\magnification=\magstep1

\tolerance=2000
%\hsize=15true cm\vsize=24true cm
\frenchspacing
\mathsurround=2pt
\parindent=0pt

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

\font\Title=cmr10 scaled\magstep2
\font\gross=cmbx10 scaled\magstep1
\font\Smallcaps=cmcsc10
\font\Minismallcaps=cmcsc5 scaled\magstep2
\font\Mini=cmr5 scaled\magstep2

\font\tenmb=cmmib10
\newfam\mbfam
\def\mb{\fam\mbfam\tenmb}
\textfont\mbfam=\tenmb
\ifnum\the\mbfam=8\def\mbfamseged{8}\fi
\def\OMEGA{{\mb \Omega}}
\def\oMEGA{{\mb \omega}}

\def\Intd{\,{\bf d}}


\def\RR{{\rm I\kern-1.6pt {\rm R}}}
\def\PP{{\rm I\kern-1.6pt {\rm P}}}
\def\CC{{\rm \kern 3.5pt \vrule height 7.5pt
        width.7pt depth0pt \kern-3pt C \kern.5pt}}
\def\TT{{\rm T}\kern-4.8pt {\rm T} \kern2pt}
\def\ZZ{{\rm Z}\kern-4.8pt {\rm Z} \kern2pt}

\def\J#1#2#3{\{ #1#2#3\} }
\def\ort{{\,\bot\,}}
\def\col{{\,\top\,}}
\def\BOX{\;\! \hbox
        {\vrule\vbox{\hrule width3.2pt\vskip3.2pt\hrule}\vrule}\> }
\def\bo#1#2{#1 {\BOX} #2 ^*}

\def\Otimes{\widehat{\otimes}}

\def\norm#1{\Vert #1 \Vert}
\def\abs#1{\vert #1 \vert}

\def\eps{\varepsilon}

\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\nnorm#1{\big\vert\big\Vert #1 \big\Vert\big\vert}
\def\spr#1#2{\big\langle #1 \big\vert #2 \big\rangle}
\def\sspr#1#2{\big\langle\big\langle #1 \big\vert #2 \big\rangle\big\rangle}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$\underline{\hbox{\bf THE JAMISON-BOTELHO CONJECTURE}}$
\bigskip

In 2008, in a finally unpublished draft [2]
F. Botelho and J. Jamison launched the following conjecture:

\medskip

\bf 1.1. Conjecture. \it Let $X_1,X_2$ be complex Banach spaces.
Suppose
$t\mapsto U_k^t$ $(k=1,2)$ are maps $\RR\to {\cal U}(X_k):=\big\{${\rm
surjective linear isometries of} $X_k \big\}$
with the one-parameter group property
$U_1^{t+s}\otimes U_2^{t+s} = [U_1^t\otimes U_2^t] [U_1^s\otimes U_2^s]$
$(t,s\in\RR)$ and such that
for every fixed bounded bilinear functional $\phi: X_1\times X_2\to\CC$
and for every couple $(x_1,x_2)\in X_1\times X_2$,
the function
$t\mapsto
%\big\langle (U^t_1\otimes U^t_2)\phi, (x_1,x_2) \big\rangle =
\phi\big( U_1^t x_1, U_2^t x_2 \big)$
is continuous. Then both $t\mapsto U^t_1$ and $t\mapsto U^t_2$ are
strongly continuous one-parameter groups.\footnote{$^*$}{\rm
That is $U_k^{t+s}=U_k^t U_k^s$ $(t,s\in\RR,\ k=1,2)$ and the Banach space valued functions
$t\mapsto U_k^t x_k$
are norm-continuous for every fixed $x_1$ resp. $x_2$.}

\medskip
\rm
Actually they even outlined a proof for 1.1 under certain additional hypothesis,
implying a requirement on the representations of the the elementary forms
which turned out to be contradictory due to the fact that we have
$(\kappa_1 x_1)\otimes (\kappa_2 x_2)= x_1\otimes x_2$ whenever $\kappa_1\kappa_2=1$.
Using a completely different probabilistic approach,
in course of the proof of
[3, Theorem 1.1] we have shown
the following slightly generalized version of the conjecture for the case with Hilbert spaces:

\medskip
\bf 1.2. Theorem. \it
Let $\HHH{1},\ldots,\HHH{N}$ be Hilbert spaces.
Assume
$\big[ {\bf U}(t):\ t\in\RR \big]$
is a one-parameter group
where
${\bf U}(t) := U_{1,t} \otimes\cdots\otimes U_{N,t}$
with unitary operators
$U_{k,t} \!\in{\cal U}(\HHH{k})$
having the
following continuity property:
$t\mapsto \prod_{k=1}^N \langle U_{k,t} {\bf x}_k \vert {\bf h}_k \rangle$
is continuous for every fixed ${\bf x}_1,{\bf h}_1\in\HHH{1},\ldots, {\bf x}_N,{\bf h}_N\in\HHH{N}$.
Then there exist functions
$\kappa_1,\ldots,\kappa_N:\RR\to\TT(:=\{ \kappa\in\CC:\ \vert \kappa\vert=1\})$
with $\prod_{k=1}^N \kappa_k \equiv 1$
such that the families $\big[ \kappa_k(t) U_{k,t} :\ t\in\RR \big]$ $(k=1,\ldots,N)$ are
strongly continuous one-parameter groups. Thus,
by Stone's classical theorem $[4,5]$
there are possibly unbounded self-adjoint operators
$A_k : {\rm dom}(A_k) \to \HHH{k}$ $(k=1,\ldots,N)$
defined on dense linear submanifolds in the respective spaces
such that that
${\bf U}(t) \!=\! \big[ \exp(itA_1) \big] \otimes\cdots\otimes \big[ \exp(itA_N) \big]$
$(t\in\RR)$.

\medskip
\rm
At first sight our arguments in [3] rely heavily upon the Hilbert space structure.
In this paper we are going to investigate how far can we get rid of the scalar product.
In Section 2 we revise the arguments on adjusted componentwise continuity of functions
of type $t\mapsto x_{1,t}\otimes\cdots\otimes x_{N,t}$. The new results extend to
to the setting of uniformly convex spaces and we get the following conclusion.

\medskip
\bf 1.3. Proposition. \it Let $\XXX{1},\ldots,\XXX{N}$ be uniformly convex Banach spaces
$($or a bit more generally: reflexive spaces where weak convergence with convergence in norm implies norm-convergence$)$.
Assume $\big[ {\bf U}(t):\ t\in\RR\big]$ is a one-parameter group
where ${\bf U}(t)= U_{1,t}\otimes\cdots\otimes U_{N,t}$
with invertible bounded linear operators $U_{k,t}\in {\cal L}(\XXX{k})$ such that
$\sup_{t\in\RR} \norm{{\bf U}(t)} = \sup_{t\in\RR} \prod_{k=1}^N \norm{U_{k,t}} < \infty$.
Then there exist functions
$\rho_1,\ldots,\rho_N: \RR\to\CC$ respectively $\widetilde{\rho}_1,\ldots,\widetilde{\rho}_N:\RR\to\CC$
such that, for each $k=1,\ldots,N$,
$$\eqalign{
&\big[ \rho_{k,t}(t) U_{k,t}:\ t\in\RR\big] \ \ \hbox{\rm is a strongly continuous family}, \cr
&\big[ \widetilde{\rho}_{k,t}(t) U_{k,t}:\ t\in\RR\big] \ \ \hbox{\rm is a one-parameter group}.\cr}$$

\rm
Unfortunately the later considerations in [3, Sections 4-5] leading to probabilistic considerations
seem to be too closely related with Hilbert space structure.
Minor modifications seem possible when replacing $L^2$-estimates with $L^p$-type estimates
resulting in perhaps technically interesting results with
not much farther generality from the original Hilbert setting of [3].
We do not intend to go into the relevant details in this work.
Nevertheless a perhaps rather aesthetical generalization
with equivalent general Banach norms in the Hilbert spaces instead of the one by inner products
can be developed due to the amenability of the
additive group of $\RR$.

