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

\address{Bolyai Institute\\
University of Szeged\\
H-6720 Szeged, Aradi V\'ertan\'uk tere 1\\ Hungary}

\email{stacho@math.u-szeged.hu}

\title[Strongly continuous one-parameter groups]
      {ON STRONGLY CONTINUOUS ONE-PARAMETER GROUPS OF AUTOMORPHISMS}


\subjclass{46G25, 47H60, 17C99}

\keywords{strongly continuous one-parameter group; N-linear
functional; surjective linear isometry; uniformly convex space;
Cartan factor; JB*-triple}

\submitted{Nov. 25, 2013}


\maketitle

    \begin{abstract}
We prove structure theorems for strongly continuous one-parameter
groups formed by surjective linear isometries of spaces of bounded
N-linear functionals over strictly convex complex Banach spaces.
Complete description is given in the case of Hilbert-equivalent
norms on the basis of probability arguments. As a consequence, we
classify the strongly continuous one-parameter automorphism groups
of all infinite-dimensional Cartan factors of Jordan theory.
    \end{abstract}



\thanks{The author was supported by the projects \
T\'AMOP-4.2.2.A-11/1/KONV-2012-0060 \ Impulse Lasers for Use in
Materials Science and Biophotonics \ and \
T\'AMOP-4.2.2.A-11/1/KONV-2012-0073 \ Telemedicine Oriented Research
in the Fields of Mathematics, Informatics and Medical Sciences of
the European Union and co-financed by the European Social Fund.}
}%ENDDEL

\begin{document}

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\def\XXXx{{\bf X}}
\def\xxx{{\bf x}}


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\headline={
\ifodd\pageno {\rm\the\pageno}\hfill {\rm Strongly continuous one-parameter groups} \hfill {\rm\the\pageno}
\else {\rm\the\pageno} \hfill {\rm L.L. Stach\'o} \hfill {\rm \the\pageno} \fi}

}%ENDDEL

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%u%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\lhead{L.L. Stach\'o}
\rhead{Strongly continuous one-parameter groups}

\centerline{\bf ON STRONGLY CONTINUOUS ONE-PARAMETER}
\centerline{\bf GROUPS OF AUTOMORPHISMS}
\bigskip
\centerline{L.L. STACH\'O}
\bigskip
\centerline{\Smallit Communicated by XXXXXXXXX YYYYYYYY}
\bigskip\bigskip
{\narrower
\ABS
\noindent
We prove structure theorems for strongly continuous one-parameter
groups formed by surjective linear isometries of spaces of bounded
N-linear functionals over strictly convex complex Banach spaces.
Complete description is given in the case of Hilbert-equivalent norms
on the basis of probability arguments. As a
consequence, we classify the strongly continuous one-parameter
automorphism groups of all infinite-dimensional Cartan factors of
Jordan theory.
\hfill\break
{\Smallit AMS 2010 Subject Classification:} 46G25, 47H60, 17C99 $\phantom{\displaystyle{\int}}$
\hfill\break
{\Smallit Key words:} strongly continuous one-parameter group; N-linear functional; surjective linear isometry;
uniformly convex space; Cartan factor; JB*-triple


\par}

\vskip10mm

%\bf 1. Introduction, main results
\centerline{\bf 1. INTRODUCTION, MAIN RESULTS}
\bigskip
\rm

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

\bigskip

%\bf 1.1. Conjecture.
\rm CONJECTURE 1.1.
\it
Let ${\bf X_1},{\bf X}_2$ 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 $\phi: {\bf X}_1\times {\bf X}_2\to\CC$
and for every couple $({\bf x}_1,{\bf x}_2)\in {\bf X}_1\times {\bf X}_2$,
the function
$t\mapsto
%\big\langle (U^t_1\otimes U^t_2)\phi, (x_1,x_2) \big\rangle =
\phi\big( U_1^t {\bf x}_1, U_2^t {\bf x}_2 \big)$
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
$t\mapsto U_k^t {\bf x}_k$
are norm-continuous for every fixed ${\bf x}_1$ resp. ${\bf x}_2$.}

\bigskip
\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
$(\kappa_1 {\bf x}_1)\otimes (\kappa_2 {\bf x}_2)= {\bf x}_1\otimes {\bf x}_2$ whenever $\kappa_1\kappa_2=1$.
Using a completely different probabilistic approach,
in course of the proof of
[10, Theorem 1.1] we have shown
the following slightly generalized version of the conjecture for the case with Hilbert spaces:

