e^\sim} : \ e \in {\bf S}\}$
is an atlas on $M$
\hfill\break
with holomorphic coordinate transitions.
\hfill\break
${\cal G} ^* \subset {\rm Aut}_{{\rm loc}}(M) \cap \{ {\cal X}$-holomorphic maps$\}$
is transitive on $M$.
\hfill\break
${\rm Aut}(M) \cap \{ {\cal X}$-holomorphic maps$\} =\{ {\rm Identity}\}$.


\bigskip
\noindent
\bf Case (E2) \rm

\noindent
$Z := \{ z\in{\bf A} : \ z^* =z\}$ self-adjoint elements.

\noindent
$\{ xyz \} := {1\over 2} xyz + {1\over 2} zyx$ \quad
$(x,y,z\in Z)$.

\noindent
$G := \{ g\in Z:\ g\ \hbox{invertible}\}$, \quad
$M = {\bf A}^+ = \{ g^*g:\ g\in G\}$.

\noindent
$\sigma: a\mapsto a^{-1}$ symmetry on $M$ with fixed point $1$.

\noindent
$L_g(a) := g^* a g$ \quad $(a\in M,\ g\in G)$; \quad
$L_g^{-1}=L_{g^{-1}}$, \ $L_g:M\leftrightarrow M, \ 1\mapsto g^* g$.

\noindent
\bf Definition. \rm
$\sigma_a := L_{a^{1/2}} \circ \sigma \circ L_{a^{-1/2}}$ \quad symmetry at $e\in M$.

\noindent
\bf Remark. \rm
$a = g^*g \Rightarrow L_g \circ \sigma \circ L_{a^{-1}} = \sigma_a$ \qquad $(g\in G)$.

\noindent
\bf Definition. \rm
Infinitesimal translation on $M$ with vector $h\in Z \equiv T_1 M$ at $1$:
$$X_h (a) := {d\over d\tau}\Big\vert_{\tau=0} \sigma_{1+(\tau/2) h} \circ \sigma_1 (a)
\qquad (a\in M).$$
\bf Lemma. \it We have
$${\rm 1)}\ X_h(a) = {1\over 2} (ha+ah) = \{ h1a\},
\qquad
{\rm 2)}\ X_h (a) = {d\over d\tau}\Big\vert_{\tau=0} \sigma_{\exp(\tau h/2)} \circ \sigma (a) .$$
\bf Proof. \rm
1) $\sigma_{1+(\tau h/2} \circ \sigma_1 (a) =
\hfill\break
= \sigma_{1+\tau h/2} (a^{-1}) =
(1+\tau h/2)^{1/2} \big[ (1+\tau h/2)^{-1/2} a^{-1} (1+\tau h/2)^{-1/2} \big]^{-1}
(1+\tau h/2)^{1/2} =
\hfill\break
= (1+\tau h/2) a (1+\tau h/2)$.

\noindent
2) $\exp(\tau h/2) = \sum\limits_{n=0}^\infty {1\over 2^n n!} \tau^n h^n =
1 + \tau h/2 + o(\tau)$.

\bigskip
\noindent
\bf Lemma. \it
$\exp X_h (a) = [\exp (h/2)] a [\exp  (h/2)]$.

\bigskip
\noindent
\bf Proof. \rm
The left and right multiplications
${\rm Mult}_\ell^h: {\bf A}\ni x\mapsto hx$ and
${\rm Mult}_r^h: {\bf A}\ni x\mapsto hx$ commute.
Observe that
$H_h = {1\over 2} [{\rm Mult}_\ell^h + {\rm Mult}_r^h]$.
Therefore
$\exp X_{2h} (a) =
\hfill\break
= \exp [{\rm Mult}_\ell^h + {\rm Mult}_r^h] a =
\exp [{\rm Mult}_\ell^h] \exp [{\rm Mult}_r^h] a =
[{\rm Mult}_\ell^{\exp h}] [{\rm Mult}_\ell^{\exp h}] a$.
% = [\exp h] a [\exp h]$.

\bigskip
\noindent
\bf Remark. \rm Since all elements from $M={\bf A}^+$ admit a spectral resolution
over some compact interval $0<[\alpha,\beta]<\infty$,
each element $a\in M={\bf A}^+$ can be written as $a=\exp h$ with a {\it unique}
self-adjoint $h\in Z$ such that $(\log\alpha) \cdot 1 \le h \le (\log \beta)\cdot 1$.