\medskip
\bf 1.4. Theorem. \it In the setting of Theorem 1.2, let
$\nnorm{\cdot}_1,\ldots,\nnorm{\cdot}_N$ denote equivalent Banach-norms on the respective spaces $\HHH{k}$
and
let each operator $U_{k,t}$ be a surjective $\nnorm{\cdot}_k$-isometry
and instead of being an $\langle \cdot \vert \cdots \rangle_k$-isometry (as supposed originally).
Then there are equivalent inner products $\sspr\cdot\cdot_k$ on the respective spaces $\HHH{k}$
such that the conclusions of Theorem 1.3 hold with suitable possibly unbounded
$\sspr\cdot\cdot_k$-selfadjoint operators $A_k$.

\medskip
\rm
The above result is an immediate consequence of the following fact.

\medskip
\bf 1.5. Proposition. \it
Given a linear automorphism of a bounded circular domain ${\bf D}$ in a Hilbert space $\HHHx$,
it is necessarily a scalar type operator as being a preserver of some equivalent inner product.
Moreover given a strongly continuous one-parameter group $\big[ U^t:\ t\in \RR\big]$ of linear automorphisms of ${\bf D}$,
or more generally a bounded strongly continuous one-parameter group of linear operators of $\HHHx$,
there exists an equivalent $\big[ U^t:\ t\in \RR\big]$-invariant inner product on $\HHHx$.

\rm
\medskip
Though it is likely that Proposition 1.5 appeared already in the literature of Banach space geometry,
as far we have not found a proper reference even after an inquiry with several colleagues.
Therefore we devote the short Section 3 to its proof relying upon Banach limits.

\medskip
\rm

In [3] we emphasized the natural connection of Theorem 1.2 with the Jordan theory of bounded symmetric domains
during the Introduction of [3]. One of the aims of this note is to work out the first step in this
direction: in Section 4,
with slight modifications of the proofs in [3], we obtain the full description of
the unbounded JB*-derivations of the infinite-dimensional Cartan factors [6,7] of Types 1,2,3.

\medskip
\bf Theorem 1.6. \it Let $\HHHx$ be a Hilbert space and let
$X\mapsto X^{{\rm T}}$ denote a transposition\footnote{$^*$}{\rm There exists an orthonormed basis
$\{ {\bf e}_j: \ j\in J\}$ in $\HHHx$ such that
$\langle X^{{\rm T}} {\bf  e}_j \vert {\bf e}_\ell \rangle =
\overline{\langle X {\bf  e}_j \vert {\bf e}_\ell \rangle}$ for all $j,\ell\in J$.
}
on ${\cal L}(\HHHx)$. Assume $\big[ {\bf U}(t): \t\in\RR\big]$ resp.
$\big[ {\bf V}(t): \t\in\RR\big]$ are one-parameter groups of surjective isometries of
${\cal A}^{(+)} := \{ X\in {\cal L}(\HHHx):\ X=X^{{\rm T}} \}$
resp.
${\cal A}^{(-)} := \{ Z\in {\cal L}(\HHHx):\ Z=-Z^{{\rm T}} \}$
such that
all the functions
$t\mapsto \langle [{\bf U}(t)X] {\bf x} \vert {\bf y}\rangle$
resp.
$t\mapsto \langle [{\bf V}(t)Z] {\bf x} \vert {\bf y}\rangle$
are continuous for all ${\bf x},{\bf y}\in\HHHx$ and
$X\in{\cal A}^{(+)}$ resp. $Z\in{\cal A}^{(-)}$.


\medskip
\rm

Recall that, by a classical representation theorem by F Riesz [4],
each Type 1 Cartan factor is isometrically isomorphic to some ${\cal L}(\HHH{1},\HHH{2})$
with suitable Hilbert spaces $\HHH{1},\HHH{2}$. Therefore Theorem 1.2 furnishes the complete description
of its unbounded JB*-derivations as operations of the form
$X\mapsto A_1X+XA_2$ with suitable possibly unbounded selfadjoint operators $A_k\in{\cal L}(\HHH{k})$.
By taking into also account that
the surjective linear isometries of a spin factor (Cartan factor of Type 4)
are automatically unitary with respect to the underlying scalar product and also that the
remaining exceptional Cartan factors (of Type 5,6) are finite dimensional (16 resp. 27 dimensions),
hence the JB*-derivations of the enveloping JB*-triple in the Friedmen-Russo type Gelfand-Neimark theorem [8]
are available for any JB*-triple.


\rm



\vfill
\eject

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$\underline{\hbox{\bf MULTILINEAR FUNCTIONALS ON UNIFORMLY SMOOTH SPACES}}$
\bigskip

\bf Assumption. \it $\XXX{1},\ldots,\XXX{N}$ smooth convex Banach spaces

\bf Remark. \rm 1) [1, p.207] $\XXXx$ is uniformly smooth if and only if
$\XXXx^*$ is uniformly convex (rotond),
$\XXXx$ is uniformly convex if and only if
$\XXXx^*$ is uniformly smooth.

2) [1, p.198] In a uniformly convex space weak convergence plus
convergence of norms implies norm (strong) convergence.

3) [1, p.196] Uniformly convex spaces (hence also uniformly smooth spaces) are reflexive.


\bigskip
\bf Lemma 1. \it Let $\phi^{(k)},\psi^{(k)}\in [\XXX{k}]^*$ with
$\norm{\phi^{(k)}}=\norm{\psi^{(k)}}=1$ $(k=1,\ldots,N)$.
Then $\phi^{(1)}\otimes\cdots\otimes\phi^{(N)} = \psi^{(1)}\otimes\cdots\otimes\psi^{(N)}$
if and only if there exist
$\kappa_1,\ldots,\kappa_N\in\TT$ such that $\psi^{(k)}=\kappa_k\phi_k$ $(k=1,\ldots,N)$.

\bigskip
\bf Proof. \rm The part
$0\ne \phi^{(1)}\otimes\cdots\otimes\phi^{(N)} = \psi^{(1)}\otimes\cdots\otimes\psi^{(N)}$
of the assumption
implies
$${\rm ker} \phi^{(k)} = {\rm ker} \psi^{(k)} \qquad (k=1,\ldots,N).$$
(Proof: Due to the reflexivity of the spaces $\XXX{k}$, the are unit vectors $\xxx_k$
with $\phi^{(k)}(\xxx_k)=1$ $(k=1,\ldots,N)$.
Given any index $\ell$, consider any vector ${\bf y}\in{\rm ker}\,\psi^{(k)}$.
With the tuple
$\overline{\xxx}:=(\xxx_1,\ldots,\xxx_{k-1},{\bf y},\xxx_{k+1},\ldots,\xxx_N)$
we have
$0=\psi^{(1)}\otimes\cdots\otimes\psi^{(N)} \big(\overline{\xxx}\big) =
\phi^{(1)}\otimes\cdots\otimes\phi^{(N)} \big(\overline{\xxx}\big) =
\phi^{(k)}({\bf y})$ that is $ {\bf y}\in{\rm ker}\,\phi^{(k)}$
showing that ${\rm ker}\,\psi^{(k)}\subset{\rm ker}\,\phi^{(k)}$.
Similarly also ${\rm ker}\,\phi^{(k)}\subset{\rm ker}\,\psi^{(k)}$.)
Hence $\psi^{(k)} \in \CC \psi^{(k)}$ for each index.
Since $\norm{\phi^{(k)}} = \norm{\psi^{(k)}}=1$
it follows even $\psi^{(k)} \in \TT \phi^{(k)}$. \quad
Qu.e.d.