\bigskip
%\bf 1.2. Theorem.
\rm THEOREM 1.2.
\it
Let $\HHH{1}\! ,\ldots,\HHH{N} \!\!\!$ be Hilbert spaces,
%Assume
$\big[ {\bf U}(t): t\!\in\!\RR \big]$
be
%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:
$t\mapsto \prod_{k=1}^N \langle U_{k,t} {\bf x}_k \vert {\bf h}_k \rangle \!$
%\break
is continuous for every fixed
${\bf x}_1,{\bf h}_1 \!\!\in\!\HHH{1} \! ,\ldots, {\bf x}_N,{\bf h}_N\!\in\!\HHH{N} \!\!$.
%$\hskip-3mm \phantom{{I^I}}$
Then
%there exist
we can find
functions
$\kappa_1,\ldots,\kappa_N \! :\RR\!\!\to\!\!\TT(:=\! \{ \kappa\!\!\in\!\!\CC: \vert \kappa\vert \!=\! 1\})$
with $\prod\limits_{k=1}^N \kappa_k \!\equiv\! 1$
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 $[8,12]$
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
${\bf U}(t) \!=\! \big[ \exp(itA_1) \big] \otimes\cdots\otimes \big[ \exp(itA_N) \big]$
$(t\in\RR)$.

\bigskip
\rm
At first sight our arguments in [10] 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 $t\mapsto {\bf x}_{1,t}\otimes\cdots\otimes {\bf x}_{N,t}$. The new results extend to
to the setting of uniformly convex spaces and we get the following conclusion.

\bigskip
%\bf 1.3. Proposition.
\rm PROPOSITION 1.3.
\it
Let $\XXX{1}\! ,\ldots,\XXX{N}$ be
uniformly convex Banach 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 invertible bounded linear operators
$U_{k,t}\!\in\! {\cal L}(\XXX{k})$ such that
the functions
$t\!\!\mapsto\! \norm{{\bf U}(t)({\bf x}_{1,t}\!\otimes\!\cdots\!\otimes\! {\bf x}_{N,t})}$
are all continuous.
Then, for each index $k \!=\! 1,\ldots,N$ there exist multipliers
$\rho_k,\widetilde{\rho}_k:\RR\!\to\!\CC$
such that,
%for each $k=1,\ldots,N$,
$$\vtop{
\hbox{$ \big[ \rho_k(t) U_{k,t}:\ t\in\RR\big] \ \
\hbox{ is a strongly continuous family}$,}
\hbox{$ \big[ \widetilde{\rho}_k(t)
U_{k,t}:\ t\in\RR\big] \ \ \hbox{ is a one-parameter group}$.}}$$

\rm
Actually, we establish 1.3 a bit more generally:
for spaces where weak convergence implies norm convergence. Also we shall see that
the multipliers can be chosen to be bounded both from above and below from $0$
(that is
$0<\inf\abs{\rho_k},\inf\abs{\widetilde{\rho}_k}$
and
$\sup\abs{\rho_k},\sup\abs{\widetilde{\rho}_k} <\infty$).
In particular, the conditions in 1.3 hold if the operators $U_{k,t}$ are all isometries,
and in this case we can choose even $\abs{\rho_k}=\abs{\widetilde{\rho}_k}\equiv 1$.

\rm
Unfortunately the later considerations in [3, Sections 4-5] leading to probabilistic arguments
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 facts with
not much farther generality from the original Hilbert setting of [3].
We do not proceed into this direction.

\rm
A perhaps rather aesthetical generalization
with equivalent general Banach norms in the Hilbert spaces instead of the ones defined by inner products
can be developed due to the amenability of the
additive group of $\RR$.

\bigskip
%\bf 1.4. Theorem.
\rm THEOREM 1.4.
\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
$($instead of being  $\big\langle \cdot \big\vert \cdot \big\rangle_k$-unitary
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
$\sspr\cdot\cdot_k$-self-adjoint operators $A_k$.

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

\bigskip
%\bf 1.5. Proposition.
\rm PROPOSITION 1.5.
\it
Any linear automorphism of a bounded circular domain ${\bf D}$ in a Hilbert space $\HHHx$
is necessarily a scalar type operator as being a preserver of some equivalent inner product.
Given a strongly continuous one-parameter group ${\cal U}:=\big[ U^t:\ t\in \RR\big]$
of linear automorphisms of ${\bf D}$,
or more generally a bounded strongly continuous one-parameter subgroup of ${\cal L}(\HHHx)$,
there exists an equivalent ${\cal U}$-invariant inner product on $\HHHx$.

\rm
\bigskip
Though it is likely that Proposition 1.5 appeared already in the literature of Banach space geometry,
as far we have not found proper references even after an inquiry with several colleagues.
Therefore we devote the short Appendix (Section 4) to its proof relying upon Banach limits.
\medskip
In [10] we emphasized the natural connection of Theorem 1.2 with the Jordan theory of bounded symmetric domains
during the Introduction of [10]. One of the aims of this note is to work out the first step in this
direction: in Section 3,
with slight modifications of the proofs in [10], we obtain the full description of
the unbounded JB*-derivations of the infinite-dimensional Cartan factors [5,6] of Types 1,2,3.