\bigskip
\noindent
\bf Definition. \rm
We write
$${\rm Exp}_Z (h) := \exp(h) \quad (h\in Z), \qquad
{\rm Log}_M a := {\rm Exp}_Z^{-1} (a) \quad (a\in M).$$
Let $h\in Z$ and set $a:=\exp(h) = {\rm Exp}_Z(h) = \exp(X_h)1$.
We call the mapping
$$M_a := \exp(X_h) = \exp X_{{\rm Log}_M a} : M \ni x \mapsto a x a$$
the
{\it pseudo-M\"obius transformation}
of the point $a$.
Natural atlas on $M$:
$$\Phi_a := M_a \circ {\rm Log}_M \circ M_a^{-1} \qquad (a\in M).$$
\bf Remark. \rm $M_a^{-1} = M_{a^{-1}}$.

\medskip
\noindent
Our next aim is to calculate the image
$\Phi_1^\# X_h : Z\to Z$
of the vector field in the atlas page of $\Phi_1$.
By definition,
$$\eqalign{
\big[ \Phi_1^\# X_h \big] (z) &:= \big[ \Phi_1^\prime \big( \Phi_1^{-1} (z)\big) \big]
X_h\big( \Phi_1^{-1}(z)\big) =\cr
&= {d\over d\tau}\Big\vert_{\tau=0}
\Big[ \Phi_1^\# \big[ \sigma_{\exp(\tau h/2)} \circ \sigma \big] \Big] (z) = \cr
&= {d\over d\tau}\Big\vert_{\tau=0}
\Big[ \Phi_1^\# \sigma_{\exp(\tau h/2)} \Big] \circ \Big[\Phi_1^\# \sigma \Big] (z) .\cr}$$
Notice that $\Phi_1^\# \sigma (z) =-z$. Therefore
$$\eqalign{
\Big[ \Phi_1^\# X_h \Big](z) &=
{d\over d\tau}\Big\vert_{\tau=0}
\Big[ \Phi_1^\# \sigma_{\exp(\tau h/2)} \Big] (-z) =\cr
&= {d\over d\tau}\Big\vert_{\tau=0}
{\rm Log}_M
\Big( \exp\big( {1\over 2} \tau h\big)
\exp\big(z\big)
\exp\big( {1\over 2} \tau h\big) \Big) .\cr}$$
Let us fix $z,h\in Z$ and define
$$w(\tau) :=
{\rm Log}_M
\Big( \exp\big( {1\over 2} \tau h\big)
\exp\big(z\big)
\exp\big( {1\over 2} \tau h\big) \Big) .$$
Since $w(0)=z$ trivially, we can write
$w(\tau) = z  + \tau b + o(\tau)$ with a unique vector $b\in Z$.
In terms of $b$, we have
$${d\over d\tau}\Big\vert_{\tau=0} \exp\big( w(\tau)\big) =
{d\over d\tau}\Big\vert_{\tau=0}
\exp\big( {1\over 2} \tau h\big)
\exp\big(z\big)
\exp\big( {1\over 2} \tau h\big) .$$
On the right hand side we simply have
$${d\over d\tau}\Big\vert_{\tau=0}
\exp\big( {1\over 2} \tau h\big)
\exp\big(z\big)
\exp\big( {1\over 2} \tau h\big) =
{1\over 2} h \exp(z) + {1\over 2} \exp(z) h \ .$$
On the left hand side the derivation can be carried out
with the power series of the exponential map termwise.
That is
$$\eqalign{
{d\over d\tau}\Big\vert_{\tau=0} \exp\big( w(\tau)\big) &=
\sum_{n=1}^\infty {1\over n!}
{d\over d\tau}\Big\vert_{\tau=0} w(\tau)^n =
\sum_{n=1}^\infty {1\over n!}
\sum_{k=1}^n z^{k-1} b z^{n-k} =\cr
&= \sum_{k,\ell=0}^\infty {1\over (k+\ell+1)!}
\big[ {\rm Mult}_\ell^{z} \big]^k \big[ {\rm Mult}_r^{z} \big]^\ell b \ .\cr}$$





\bye

\bigskip
\bigskip
\noindent
\bf Axioms
\rm
\bigskip

\item{(A1)} ${\cal G}^*$ is transitive in the following sense
\hfill\break
$\forall\ p,q\in M \quad \exists\ G\in {\cal G}^* \qquad
q = \Phi_q^{-1} \circ G \circ \Phi_p$ .

\medskip
\item{(A2)}
$\forall\ p\in M \quad \forall\ u,v,w\in T_p M$
$\phantom{X_{X_{X_{X_X}}}}$
\hfill\break
$\{ uvw \}_p = [\Phi_p^\prime(p)]^{-1} \{ [\Phi_p^\prime(p) u][\Phi_p^\prime(p) v][\Phi_p^\prime(p) w] \}$ .

\medskip
\item{(A3)} The mapping $(p,u,v,w)\mapsto \{ uvw\}_p$ is continuously differentiable


%\bigskip
%\bigskip
%\noindent
%\bf Definition.
%\rm
%$(M,{\cal A},{\cal P})$ is a {\it Jordan manifold} if (A1)+(A2)+(A3) hold.









\bye