\bigskip
\bf Lemma 2. \it Let $\phi^{(k)}_j\in[\XXX{k}]^*$ with $\norm{\phi^{(k)}_j}=1$  $(j\in{\cal J},\ k=1,\ldots,N)$
and suppose
$$\phi^{(k)}_j\otimes\cdots\otimes\phi^{(N)}_j(\xxx_1,\ldots,\xxx_N)\to
\phi^{(k)}\otimes\cdots\otimes\phi^{(N)}(\xxx^{(1)},\ldots,\xxx^{(N)})$$
for any
%fixed
$(\xxx^{(1)},\ldots,\xxx^{(N)}) \in \XXX{1}\times\cdots\times\XXX{N}$.
Then there are constants
$\big[ \kappa_j^{(k)} : j\in{\cal J},\ k=1,\ldots,N\big]$ in $\TT$
such that
$$\norm{\kappa_j^{(k)}\phi_j^{(k)} - \phi^{(k)}} \to 0 \qquad (k=1,\ldots,N).$$
\bf Proof. \rm
For any index $j\in{\cal J}$ let
$\eps_j:=
\sum_{k=1}^N \inf_{\mu\in\TT} \norm{ \mu \phi_j^{(k)} -\phi^{(k)} }$.
By the compactness of $\TT$ we can
choose $\mu_j^{(1)},\ldots,\mu_j^{(N)}\in\TT$
such that
$\eps_j:=
\sum_{k=1}^N \norm{ \mu_j^{(k)} \phi_j^{(k)} -\phi^{(k)} }$.
We show $\eps_j\to 0$ as follows:
in the contrary case $\eps:=\lim\sup_{j\in{\cal J}} \eps_j >0$ there would be a
subnet $j_\nu$ $\nu\in{\cal N}$ along with functionals $\psi^{(k)}$ of norm one
and constants $\mu^{(k)}\in\TT$
such that
$\mu_j^{(k)}\to\mu^{(k)}, \norm{\phi_j^{(k)}-\psi^{(k)}} \to 0$ $(k=1,\ldots,N)$.
Then, however, we would have the contradiction (to the previous Lemma)
$$\big[\mu^{(1)} \psi^{(1)}\big]\otimes\cdots\otimes\big[\mu^{(1)} \psi^{(1)}\big] =
\phi^{(1)}\otimes\cdots\otimes\phi^{(N)}, \quad
\exists\, k \quad \mu^{(k)} \psi^{(k)} \ne \phi^{(k)} .$$
Hence it follows immediately that the choice
$$\kappa_j^{(k)} := \mu_j^{(k)} \quad \hbox{for $k<N$ and} \quad
\kappa_j^{(k)} := \overline{\mu^{(1)}\cdots\mu^{(N-1)}}$$
suits the requirements of the lemma. \quad
Qu.e.d.

\bigskip
\bf Proposition. \it
Suppose the functionals $\phi_t^{(k)} \in [\XXX{k}]^*$ $(t\in\RR,\ k=1,\ldots,N)$
are of norm one and the functions
$$t\mapsto \phi^{(k)}_t\otimes\cdots\otimes\phi^{(N)}_t(\xxx_1,\ldots,\xxx_N) $$
are continuous for all fixed
$(\xxx^{(1)},\ldots,\xxx^{(N)}) \in \XXX{1}\times\cdots\times\XXX{N}$.
Then there are functions $t\mapsto \kappa_t^{(k)}$ $(k=1,\ldots,N)$ from $\RR\to\TT$
such that
$$\prod_{k=1}^N \kappa_t^{(k)} =1 \quad (t\in\RR), \qquad
t\mapsto \kappa_t^{(k)} \phi_t^{(k)} \ \hbox{\rm is norm-continuous}\ (k=1,\ldots,N).$$
\bf Proof. \rm
It suffices to establish the statement locally (i.e.
for any $\tau\in\RR$ there is an open interval $I_\tau$ around $\tau$ where the statement holds and
we may assume $\tau=0$ without loss of generality).
%
Since the spaces $\XXX{k}$ are reflexive, we can choose unit vectors $\xxx^{(k)}$
such that $\phi_0^{(1)}(\xxx^{(1)}) = \cdots = \phi_0^{(N)}(\xxx^{(N)}) =1$.
By their aid we define the interval $I_0$ as the connected component of the open set
$\big\{ t\in\RR:\ \phi_t^{(1)}(\xxx^{(1)}),\ldots,\phi_t^{(N)}(\xxx^{(N)}) \ne 0 \big\}$.

For the parameters $t\in I_0$, we define the functions $t\mapsto\kappa_t^{(k)}$ as
$$\kappa_t^{(k)} := \overline{\phi_t^{(k)} (\xxx^{(k)})}/\abs{\phi_t^{(k)} (\xxx^{(k)}} \quad (k<N), \qquad
\kappa_t^{(N)} :=\overline{\kappa_t^{(1)}\cdots\kappa_t^{(N-1)}} .$$

To establish the norm continuity of the functions $t\mapsto \kappa_t^{(k)} \phi_t^{(k)}$ (on $I_0$)
we consider a convergent sequence $t_n \to t_0$ in $I_0$ and show that
$(*)$ $\norm{ \kappa_{t_n}^{(k)} \phi_{t_n}^{(k)} - \kappa_{t_0}^{(k)} \phi_{t_0}^{(k)}}\to 0$.
In view of Lemma 1, there are convergent sequences $\mu_n^{(k)}\to 1$ $(k=1,\ldots,n)$ in $\TT$
such that $\norm{ \mu_{t_n}^{(k)} \phi_{t_n}^{(k)} - \phi_{t_0}^{(k)}}\to 0$  $(k=1,\ldots,n)$.
Fix any index $k$. To prove $(*)$ it suffices to show that
$(**)$ $\kappa_{t_n}^{(k)}/\mu_{t_n}^{(k)}\to 1$.
Observe that
$\kappa_t^{(k)} \phi_t^{(k)} (\xxx_k) =
\abs{\phi_t^{(k)} (\xxx_k)}$ for any $t\in I_0$.
Therefore
$${\kappa_{t_n}^{(k)} \phi_t^{(k)} \over \mu_{t_n}^{(k)}} =
{\kappa_{t_n}^{(k)} \phi_t^{(k)} (\xxx_k) \over \mu_{t_n}^{(k)} \phi_{t_n}^{(k)}(\xxx_k) } =
{\abs{\phi_{t_n}^{(k)} (\xxx_k)} \over \mu_{t_n}^{(k)} \phi_{t_n}^{(k)}(\xxx_k) } =
{\abs{\mu_{t_n}^{(k)}\phi_{t_n}^{(k)} (\xxx_k)} \over \mu_{t_n}^{(k)} \phi_{t_n}^{(k)}(\xxx_k) }
\to { \abs{\phi_{t_0}(\xxx_k)} \over \abs{\phi_{t_0}(\xxx_k)}} =1.$$
Qu.e.d.

\rm

\bigskip

\bf Definition. \rm The {\it injective tensor product} is the space
$$\XXX{1}\otimes\cdots\otimes\XXX{N}  :=
\big\{ \hbox{$N$-linear functionals $\XXX{1}\times\cdots\times\XXX{N}\to\CC$} \big\}$$
with its natural norm.