\bigskip
\rm

Recall that, by a classical representation theorem by F Riesz [8],
each Type 1 Cartan factor is isometrically isomorphic to some ${\cal L}(\HHH{1},\HHH{2})$
with suitable Hilbert spaces $\HHH{1},\HHH{2}$.
Theorem 1.2 furnishes the complete description
of their unbounded JB*-derivations as operations of the form
$X\mapsto A_1X+XA_2$ with suitable possibly unbounded self-adjoint operators $A_k\in{\cal L}(\HHH{k})$.
%
The next theorem provides the full description of the unbounded JB*-derivations
of the Cartan factors of types 2 and 3 which can be represented
without loss of generality in the form
${\cal F}_2$ resp. ${\cal F}_3$ below.


\bigskip
%\bf Theorem 1.6.
\rm THEOREM 1.6.
\it
Let $\HHHx$ be a Hilbert space with a fixed orthonormed basis
${\bf E} =\{ {\bf e}_j: \ j\in J\}$ and let
$X\mapsto X^{{\rm T}},\overline{X}$
denote the operator transposition resp. conjugation
associated to ${\bf E}$.\footnote{\rm The matrices of $X^{{\rm T}}$ resp. $\overline{X}$ are the
transpose resp. conjugate of the matrix of $X$ with respect to ${\bf E}$:
$\langle X^{{\rm T}} {\bf  e}_j \vert {\bf e}_\ell \rangle =
\langle X {\bf  e}_\ell \vert {\bf e}_j \rangle$ and
$\langle \overline{X} {\bf  e}_j \vert {\bf e}_\ell \rangle =
\overline{ \langle X {\bf  e}_j \vert {\bf e}_\ell \rangle}$
for all $j,\ell\in J$.
}
Assume $\big[ {\bf U}(t): \ t\in\RR\big]$ resp.
$\big[ {\bf V}(t): \ t\in\RR\big]$ are one-parameter groups of surjective
linear
 isometries of
${\cal F}_2 := \{ X\in {\cal L}(\HHHx):\ X=X^{{\rm T}} \}$
resp.
${\cal F}_3 := \{ Z\in {\cal L}(\HHHx):\ Z=-Z^{{\rm T}} \}$
such that
all the functions
\break
$t\mapsto \langle [{\bf U}(t)X] {\bf x} \vert {\bf y}\rangle \!$
resp.
$t\mapsto \langle [{\bf V}(t)Z] {\bf x} \vert {\bf y}\rangle$
are continuous for all fixed ${\bf x},{\bf y}\!\in\!\HHHx$ and
$X\in{\cal F}_2$ resp. $Z\in{\cal F}_3$.
Then
$$\vtop{
\hbox{${\bf U}(t)X = [\exp(it A)] X [\exp(it \overline{A})] \quad (X\in {\cal F}_2,\ t\in\RR)$,}
\hbox{${\bf V}(t)Z = [\exp(it B)] Z [\exp(it \overline{B})] \quad \ (Z\in {\cal F}_3,\ t\in\RR)$ .}
}$$
Thus
the possibly unbounded JB*-derivations of ${\cal F}_3,{\cal F}_3$ are
of the form $X\mapsto AX+X\overline{A}$ resp.
$Z\mapsto BZ+Z\overline{B}$
for some possibly unbounded self-adjoint operators $A,B$ on $\HHHx$.

\bigskip
\rm


Since
the surjective linear isometries of a spin factor (Cartan factor of Type 4)
are automatically unitary with respect to the underlying Hilbert space scalar product.
and since the
exceptional Cartan factors (of Type 5,6) are finite dimensional (16 resp. 27 dimensions),
Theorem 1.6 completes the description of the possibly unbounded derivations of any Cartan factor.
%
%\bigskip
%\rm
%
One may intend to apply this result to get the description of the possibly unbounded JB*-derivations
for all JB*-triples\footnote{\rm Banach spaces with holomorphically symmetric unit ball.
They were introduced and axiomatized algebraically by means of a three-variable product in 1983 by Kaup [7].
For an elementary introduction see e.g. [6].}
via the Gelfand-Neimark type theorem by Friedmen-Russo [3].
This latter asserts that any JB*-triple be embedded into a suitable $\ell^\infty$-direct sum of
Cartan factors as a weak*-dense norm-closed submanifold.
We finish by raising the following two related open problems:

\bigskip
%\bf 1.7. Problems.
\it Problems \rm 1.7. a) Describe the pointwise continuous $\!( \!\!$resp. %pointwise
weakly con\-tinuous$\!\! )$
one-parameter groups of surjective linear isometries for any JB*-triple.
%\smallskip

\rm b) Describe the pointwise weak*-continuous one-parameter groups of surjective linear isometries
for all JBW*-triples $(\!$JB*-triples with predual$)$.