The {\it projective tensor product} $\XXX{1}\Otimes\cdots\Otimes \XXX{N}$ is the
space of the linear functionals on $\XXX{1}\otimes\cdots\otimes\XXX{N}$
of the form
$$\sum_{n=1}^\infty \delta_{\xxx^{(1)}_n,\ldots,\xxx^{(N)}_n},
\qquad \delta_{\xxx^{(1)},\ldots,\xxx^{(N)}}: \phi\mapsto \Phi(\xxx^{(1)},\ldots,\xxx^{(N)})$$
with the norm
$$\norm{\varphi} := \inf\Big\{ \sum_{n=1}^\infty  \prod_{k=1}^N \norm{\xxx^{(k)}_n} : \
\varphi = \sum_{n=1}^\infty \delta_{\xxx^{(1)}_n,\ldots,\xxx^{(N)}_n}\Big\} .$$
\bf Remark. \rm
$\XXX{1}\Otimes\cdots\Otimes \XXX{N}$ is a
(metric) predual of $\XXX{1}\otimes\cdots\otimes\XXX{N}$.
\rm

\bigskip
\bf Lemma 3. \it Let $t\mapsto L_t$ be a mapping $\RR\to {\cal L}(\XXX{1}\Otimes\cdots\Otimes \XXX{N})$
such that all the functions $t\mapsto L_t \delta_{\xxx^{(1)},\ldots,\xxx^{(N)}}$
are continuous and $\norm{L_t}\le 1$ $(t\in\RR)$.
Then also all the functions $t\mapsto L_t\varphi$ $(\varphi\in\XXX{1}\Otimes\cdots\Otimes \XXX{N})$
are continuous.

\bigskip
\bf Proof. \rm Let $\varphi\in\XXX{1}\Otimes\cdots\Otimes \XXX{N}$ be arbitrarily fixed.
By definition we can represent it in the form
$$\varphi = \sum_{n=1}^\infty \delta_{\xxx^{(1)}_n,\ldots,\xxx^{(N)}_n} , \quad
\sum_{n=1}^\infty  \prod_{k=1}^N \norm{\xxx^{(k)}_n} < \infty .$$
Therefore the statement follows from the observation that the partial sums
$t\mapsto \sum_{n=1}^\nu L_t \delta_{\xxx^{(1)}_n,\ldots,\xxx^{(N)}_n}$ $(\nu=1,2,\ldots)$
form a uniformly convergent sequence of continuous functions $\RR\to\XXX{1}\Otimes\cdots\Otimes \XXX{N}$
due to the inequalities
$$\norm{\sum_{n=1}^\nu L_t \delta_{\xxx^{(1)}_n,\ldots,\xxx^{(N)}_n} -
\sum_{n=1}^\mu L_t \delta_{\xxx^{(1)}_n,\ldots,\xxx^{(N)}_n}} \le
\sum_{n=\min\{\mu,\nu\}}^{\max\{\mu,\nu\}} \prod_{k=1}^N \norm{\xxx^{(k)}_n} \to 0
\qquad (\mu,\nu\to\infty). $$




\vfill \eject

$\underline{\hbox{\bf TYPES 2,3}}$
\bigskip

$\CC:=\{$complex numbers$\}$, $\TT:=\{ \kappa\in\CC:\ \abs{\kappa}=1\}$.
\smallskip
$\HHHx$ complex Hilbert space,
inner product $\langle . \vert . \rangle$, norm $\norm{.}$, \quad
$\hhh^* :=[\HHHx \ni{\bf x}\mapsto \langle {\bf x}\vert \hhh\rangle]$
\smallskip
${\rm Ball}(\HHHx):= \{ \hhh\in\HHHx:\ \norm{\hhh}<1$ unit ball, \quad
$\partial{\rm Ball}(\HHHx):= \{ \hhh\in\HHHx:\ \norm{\hhh}=1$ unit sphere.
\smallskip
${\bf g}^* \otimes \hhh^* := [({\bf x},{\bf y})\mapsto {\bf g}^* ({\bf x}) \hhh^*({\bf y})]$, \quad
${\bf g}^* \land \hhh^* := {\bf g}^* \otimes \hhh^* - \hhh^* \otimes {\bf g}^*$.

$$\vtop{
\hbox{$F_2:=\{$symmetric continuous bilinear functionals $\HHHx \times \HHHx \to \CC\} $}
\hbox{$F_3:=\{$antisymmetric continuous bilinear functionals $\HHHx \times \HHHx \to \CC\} $}}$$



\bf Lemma 0. \rm [Stach\'o 2010, Lemma 2.3]. \it
Suppose ${\bf F}: \RR\to \PP(\HHHx) :=
\{ \TT {\bf g}: \ \langle {\bf g}\vert{\bf g}\rangle=1 \}$
is a continuous mapping with respect to the distance
%$\eqii$.
${\rm dist}(\TT{\bf g} ,\TT\hhh) :=
\min_{\abs{\kappa}=\abs{\lambda}=1} \abs{\kappa {\bf g} - \lambda\hhh}$.
Then ${\bf F}(t) =\TT \hhh_t$ $(t\in\RR)$ for some
continuous function $t\mapsto \hhh_t \in \partial {\rm Ball}(\HHHx)$.
\rm

\bigskip

\bf Lemma 1. \it Let $t\mapsto {\bf e}_t$ be a function $\RR\to \partial {\rm Ball}(\HHHx)$
ranging in unit vectors. If
for any convergent net $t_\nu \to t\in\RR$ there exists a constant $\sigma_*\in\{ -1,1\}$
along with a subnet $t_{\nu_\alpha}$ such that
${\bf e}_{t_{\nu_\alpha}} \to \sigma_* {\bf e}_t$
then there also exists a function $\sigma:\RR\to\{-1,1\}$ such that
$t\mapsto \sigma(t) {\bf e}_t$ is continuous.

\bigskip
\bf Proof. \rm
We can apply [Stach\'o 2010, Lemma 2.3] as follows. Observe that
$t\mapsto \TT {\bf e}_t$ is necessarily continuous.
Indeed, if $t_\nu\to t$ then
$\liminf_\nu {\rm dist}(\TT {\bf e}_{t_\nu}, \TT {\bf e}_t) \le
\liminf_\nu \min_{\sigma_*=\pm1} \abs{{\bf e}_{t_\nu} - \sigma_* {\bf e}_t} =0$.
Thus $\TT {\bf e}_t = \TT \hhh_t$ $(t\in\RR)$ for some continuous function
$t\mapsto \hhh_t\in\partial{\rm Ball}(\HHHx)$,
that is $t\mapsto\kappa(t){\bf e}_t$ is continuous for some $\kappa:\RR\to\TT$.
For any point $t\in\RR$, we can choose an open interval $I_t$ around it such that
$\big\langle \kappa(s){\bf e}_s \big\vert \kappa(t){\bf e}_t \big\rangle >0$ for all $s\in I_t$.
Each function $\sigma_t(s) := {\rm sign}\, {\rm Re} \big( \kappa(s)\overline{|\kappa(t)}\big)$
is continuous on the the interval $I_t$.
By the $\sigma$-compactness of $\RR$, we can find a sequence
$\cdots< \tau_{-2}<\tau_{-1}< \tau_0< \tau_1< \tau_2<\cdots$ such that
each interval $[\tau_{n-1},\tau_n]$ is included in some of member of
the family $\{ I_t:\ t\in\RR\}$, say $[\tau_{n-1},\tau_n] \subset I_{t_n}$ $(n=0,\pm1,\pm2,\ldots)$.
The we obtain a function $\sigma$ suiting the requirements of the lemma
by letting $\sigma(t):=\sigma_{t_0}(t)$ for $t\in[\tau_{-1},\tau_0]$,
and then, recursively for $k=1,2,\ldots$, \
$\sigma(t):=\sigma(\tau_{-k})\sigma_{t_{-k}} (\tau_{-k}) \sigma_{t_{-k}}(t)$
for $t\in[\tau_{-k-1},\tau_{-k}]$ and
$\sigma(t):=\sigma(\tau_{k-1})\sigma_{t_{k}} (\tau_{k-1}) \sigma_{t_{k}}(t)$
for $t\in[\tau_{k-1},\tau_{k}]$, respectively,


\bigskip
\bf Corollary 1. \it If $t\mapsto \phi({\bf e}_t,{\bf e}_t)$ is continuous for all $\phi\in F_2$
then there is a function $\sigma:\RR\to\{-1,1\}$
such that $t\mapsto \sigma(t) {\bf e}_t$ is continuous.

\bigskip
\bf Proof. \rm Given any vector $\hhh\in\HHHx$, with the functional
$\phi_\hhh ({\bf x},{\bf y}):= \hhh^* \otimes \hhh^* (\in F_2)$
%$\big\langle {\bf x} \big\vert \hhh \big\rangle \big\langle {\bf y} \big\vert \hhh \big\rangle$
we see that the function
$t\mapsto \big\langle {\bf e}_t \big\vert \hhh \big\rangle^2$ is continuous.
Consider a convergent net $t_\nu\to t\in\RR$ and let $t_{\nu_\alpha} (\to t)$ be its
{\it universal subnet}.
Since the range $\{ {\bf e}_t: \ t\in\RR\}$ is contained in the weakly compact
closed unit ball $\overline{\rm Ball}(\HHHx)$,
for some vector  ${\bf g}\in \overline{\rm Ball}(\HHHx)$ we have
$$\big\langle {\bf e}_{t_{\nu_\alpha}} \big\vert \hhh \big\rangle \to
\big\langle {\bf g} \big\vert \hhh \big\rangle , \quad
\big\langle {\bf g} \big\vert \hhh \big\rangle^2 = \big\langle {\bf e}_t \big\vert \hhh \big\rangle^2
\qquad (\hhh\in\HHHx) .$$
Since the mapping $\hhh\to \big\langle {\bf g} \big\vert \hhh \big\rangle$ is conjugate-linear,
it follows that ${\bf g}\in \big\{ \pm {\bf e}_t \big\}$.
That is, for some $\sigma_* \in\{ \pm 1\}$ we have
$\sigma_* {\bf e}_t = {\rm weak}\,\lim_{\nu_\alpha} {\bf e}_{t_{\nu_\alpha}}$.
It is well-known that weak convergence entails norm convergence for nets of unit vectors in Hilbert spaces.
Therefore we even have $\sigma_* {\bf e}_t = {\rm norm}\,\lim_{\nu_\alpha} {\bf e}_{t_{\nu_\alpha}}$
whence Lemma 1 applies.

\bigskip
\bf Corollary 1$^\prime$. \it Let $t\mapsto [{\bf e}_t,{\bf f}_t]$ be a function
$\RR\to\{${\rm orthonormed 2-frames}$\}$. If
$t\mapsto \phi(\xi {\bf e}_t+\eta {\bf f}_t,\xi {\bf e}_t+\eta{\bf f}_t)$
is continuous for all $\phi\in F_2$ and $\xi,\eta\in \{0,\pm 1\}$ then there is a function
$\sigma:\RR\to\{-1,1\}$ such that $t\mapsto \sigma(t) [{\bf e}_t,{\bf f}_t]$ is
continuous.

\bigskip
\bf Proof. \rm
By Corollary 1, for some $\{-1,1\}$-valued functions
$t\mapsto \kappa(t), t\mapsto \lambda(t), t\mapsto\mu(t)$,
the functions $t\mapsto \lambda(t) {\bf e}_t, t\mapsto \kappa(t){\bf f}_t$,
$t\mapsto \mu(t) 2^{-1/2} [{\bf e}_t + {\bf f}_t]$ are continuous.
Therefore also $t\mapsto \big\langle \lambda(t) \big\vert {\bf e}_t \big\rangle = \lambda(t)\mu(t)$ and
similarly $t\mapsto \kappa(t)\mu(u)$ are continuous and necessarily constant with value $1$ or $-1$.
Hence the statement is immediate.

\bigskip
\bf Corollary 2. \it Suppose $t\mapsto [{\bf e}_t,{\bf f}_t,{\bf g}_t]$
is a function
$\RR\to\{${\rm orthonormed 3-frames in }$\HHHx\}$  such that
the functions
%$t\mapsto {\bf e}_t^* \land {\bf f}_t^*$, $t\mapsto {\bf e}_t^* \land {\bf g}_t^*$
$t\mapsto\phi({\bf e}_t,{\bf f}_t)$, $t\mapsto\phi({\bf e}_t,{\bf g}_t)$, $t\mapsto\phi({\bf f}_t,{\bf g}_t)$
are continuous for all $\phi\in F_3$. Then there exists a function $\sigma:\RR\to\{-1,1\}$
such that $t\mapsto \sigma(t)[{\bf e}_t,{\bf f}_t,{\bf g}_t]$ is continuous.

\bigskip
\bf Proof. \rm
Consider a convergent net $t_\nu\to t\in\RR$ and let $t_{\nu_\alpha} (\to t)$ be again its
universal subnet.
By the weak compactness of $\overline{\rm Ball}(\HHHx)$,
for some vectors  ${\bf e}_*, {\bf f}_* \in \overline{\rm Ball}(\HHHx)$ we have
$\big\langle {\bf e}_{t_{\nu_\alpha}} \big\vert \hhh \big\rangle \to
\big\langle {\bf e}_* \big\vert \hhh \big\rangle$,
$\big\langle {\bf f}_{t_{\nu_\alpha}} \big\vert \hhh \big\rangle \to
\big\langle {\bf f}_* \big\vert \hhh \big\rangle$,
$\big\langle {\bf g}_{t_{\nu_\alpha}} \big\vert \hhh \big\rangle \to
\big\langle {\bf g}_* \big\vert \hhh \big\rangle$
for any fixed vector $\hhh\in\HHHx$.
Given any couple of vectors ${\bf x},{\bf y}\in\HHHx$, with the functional
$\phi_{{\bf x},{\bf y}} := {\bf x}^* \land {\bf y}^* (\in F_3)$
we see that the function
$\tau\mapsto
\big\langle {\bf e}_\tau \big\vert {\bf x} \big\rangle  \big\langle {\bf f}_\tau \big\vert {\bf y} \big\rangle -
\big\langle {\bf f}_\tau \big\vert {\bf x} \big\rangle  \big\langle {\bf e}_\tau \big\vert {\bf y} \big\rangle$
is continuous. Hence we conclude that
$$\big\langle {\bf e}_\tau \big\vert {\bf x} \big\rangle  \big\langle {\bf f}_t \big\vert {\bf y} \big\rangle -
\big\langle {\bf f}_t \big\vert {\bf x} \big\rangle  \big\langle {\bf e}_t \big\vert {\bf y} \big\rangle =
\big\langle {\bf e}_* \big\vert {\bf x} \big\rangle  \big\langle {\bf f}_* \big\vert {\bf y} \big\rangle -
\big\langle {\bf f}_* \big\vert {\bf x} \big\rangle  \big\langle {\bf e}_* \big\vert {\bf y} \big\rangle
\qquad ({\bf x},{\bf y}\in\HHHx) .$$
That is we have ${\bf e}_t^* \land {\bf f}_t^* = {\bf e}_*^* \land {\bf f}_*^*$.
It is well-known from classical linear algebra
that then for some constants $\lambda_{11},\lambda_{12},\lambda_{21},\lambda_{22}\in\CC$
we have
$${\bf e}_* = \lambda_{11} {\bf e}_t + \lambda_{12} {\bf f}_t, \quad
{\bf f}_* = \lambda_{21} {\bf e}_t + \lambda_{22} {\bf f}_t, \quad
{\rm det}\big[ \lambda_{k,\ell} \big] =1 .$$
In particular ${\bf e}_* \in \CC {\bf e}_t + \CC {\bf f}_t$.
Similarly (with ${\bf g}_{t_{\nu_\alpha}}$ in place of ${\bf f}_{t_{\nu_\alpha}})$,
also ${\bf e}_* \in \CC {\bf e}_t + \CC {\bf g}_t$
and therefore ${\bf e}_* \in \CC {\bf e}_t$ that is
${\bf e}_*=\lambda {\bf e}_t$ with $\abs{\lambda} \le 1$.
Analogously ${\bf f}_* =\mu {\bf f}_t$ and ${\bf g}_* =\kappa {\bf g}_t$ with $\abs{\mu},\abs{\kappa}\le 1$ as well.
Since ${\bf e}_t^* \land {\bf f}_t^* = {\bf e}_*^* \land {\bf f}_*^*$,
it follows $\lambda\mu=1$. Similarly $\lambda\kappa=\mu\kappa=1$.
Therefore $\kappa=\lambda=\mu \in\{ -1,1\}$. By writing
$\sigma_t$ for the common value of $\kappa,\lambda,\mu$,
we get
$\lim_{\nu_\alpha} {\bf e}_{t_{\nu_\alpha}} = \sigma_t {\bf e}_t$.
According to Lemma 1, for suitable functions $t\mapsto \lambda(t), t\mapsto \mu(t), t\mapsto \kappa(t)$
ranging in $\{-1,1\}$ the functions
$t\mapsto \lambda(t) {\bf e}_t, t\mapsto \mu(t) {\bf f}_t, t\mapsto \kappa(t) {\bf g}_t$
are continuous. In particular also both
$t\mapsto [\lambda(t) {\bf e}_t]^* \land [\lambda(t) {\bf f}_t]^*$ and
$t\mapsto {\bf e}_t^* \land {\bf f}_t^*$ are continuous entailing
the continuity and hence the constant being of the $\{-1,1\}$-valued function $t\mapsto \lambda(t)\mu(t)$.
$t\mapsto [\lambda(t) {\bf e}_t]^* \land [\lambda(t) {\bf f}_t]^*$.
Thus $\mu(t)\equiv \lambda(t)$ or $\mu(t)\equiv -\lambda(t)$.
Similarly $\kappa(t)\equiv \lambda(t)$ or $\kappa(t)\equiv -\lambda(t)$.
In any case
%the function
$t\mapsto\lambda(t)[{\bf e}_t,{\bf f}_t,{\bf g}_t]$
must be continuous.