%\bigskip
%\bf 1.8. Problem.
%\it Problem \rm 1.8.
\rm c)
\rm (Non-linear issue of 1.7).
Describe the pointwise continuous $($resp. pointwise weakly-, resp. weak*-continuous in the JBW*-case$)$
one-parameter groups of holomorphic automorphisms of the unit ball of any JB*-triple.
In particular, in the case of the unit ball of a Hilbert space,
are they all of the form
$\exp\Big( \Big[ a- \langle {\bf x} \vert {\bf a} \rangle {\bf x} + iA {\bf x} \Big]
{\partial \over \partial {\bf x}} \Big)$ with suitable vector ${\bf a}$ and a possibly
unbounded self-adjoint operator $A$.



\vskip10mm
\rm



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\bf 2. Adjusted piecewise continuity of tensor-isometries
\centerline{\bf 2. ADJUSTED CONTINUITY OF TENSOR MAPS}
\bigskip
\rm

Henceforth, throughout the whole paper, let
$\XXX{1},\ldots,\XXX{N}$ be reflexive Banach spaces with duals denoted by
$[\XXX{k}]^*,\ldots,[\XXX{N}]^*$
where weak convergence along with convergence in norm entails norm convergence.
That is, by assumption,
for each fixed index $k=1,\ldots,N$ and for any net $\big[ {\bf x}_j :\ j\in J\big]$ in $\XXX{k}$
we have
$$\norm{ {\bf x}_j \!-\! {\bf x} }\to 0 \ \hbox{whenever} \
\norm{{\bf x}_j} \!\to\! \norm{{\bf x}} \ \hbox{and} \
\big\langle {\bf x}_j \!-\! {\bf x}, \phi \big\rangle \!\to\! 0 \ \ (\phi\!\in\![\XXX{k}]^*) .
\leqno(2.1)$$
Notice [1] that {\it uniformly convex} \ spaces are reflexive satisfying (2.1),
and a Banach space is uniformly convex if and only if its duals  uniformly smooth
(and vice versa, via reflexivity).

\smallskip
As usually, for the natural coupling between $\XXX{k}$ and $[\XXX{k}]^*$
we write $\langle {\bf x}, \phi \rangle \big(\! := \phi({\bf x})\big)$.
Given any family
${\bf x}^{(k)}\!\in\!\XXX{k}, \phi^{(k)}\!\in\![\XXX{k}]^*$
$(k \!=\! 1,\ldots,N)$,
${\bf x}^{(1)}\otimes\cdots\otimes{\bf x}^{(N)}$
resp.
$\phi^{(1)}\otimes\cdots\otimes\phi^{(N)}$
will denote
the elementary $N$-linear functionals \
$\big( \psi^{(1)}, \ldots, \psi^{(N)} \big) \mapsto \prod_{k=1}^N \langle {\bf x}^{(k)}, \psi^{(k)} \rangle$ \
resp. \
$\big( {\bf y}^{(1)}, \ldots, {\bf y}^{(N)} \big) \mapsto \prod_{k=1}^N \langle {\bf y}^{(k)}, \phi^{(k)} \rangle$.
%We emphasize the following elementary observation:
Remark that
$$\vcenter{
\hbox{
$0\ne {\bf x}^{(1)}\otimes\cdots\otimes{\bf x}^{(N)} = {\bf y}^{(1)}\otimes\cdots\otimes{\bf y}^{(N)}$
\hskip1mm if and only if}
\hbox{
${\bf x}^{(k)}\!=\!\rho_k{\bf y}_k \ (k \!=\!1,\ldots,N)
 \ {\rm with} \ \prod\limits_{k=1}^N \rho_k \!=\! 1$ for some
$\rho_1,\ldots,\rho_N\!\in\!\!\CC$.}
}
\leqno(2.2)$$
It is well-known [9] that the tensor coupling
$$\hbox{$\Big\langle {\bf x}^{(1)}\otimes\cdots\otimes{\bf x}^{(N)} ,
\phi^{(1)}\otimes\cdots\otimes\phi^{(N)} \Big\rangle :=
\prod\limits_{k=1}^N \langle {\bf x}^{(k)}, \phi^{(k)} \rangle$}$$
admits a bounded $N$-linear extension to the projective tensor product of the
spaces $\XXX{k}$ with the injective tensor product of the dual spaces,
and the latter is a closed subspace in
${\cal B}(\XXX{1},\ldots,\XXX{N}) :=\big\{$bounded $N$-linear functionals
$\XXX{1}\times\cdots\XXX{N}\to\CC \big\}$ with its natural $\sup$-norm.
We shall speak of {\it weak convergence} in $\XXX{1}\otimes\cdots\otimes\XXX{N}$
in the sense of the tensor coupling. In particular
${\bf x}_j^{(1}\otimes\cdots\otimes{\bf x}_j^{(N)} \Tow
{\bf x}^{(1)}\otimes\cdots\otimes{\bf x}^{(N)}$
if $\prod_k \phi^{(k)}({\bf x}_j^{(k)} \to \prod_k \phi^{(k)}({\bf x}^{(k)}$
for fixed $\phi^{(k)}\in[\XXX{k}]^*$ $(k=1,\ldots,N)$.