\bigskip
\hrule
\bigskip

\bf Example of prebidual

\bigskip
\rm
Let $F:={\cal L}({\bf H}^{(1)},{\bf H}^{(2)})$ be a Cartan factor of type 1,
$E$ a weak*-dense subtriple of $F$.

QUESTION: Is $E$ a TRO (ternary ring of operators)? \quad NO.

\bigskip
\bf Example. \rm
Let ${\bf H}^{(1)}={\bf H}^{(2)}:={\bf H}$ an infinite dimensional complex Hilbert space with conjugation.
Consider
$$\hbox{$E:=F_0 + F_1 = \{$compact op.-s in $F\} + \{$antisymm. op.-s in $F\}$ .}$$
If $a,b,c\in F_0$ and $A,B,C\in F_1$ then
$\{ (a+A)(b+B)(c+C)\} =
{1\over 2}\big[(a+A)(b+B)^*(c+C) + (c+C)(b+B)^*(a+A) \big] \in
\{ ABC \} + F_0$ because $F_0$ is a self-adjoint multiplicative ideal of $F$.
On the other hand,
$[ (a+A)(b+B)(c+C)] = (a+A)(b+B)^*(c+C) \in AB^*C + F_0$.
Let ${\bf H} = \oplus_{k=1}^3 {\bf H}_k$ with ${\rm dim}({\bf H}_k)={\rm dim}({\bf H})$
with self-conjugate components ${\bf H}_k = \overline{{\bf H}_k}$.
If $A,B,C$ have the (antisymmetric) matrix forms
$\big[ \alpha_{jk} I\big]_{j,k=1}^3,\ \big[ \beta_{jk} I\big]_{j,k=1}^3,\ \big[ \gamma_{jk} I\big]_{j,k=1}^3$
with respect to the decomposition ${\bf H} = \oplus_{k=1}^3 {\bf H}_k$,
then
$$\{ ABC \} + F_0 = [ ABC ] + F_0 \iff
\Big\{ \big[ \alpha_{jk} \big] \big[ \beta_{jk} \big] \big[ \gamma_{jk} \big]\Big\} =
\Big[ \big[ \alpha_{jk} \big] \big[ \beta_{jk} \big] \big[ \gamma_{jk} \big]\Big] .$$
We have no equality e.g. with the matrices
$A \!=\!B \!:= \!\left[ \matrix{0 & 1 & 1 \cr -1 & 0 &1 \cr -1 &-1 &0 \cr} \right]$,
$C \!:=\! \left[ \matrix{0 & 1 & 0 \cr -1 & 0 &0 \cr 0 &0 &0 \cr} \right]$.


\vfill
\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vtop{
\hbox{
$\underline{\hbox{\bf BOUNDED STROGLY CONTINUOUS 1-PARAMETER GROUPS}}$}
\hbox{$\underline{\hbox{\bf IN HILBERT SPACES}}$}
}

\bigskip



Let $\HHHx$ be a Hilbert space with scalar product $\langle . \vert . \rangle$.
Suppose $\vert\Vert . \Vert\vert$ is an equivalent norm on $\HHHx$
that is $M^{-1} \norm{{\bf x}}\le \nnorm{{\bf x}}\le M\norm{{\bf x}}$ $({\bf x}\in\HHHx$).


\bigskip
\bf Proposition. \it
Assume
${\bf U}$ is a
surjective isometry $\HHHx\to\HHHx$ with respect to the norm $\nnorm.$.
Then there exists a ${\bf U}$-invariant equivalent scalar product $\sspr..$.\footnote{$^*$}{\rm That
is $\norm{{\bf x}}^2/ M \le \sspr{{\bf x}}{{\bf x}} \le M \norm{{\bf x}}^2$ and
$\sspr{U{\bf x}}{U{\bf y}} = \sspr{{\bf x}}{{\bf y}}$
for some $M>0$ and for all ${\bf x},{\bf y}\in\HHHx$.}

\bigskip
\bf Proof. \rm
By assumption we have
$$M^{-1}\norm{{\bf x}} \le \nnorm{{\bf x}} \le M \norm{{\bf x}} \ , \quad
M^{-2}\norm{{\bf x}} \le \norm{{\bf U} {\bf x}} \le M^2 \norm{{\bf x}} \qquad
({\bf x}\in\HHHx,\ t\in\RR) \ .$$
Consider a Banach limit $\Lambda_{N\to\infty}$ (for its existence see [??])
on the space $ c_{{\rm b}}$
of all bounded bisequences as functions $\ZZ=\{ 0,\pm1,\pm2,\ldots\} \!\to\!\CC$.
Thus $\Lambda_{N\to\infty}$ is a linear functional $c_{{\rm b}} \!\to\! \CC$
such that
$$\liminf_{N\to\infty} f(N) \le \Lambda_{N\to \infty} f(N) \le \limsup_{N\to\infty} f(N) \qquad
\hbox{whenever} \quad {\rm range}(f) \subset\RR \ .
\leqno(1)$$
Since the functions $n\mapsto {\bf U}^n{\bf x}$ are bounded with respect to both equivalent norms
$\norm.$ and $\nnorm.$, the operation
$$\sspr{{\bf x}}{{\bf y}} := \Lambda_{N\to\infty} \left(
{1\over 2N+1} \sum_{n=-N}^N \spr{{\bf U}^u{\bf x}}{{\bf U}^n{\bf y}}\ dt \right) $$
is well-defined for every couple of vectors ${\bf x},{\bf y}\in\HHHx$.
The sesquilinearity of the product $\spr..$ along with the inequalities (1)
entail immediately that
$\sspr..$ is also a scalar product on $\HHHx$ with
$$M^{-4} \spr{{\bf x}}{{\bf x}}  \le \sspr{{\bf x}}{{\bf x}} \le M^{4} \spr{{\bf x}}{{\bf x}}
\qquad ({\bf x}\in\HHHx)\ .$$
We complete the proof by showing the $U$-invariance of the new scalar product as follows.
Indeed, we have
%$$\left\vert \sum_{n=k}^\ell \spr{U^n{\bf x}}{U^n{\bf y}} \right\vert \le M^4 (\ell-k+1) \norm{{\bf x}}^2 \qquad
%\hbox{whenever $k,\ell\in \ZZ$ with $k\le \ell$}.$$
%Therefore
$$\eqalign{
&\sspr{U{\bf x}}{U{\bf y}} =
\Lambda_{N\to\infty} \left(  {1\over 2N+1} \sum_{n=-N}^N
\spr{U^{n+1}{\bf x}}{U^{n+1}{\bf y}} \right) =\cr
%&= \Lambda_{N\to\infty} \left(  {1\over 2N+1} \sum_{n=-N+1}^{N+1}
%\spr{U^n{\bf x}}{U^n{\bf y}} \right) =\cr
&= \Lambda_{N\to\infty} \left(  {1\over 2N+1} \left[ -\spr{U^{-N}{\bf x}}{U^{-N}{\bf y}}
+ \sum_{n=-N}^N \spr{U^n{\bf x}}{U^n{\bf y}}
+ \spr{U^{N+1}{\bf x}}{U^{N+1}{\bf y}} \right] \right) =\cr
&= \Lambda_{N\to\infty} \left(  {1\over 2N+1} \sum_{n=-N}^N \spr{U^n{\bf x}}{U^n{\bf y}} + O(N^{-1}) \right) =\cr
&= \Lambda_{N\to\infty} \left(  {1\over 2N+1} \sum_{n=-N+m}^{N+m} \spr{U^n{\bf x}}{U^n{\bf y}} \right) =\cr
&= \sspr{{\bf x}}{{\bf y}} \ . \qquad {\rm Qu.e.d.}\cr}$$