\bigskip
%\bf 2.3. Lemma.
\rm LEMMA 2.3.
\it Let
$\big[ {\bf e}^{(k)}_j :\ j\in {\cal J}\big]$
be nets of unit vectors
in the respective spaces $\XXX{k}$ such that
$${\bf e}^{(1)}_j\otimes\cdots\otimes{\bf e}^{(N)}_j \Tow {\bf e}^{(1)}\otimes\cdots\otimes{\bf e}^{(N)}$$
where ${\bf e}^{(k)} \in\XXX{k}$ $(k=1,\ldots,N)$ are also vectors with norm $1$.
Then there are nets of constants
$\big[ \kappa_j^{(k)} : j\in{\cal J} \big]\ (k=1,\ldots,N\big)$
in $\TT :=\{ \kappa\!\in\!\CC : \abs{\kappa} \!=\! 1\}$
such that
$$\prod_{\ell=1}^N \kappa_j^{(\ell)} =1, \quad
\norm{\kappa_j^{(k)}{\bf e}_j^{(k)} - {\bf e}^{(k)}} \to 0 \qquad (k=1,\ldots,N).$$

\it Proof. \rm
Let
$\TT\!_0^N:=\big\{ (\kappa_1,\ldots,\kappa_N)\!\in\!\TT^N:\, \prod_k \kappa_k \!=\! 1\big\}$,
and
for any index $j\!\in\!{\cal J}$ define $\eps_j$ resp.
$\big( \kappa^{(1)}_j,\ldots,\kappa^{(N)}_j \big)$ as the minimal value resp.
a minimum location of the continuous function
$\Delta_j (\kappa_1,\ldots,\kappa_N) :=
\sum\limits_{k=1}^N \norm{ \kappa_k {\bf e}^{(k)}_j - {\bf e}^{(k)} }$ on
the compact domain $\TT\!_0^N$.
We have to show $\eps_j\to 0$.
%
As a consequence of the compactness of $\TT$ along with the
weak compactness of the closed unit ball in each $\XXX{k}$,
we can choose a
subnet $\big[j_\nu:\ \nu\in{\cal N}\big]$ of ${\cal J}$
along with vectors ${\bf y}^{(k)}$ of norm $\le 1$
%and constants $\kappa^{(k)}\in\TT$
such that
%$\kappa_{j_\nu}^{(k)}\to\kappa^{(k)},
$$\hbox{$\eps_{j_\nu}\!\to\!\limsup_j \eps_j$, \
$\big\langle \kappa^{(k)}_j {\bf e}_{j_\nu}^{(k)} \!-\! {\bf y}^{(k)}, \phi^{(k)} \big\rangle \!\to\! 0$
for all $\phi^{(k)}\!\in\![\XXX{k}]^*$}
\leqno(2.4)$$
for $k \!=\! 1,\ldots,N$. By passing to $j_\nu$-limits we have
$${\bf y}^{(1)} \otimes\cdots\otimes {\bf y}^{(N)} =
{\bf e}^{(1)}\otimes\cdots\otimes{\bf e}^{(N)}.$$
Since
$\prod_k \norm{{\bf y}^{(k)}} = \norm{{\bf y}^{(1)} \otimes\cdots\otimes {\bf y}^{(N)}} =
\norm{{\bf e}^{(1)} \otimes\cdots\otimes {\bf e}^{(N)}}$ and
$\norm{{\bf y}^{(1)}},\ldots,\norm{{\bf y}^{(1)}} \!\le\! 1$,
all the $ \!{\bf y}^{(k)}$ must be unit vectors. Since also $\norm{{\bf e}^{(k)}} \!= \!1$,
in view of (2.2) here we even have
${\bf y}^{(k)} =\rho^{(k)} {\bf e}^{(k)}$ with
suitable constants $\rho^{(k)}\in\TT$.
 Thus,
by our basic topological assumption on the equivalence of weak and norm convergence on the unit sphere
of the spaces $\XXX{k}$, from (2.4) it follows
%$$\norm{ \kappa^{(k)}_{j_\nu} {\bf e}^{(k)}_{j_\nu} - \rho^{(k)} {\bf e}^{(k)} }  =
%\norm{ \kappa^{(k)}_{j_\nu} {\bf e}^{(k)}_{j_\nu} - {\bf y}^{(k)} } \to 0 \qquad (k=1,\ldots,N).$$
$$\vtop{
\hbox{$\kappa^{(k)}_{j_\nu} {\bf e}^{(k)}_{j_\nu} \to {\bf y}^{(k)} = \rho^{(k)} {\bf e}^{(k)}$
\ \ in norm,}
\hbox{$\Delta_{j_\nu} (\kappa^{(1)}_{j_\nu},\ldots,\kappa^{(N)}_{j_\nu})
\to \!\! \sum\limits_{k=1}^N \!\! \norm{{\bf y}^{(k)} \!\!-\! {\bf e}^{(k)}} \!=\!
 \sum\limits_{k=1}^N \!\! \abs{\rho^{(k)} \!\!-\! 1}$.} }$$
%On the other hand,
By the definition of the terms $\kappa^{(k)}_j$ by means of $\Delta_j$,
for any fixed tuple
$( \kappa_1,\ldots,\kappa_N ) \in\TT\!_0^N$ we have
$\sum_k  \norm{ \kappa_k {\bf e}^{(k)}_j - {\bf e}^{(k)} }  \ge
\sum_k \norm{ \kappa^{(k)}_j {\bf e}^{(k)}_j  - {\bf e}^{(k)} }$.
Since, by taking a convergent subnet
%for any (necessarily existing) cluster point
$\big[ (\kappa^{(1)}_{j_{\nu_\tau}},\ldots,\kappa^{(N)}_{j_{\nu_\tau}} ) :\ \tau\in{\cal T}\big]$
with limit point $(\xi_1,\ldots,\xi_N)\in\TT\!_0^N$,
we have
${\bf e}^{(k)}_{j_{\nu_\tau}} =
[\kappa^{(k)}_{j_{\nu_\tau}}]^{-1} \kappa^{(k)}_{j_{\nu_\tau}} {\bf e}^{(k)}_{j_{\nu_\tau}}
\to \overline{\xi_k}\rho^{(k)} {\bf e}^{(k)}$.
Hence we conclude that
$$\sum_{k=1}^N  \abs{ \kappa_k \overline{\xi_k} -1 } =
\sum_{k=1}^N  \norm{ \kappa_k \overline{\xi_k}\rho^{(k)} {\bf e}^{(k)} - {\bf e}^{(k)} } \ge
\sum_{k=1}^N  \norm{ \rho^{(k)} {\bf e}^{(k)} - {\bf e}^{(k)} } $$
for any $(\kappa_1,\ldots,\kappa_N)\in\TT\!_0^N $.
With the choice $\kappa_k:=\xi_k$ $(k=1,\ldots,N)$ we get
$\rho^{(k)}=1$  $(k=1,\ldots,N)$ and
$\limsup_j \eps_j =
\lim_\nu \Delta_{j_\nu} ( \kappa^{(1)}_{j_\nu},\ldots,  \kappa^{(N)}_{j_\nu} ) =0$
which completes the proof.

\bigskip
%\bf 2.5. Proposition.
\rm PROPOSITION 2.5.
\it
Suppose ${\bf e}_t^{(k)} \in \XXX{k}$ $(t\in\RR,\ k=1,\ldots,N)$
are unit vectors such that function
$t\mapsto {\bf e}_t^{(1)} \otimes\cdots\otimes {\bf e}_t^{(N)}$
is weakly continuous.
Then one can find a function
$t\mapsto (\kappa_t^{(k)},\ldots, \kappa_t^{(k)} )$
from $\RR$ to $\TT\!_0^N :=\big\{(\kappa_1,\ldots\kappa_N)\in\TT:\ \prod_k \kappa_k=1 \big\}$
such that the  functions
$t\mapsto \kappa_t^{(k)} {\bf e}_t^{(k)}$ $(k=1,\ldots,N)$ are norm-continuous.
\bigskip
\it Proof. \rm
An analogous argument with the $\sigma$-compctness of the real line
as in [10, proof of Lemma 2.3] shows: it suffices to see that
for any $\tau\in\RR$ there is an open interval $I_\tau$ around $\tau$ where the statement holds.
%We may assume $\tau=0$ without loss of generality.
Let us fix $\tau$ arbitrary.
By choosing a family
$\phi^{(k)} \in [\XXX{k}]^*$ $(k=1,\ldots,N)$
of functionals
with
$\big\langle {\bf e}^{(k)}_\tau , \phi^{(k)} \big\rangle = \norm{\phi^{(k)}} =1$
(guaranteed by the Hahn-Banach theorem),
define the interval $I_\tau$ as the connected component of $\tau$ in the open set
$\big\{ t\in\RR:\ \prod_{k=1}^N \langle {\bf e}_t^{(k)}, \phi^{(k)} \rangle \ne 0 \big\}$.
%
For the parameters $t\in I_\tau$, we define the functions $t\mapsto\kappa_t^{(k)}$ as
$$\kappa_t^{(k)} := \abs{ \big\langle {\bf e}^{(k)}_t , \phi^{(k)} \big\rangle } /
 \big\langle {\bf e}^{(k)}_t , \phi^{(k)} \big\rangle
%\overline{\phi_t^{(k)} (\xxx^{(k)})}/\abs{\phi_t^{(k)} (\xxx^{(k)}}
\quad (k<N), \qquad
\kappa_t^{(N)} :=\overline{\kappa_t^{(1)}\cdots\kappa_t^{(N-1)}} .$$
To establish the norm continuity of the functions $t\mapsto \kappa_t^{(k)} \phi_t^{(k)}$ (on $I_\tau$)
we consider a convergent sequence $t_n \to t_0$ in $I_\tau$ and show that
$$\norm{ \kappa_{t_n}^{(k)} {\bf e}_{t_n}^{(k)} - \kappa_{t_0}^{(k)} {\bf e}_{t_0}^{(k)}}\to 0 \ .
\leqno(2.6)$$
In view of Lemma 2.3, there are convergent sequences
$\big[\mu_{t_n}^{(k)} :\ n=1,2,\ldots\big]$ $(k=1,\ldots,N)$ in $\TT$
such that $\norm{ \mu_{t_n}^{(k)} {\bf e}_{t_n}^{(k)} - {\bf e}_{t_0}^{(k)}}\to 0$  $(k=1,\ldots,N)$.
Fix any index $k$. To prove (2.6)
we have
%it suffices
to see that
$$\kappa_{t_n}^{(k)}/\mu_{t_n}^{(k)}\to \kappa_{t_0}^{(k)} \ .
$$
Observe that
$\kappa_t^{(k)} \big\langle {\bf e}_t^{(k)} , \phi^{(k)} \big\rangle =
\abs{ \big\langle {\bf e}_t^{(k)} , \phi^{(k)} \big\rangle }$ for any $t\in I_0$.
Therefore
$${\kappa_{t_n}^{(k)} \over \mu_{t_n}^{(k)}}
\vtop{
\hbox{$= \displaystyle{
{\kappa_{t_n}^{(k)} \big\langle {\bf e}_{t_n}^{(k)} , \phi^{(k)} \big\rangle
\over
\mu_{t_n}^{(k)} \big\langle {\bf e}_{t_n}^{(k)} , \phi^{(k)} \big\rangle } =
{\abs{  \big\langle {\bf e}_{t_n}^{(k)} , \phi^{(k)} \big\rangle }
\over
\mu_{t_n}^{(k)}  \big\langle {\bf e}_{t_n}^{(k)} , \phi^{(k)} \big\rangle } =}$}
\hbox{$= \displaystyle{
{\abs{\mu_{t_n}^{(k)}  \big\langle {\bf e}_{t_n}^{(k)} , \phi^{(k)} \big\rangle }
\over
\mu_{t_n}^{(k)}  \big\langle {\bf e}_{t_n}^{(k)} , \phi^{(k)} \big\rangle }
\longrightarrow { \abs{  \big\langle {\bf e}_{t_0}^{(k)} , \phi^{(k)} \big\rangle }
\over  \big\langle {\bf e}_{t_0}^{(k)} , \phi^{(k)} \big\rangle } = \kappa_{t_0}^{(k)}.}$}
}$$
%Qu.e.d.