\DEL{
Given any $m\in\ZZ$, we have
$$\left\vert \sum_{n=k}^\ell \spr{{\bf U}^n{\bf x}}{U^n{\bf y}} \right\vert \le M^4 (\ell-k+1) \norm{{\bf x}}^2 \qquad
\hbox{whenever $k,\ell\in \ZZ$ with $k\le \ell$}.$$
Therefore
$$\eqalign{
&\sspr{U^m{\bf x}}{U^m{\bf y}} =
\Lambda_{N\to\infty} \left(  {1\over 2N+1} \sum_{n=-N}^N
\spr{U^{n+m}{\bf x}}{U^{n+m}{\bf y}} \right) =\cr
&= \Lambda_{N\to\infty} \left(  {1\over 2N+1} \sum_{n=-N+m}^{N+m}
\spr{U^n{\bf x}}{U^n{\bf y}} \right) =\cr
&= \Lambda_{N\to\infty} \left(  {1\over 2N+1} \left[
\sum_{n=-N}^N +{\rm sgn}(m)\sum_{n=-N+m}^{-M} + {\rm sgn}(m)\sum_{n=N}^{N+m} \right]
\spr{U^n{\bf x}}{U^n{\bf y}} \right) =\cr
&= \Lambda_{N\to\infty} \left(  {1\over 2N+1} \sum_{n=-N+m}^{N+m} \spr{U^n{\bf x}}{U^n{\bf y}} + O(N^{-1}) \right) =\cr
&= \Lambda_{N\to\infty} \left(  {1\over 2N+1} \sum_{n=-N+m}^{N+m} \spr{U^n{\bf x}}{U^n{\bf y}} \right) =\cr
&= \sspr{{\bf x}}{{\bf y}} \ . \qquad {\rm Qu.e.d.}\cr}$$
}%ENDDEL

\bigskip
\bf Proposition. \it
Suppose $\vert\Vert . \Vert\vert$ is an equivalent norm on $\HHHx$ and
let ${\bf U} :=\big[ {\bf U}^t:\ t\in \RR\big]$ be a strongly continuous one-parameter group of
surjective isometries $\HHHx\to\HHHx$ with respect to the norm $\nnorm.$.
Then there exists a ${\bf U}$-invariant equivalent scalar product $\sspr..$.\footnote{$^*$}{\rm That
is $\norm{{\bf x}}^2/ M \le \sspr{{\bf x}}{{\bf x}} \le M \norm{{\bf x}}^2$ and
$\sspr{U^t{\bf x}}{U^t{\bf y}} = \sspr{{\bf x}}{{\bf y}}$
for some $M>0$ and for all ${\bf x},{\bf y}\in\HHHx$.}

\bigskip
\bf Proof. \rm
Since $\vert\Vert . \Vert\vert$ is an equivalent norm on $\HHHx$
and since all the operators ${\bf U}^t,{\bf U}^{-t}$ are its isometries,
for some constant $M\in[1,\infty)$ we have
$$M^{-1}\norm{{\bf x}} \le \nnorm{{\bf x}} \le M \norm{{\bf x}} \ , \quad
M^{-2}\norm{{\bf x}} \le \norm{{\bf U}^t {\bf x}} \le M^2 \norm{{\bf x}} \qquad
({\bf x}\in\HHHx,\ t\in\RR) \ .$$

Consider a Banach limit $L_{R\to\infty}$ (see [??]) on the space ${\cal C}_{{\rm b}} [0,\infty)$
of all bounded continuous functions $[0,\infty)\!\to\!\CC$.
Thus $L_{R\to\infty}$ is a linear functional ${\cal C}_{{\rm b}} [0,\infty) \!\to\! \CC$
such that
$$\liminf_{R\to\infty} f(R) \le L_{R\to \infty}f(R) \le \limsup_{R\to\infty} f(R) \qquad
\hbox{whenever} \quad {\rm range}(f) \subset\RR \ .
\leqno(1')$$
Since the functions $t\mapsto {\bf U}^t{\bf x}$ are continuous and bounded with respect to both equivalent norms
$\norm.$ and $\nnorm.$, the operation
$$\sspr{{\bf x}}{{\bf y}} := L_{R\to\infty} \left(
{1\over 2R} \int_{-R}^R \spr{{\bf U}^t{\bf x}}{{\bf U}^t{\bf y}}\ dt \right) $$
is well-defined for every couple of vectors ${\bf x},{\bf y}\in\HHHx$.
The sesquilinearity of the product $\spr..$ along with the inequalities (1')
entail immediately that
$\sspr..$ is also a scalar product on $\HHHx$ with
$$M^{-4} \spr{{\bf x}}{{\bf x}}  \le \sspr{{\bf x}}{{\bf x}} \le M^{4} \spr{{\bf x}}{{\bf x}}
\qquad ({\bf x}\in\HHHx)\ .$$
We complete the proof by showing the ${\bf U}$-invariance of the new scalar product as follows.
Given any parameter $s\in\RR$, we have
$$\left\vert \int_I \spr{{\bf U}^t{\bf x}}{{\bf U}^t{\bf y}}\ dt \right\vert \le M^4 s \norm{{\bf x}}^2 \qquad
\hbox{whenever $I$ is an interval of length $\abs{s}$}. $$
Therefore
$$\eqalign{
&\sspr{{\bf U}^s{\bf x}}{{\bf U}^s{\bf y}} =
L_{R\to\infty} \left(  {1\over 2R} \int_{-R}^R \spr{{\bf U}^{t+s}{\bf x}}{{\bf U}^{t+s}{\bf y}}\ dt \right) =\cr
&= L_{R\to\infty} \left(  {1\over 2R} \int_{-R+s}^{R+s} \spr{{\bf U}^t{\bf x}}{{\bf U}^t{\bf y}}\ dt \right) =\cr
&= L_{R\to\infty} \left(  {1\over 2R} \left[\int_{-R}^R +\int_{-R+s}^{-R} + \int_R^{R+s} \right]
\spr{{\bf U}^t{\bf x}}{{\bf U}^t{\bf y}}\ dt \right) =\cr
&= L_{R\to\infty} \left(  {1\over 2R} \int_{-R+s}^{R+s} \spr{{\bf U}^t{\bf x}}{{\bf U}^t{\bf y}}\ dt + O(R^{-1}) \right) =\cr
&= L_{R\to\infty} \left(  {1\over 2R} \int_{-R+s}^{R+s} \spr{{\bf U}^t{\bf x}}{{\bf U}^t{\bf y}}\ dt \right) =\cr
&= \sspr{{\bf x}}{{\bf y}} \ . \qquad {\rm Qu.e.d.}\cr}$$

\bigskip
\bf Proposition. \it In general, if $[{\bf U}^t:\ t\in\RR]$ is a strongly continuous one-parameter group
of bounded linear operators on a Banach space $E$ such that $M:=\sup_t \norm{{\bf U}^t} <\infty$,
then there exists an equivalent norm $\Vert\vert . \vert\Vert$ on $E$
such that $M^{-1} \norm{.} \le \Vert\vert .\vert\Vert \le M$ with respect to which
all the operators ${\bf U}^t$ are isometries.