%\bf 2.7. Corollary.
\rm COROLLARY 2.7.
\it
If $t\mapsto {\bf x}^{(k)}_t$ $(k=1,\ldots,N)$ are nowhere vanishing functions $\RR\to\XXX{k}$
with weakly continuous tensor product
$t\mapsto {\bf x}^{(1)}_t\otimes\cdots\otimes{\bf x}^{(N)}_t$
\break
whose norm $t\mapsto \prod_{k=1}^N \norm{ {\bf x}^{(k)}_t}$ is also continuous,
then there are multipliers $\kappa^{(k)}_t \in\TT$ \ $(t\in\RR,\ k=1,\ldots,N)$
making the vector valued functions
$t\mapsto \kappa^{(k)}_t \norm{ {\bf x}^{(k)}_t }^{-1} \prod_{j=1}^N \norm{ {\bf x}^{(j)}_t}^{1/N}
{\bf x}^{(k)}_t$
norm-continuous.

\bigskip
\it Proof. \rm
A choice in accordance with Proposition 2.5
for the multipliers $\kappa^{(k)}_t$ to the functions
$t\mapsto {\bf e}^{(k)}_t := \norm{ {\bf x}^{(k)}_t }^{-1} {\bf x}^{(k)}_t $
suits the statement.

\bigskip
\rm
For the next two lemmas let ${\bf Z}$ denote a complex topological vector space
with separating topological dual
(thus for any vector $0 \ne {\bf z}\in{\bf Z}$ there is a continuous linear functional
$\psi: {\bf Z}\to\CC$ with $\psi({\bf z})=1$).