\bigskip
\bf Proof. \rm
By the aid of any Banach limit $L_{R\to\infty}$, a similar argument as in the previous proof shows that
the functional
$$\Vert\vert {\bf x} \vert\Vert := L_{R\to \infty} \left( R^{-1} \int_{-R} \norm{{\bf U}^t {\bf x}} \ dt \right)
\qquad ({\bf x}\in E)$$
is a norm suiting the requirements of the proposition.

\bigskip
\bf Remark. \rm %Nevertheless,
It is easy to construct a bounded
one-parameter group $[{\bf U}^t:\ t\in\RR]$ even on the 4-dimensional space ${\rm Mat}(2,2,\CC)$  which cannot be
of the form ${\bf U}^t X = \exp(it A_1) X \exp(it A_2)$.

\bigskip
\bf Example. \rm Let
$\Pi_1:X\mapsto {1\over 2}(X+X^{{\rm T}})$,
$\Pi_2:X\mapsto {1\over 2}(X-X^{{\rm t}})$
denote the bicontractive projections onto the symmetric and antisymmetric parts of ${\rm Mat}(2,2,\CC)$,
respectively.
Clearly, the operators
$${\bf U}^t X := \Pi_1 X + e^{it} \Pi_2 X \qquad \big( X\in {\rm Mat}(2,2,\CC)\big)$$
form a bounded (necessarily norm continuous) one-parameter group in
${\cal L}\big( {\rm Mat}(2,2,\CC)\big)$. Suppose ${\bf U}^t X= U_1 X U_2$
where
$U_k^t =\exp(itA_k)$.
Given any parameter $t\in\RR$,
since ${\bf U}^t$ preserves the manifold ${\rm ran}(\Pi_1)$ of symmetric matrices,
necessarily $U_1^t=U_2^{-t}=\exp(ta)\big[ {1\ 0 \atop 0\ 1}\big]$
for some $a\in\CC$.\footnote{$^*$}{Proof:
We have
${d\over dt}\vert_{t=0} U_1^t X U_2^t = A_1 X + X A_2$
for the matrices
$X=\big[ {1\ 0 \atop 0\ 1 }\big],\big[ {0\ 0 \atop 0\ 1}\big]$,
resp. $X=\big[ {0\ 1 \atop 1\ 0 }\big]$.
Hence, for the entries we get
$[A_1]_{11}=-[A_2]_{11}$, $[A_1]_{22}=-[A_2]_{22}$,
$[A_1]_{12}=[A_2]_{12}=[A_1]_{21}=-[A_2]_{21}=0$,
resp.
$[A_1]_{11}=-[A_2]_{22}$, $[A_1]_{22}=-[A_2]_{11}$.
By setting $a:=[A_1]_{11}$, this means that
$A_1= a \big[ {1\ 0 \atop 0\ 1 }\big]$, $A_2= -a \big[ {1\ 0 \atop 0\ 1 }\big]$.
}
Hence, with the antisymmetric matrix $Y:=\big[ {0\ -1 \atop -1\ 0 }\big]$
we get the contradiction
$e^{it} Y  ={\bf U}^t Y = \exp(ta)Y\exp(-ta) =Y$.


\bigskip
\bf Example. \rm Let $E:={\rm Mat}(2,2,\CC)$.
Fix any real number $\omega\ne 0$ and, in terms of the entries,
for any $\big( X=\big[ x_{jk} \big]$ and $t\in\RR$ define
$$\big[ U^t X \big]_{jk} := x_{jk} \ \ \hbox{if $(j,k)\not\in\{ (1,2),(2,1)\}$},
\quad
\big[ U^t X \big]_{12}:= e^{i\omega t} x_{12}, \
\big[ U^t X \big]_{21}:= e^{i\omega t} x_{21} .$$
Clearly $[U^t:\ t\in\RR]$ is a bounded (necessarily norm continuous) one-parameter subgroup of
${\cal L}( E )$.
The assumption $(*)$ $U^t X = U_1^t X U_2^t$ $(t\in\RR,\ X\in E)$ leads to the
following contradiction.
Given any parameter $t\in\RR$ with $e^{i\omega t}\ne \pm$,
by considering the matrices
%$X:=\big[ \delta_{jm}\delta_{nk}\big]_{j=1}^M\strut_{k=1}^N$ $(m,n=1,2)$,
$X:=\left[{1\ 0 \atop 0\ 0} \right],\left[{0\ 0 \atop 0\ 1} \right],
\left[{0\ 1 \atop 0\ 0} \right],\left[{0\ 0 \atop 1\ 0} \right], $
with the non-zero entries in $(*)$
we get the contradictory system of equations
$1= \big[ U_1^t \big]_{11} \big[ U_2^t \big]_{11} =\big[ U_1^t \big]_{22} \big[ U_2^t \big]_{22}$
resp.
$e^{i\omega}= \big[ U_1^t \big]_{11} \big[ U_2^t \big]_{22} = \big[ U_1^t \big]_{22} \big[ U_2^t \big]_{11}$
(implying indeed
$\big[ U_2^t \big]_{11} = 1/\big[ U_1^t \big]_{11}$,
$\big[ U_2^t \big]_{22} = 1/\big[ U_1^t \big]_{22}$
and hence
$\big[ U_1^t \big]_{11} / \big[ U_1^t \big]_{22} =
\big[ U_1^t \big]_{22} / \big[ U_1^t \big]_{11} = e^{i\omega}$
with the consequence
$\big[ U_1^t \big]_{11} / \big[ U_1^t \big]_{22} =1 = e^{2i\omega} \ne (\pm 1)^2$).


\vfill
\eject
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$\underline{\hbox{\bf REFERENCES}}$

\bigskip
\item{[1]} B. BEAUZAMY,
Introduction to to Banach Spaces and their Geometry (2nd rev. edition),
North Holland Mat. Studies 68,  Elsevier Science Publishers, Amsterdam--New York, 1991.

\bigskip
\item{[2]} Preliminary draft of
"F. BOTELHO - J. JAMISON, Projections on tensor products of Banach spaces,
Arch. Math. 90 (2008), 341–352".

\bigskip
\item{[3]} L.L. STACH\'O,
On strongly continuous one-parameter groups of automorphisms of multilinear functionals,
J. Math. Anal. Appl. 363 (2010) 419–430.

\bigskip
\item{[4]}
F. RIESZ - B. Sz.-NAGY,
Vorlesungen über Funktionalanalysis,
VEB Deutscher Verlag der Wissenschaften, Berlin, 1968.

\bigskip
\item{[5]}
K. YOSIDA,
Functional Analysis, Classics Math.,
Springer-Verlag, Berlin, 1995.

\bigskip
\item{[6]} H. UPMEIER,

\bigskip
\item{[7]} R. IORDANESCU,
Jordan structures in analysis, geometry and physics,
Editura Academiei Romane, Bucharest, 2009.

\bigskip
\item{[8]} Y. FRIEDMAN - B. RUSSO,
The Gelfand-Naimark theorem for JB*-triples,
Duke Math. J. 53 (1986), 139-148.









\bye