\bigskip
%\bf 2.8. Lemma.
LEMMA 2.8.
\it
Assume
${\bf z}_1(t),{\bf z}_2(t)$ are linearly independent vectors in ${\bf Z}$
for all $t\in\RR$ and let
$\rho_1,\rho_2,\mu: \RR\to\CC\setminus\{ 0\}$
be functions making the maps
$\zeta_1 {\bf z}_1, \zeta_2 {\bf z}_2, \mu ({\bf z}_1+{\bf z}_2)$
continuous. Then the functions $\zeta_2 {\bf z}_1,\zeta_1 {\bf z}_2$ are also continuous.

\bigskip\it Proof. \rm
Let us fix any point $t_0\in\RR$.
It suffices to see the continuity of $\zeta_1/\zeta_2$ at $t_0$.
To this aim let us choose a couple $\psi_1,\psi_2$ of continuous linear functionals
${\bf Z}\to\CC$ with
${\rm det}\big[ {
\psi_1 {\bf z}_1(t_0) \ \psi_1 {\bf z}_2(t_0)
\atop
\psi_2 {\bf z}_1(t_0) \ \psi_2 {\bf z}_2(t_0)
} \big] \ne 0$
and define
$P{\bf z}:= \big[ { \psi_1{\bf z} \atop  \psi_2{\bf z} }\big]$ $({\bf z}\in{\bf Z})$.
This can be done by the linear independence of
${\bf z}_1(t_0),{\bf z}_2(t_0)$.
Then, by the continuity of both $t\mapsto \zeta_k(t) {\bf z}_k(t)$ $(k=1,2)$,
there is a neighborhood $I$ of $t_0$ where
${\rm det} \big[ P\zeta_1(t) {\bf z}_1(t), \  P\zeta_2(t) {\bf z}_2(t) \big] \ne 0$
$(t\in I)$ and the
matrix valued function
$t\mapsto L_t :=  \big[ P\zeta_1(t) {\bf z}_1(t), \  P\zeta_2(t) {\bf z}_2(t) \big] ^{-1}$
is continuous on $I$.
The continuity of $t\mapsto \mu(t) {\bf z}_1(t) + {\bf z}_2(t)]$
entails the continuity of the function
$t\mapsto L_t P \mu(t) [ {\bf z}_1(t) + {\bf z}_2(t)] =
\big[ { \mu(t)/\zeta_1(t) \atop \mu(t)/\zeta_2(t) }\big]$
on $I$
whence the continuity of $\zeta_1/\zeta_2 = [\mu/\zeta_1]/[\mu/\zeta_2]$ at $t_0$ is immediate.

\bigskip
%\bf 2.9. Lemma.
LEMMA 2.9.
\it
Let $t\!\mapsto\! U_t$ be a map $\RR \!\to\! {\rm GL}({\bf Z})$
and let $\rho:\RR\!\to\!\CC\!\setminus\!\{ 0\}$ resp.
$\lambda: \RR^2 \to \CC\setminus \{ 0\}$ be functions such that
all the maps $t\!\mapsto\! \rho(t)U_t {\bf z}$ $({\bf z}\!\in\!{\bf Z})$ are continuous and we have
$U_{s+t} = \lambda(s,t) U_s U_t$ for all $s,t\in\RR$.
Then there exists a function $\widetilde{\rho}:\RR\to\CC\setminus\{ 0\}$
making $\big[ \widetilde{\rho}(t) U_t:\ t\in\RR\big]$ a one-parameter group.

\bigskip\it Proof. \rm
By induction on $n$, we see that there are functions
$\lambda_n:\RR^n\to\CC\setminus \{ 0\}$
with
$$U_{s_1+\cdots+s_n} = \lambda_n(s_1,\ldots,s_n) U_{s_1}\cdots U_{s_n}$$
for all possible choices.
We establish the {\it symmetry} of each $\lambda_n$
by showing that the family $\{ U_t:\ t\in\RR\}$ is Abelian.
Since $U_0 = U_{0+0}= \lambda(0,0) U_0^2$, necessarily
$U_0 =\lambda(0,0)^{-1}{\rm Id}$.
Hence the identity $\lambda(t,-t) U_t U_{-t} =U_0$ implies that
$U_{-t}$ is always a multiple of $U_t^{-1}$.
Since, for $n=2,3,\ldots$ we have
$U_{nt} = \lambda_n(t,\ldots,t) U_t^n$,
the set
$\{ U_q:\ q\!\in\!\QQ\} = \bigcup_{m=1}^\infty \{ U_{n/m!} :\ n\!\in\!\ZZ\}$
is Abelian as being the union of an increasing sequence of Abelian families.
By the density of the rational numbers within $\RR$,
our strong continuity assumption entails that
$\{ \rho(t) U_t :\ t\in\RR\}$ and hence
$\{ U_t:\ t\in\RR\}$ are Abelian.

From the relation $U_{nt} \in\CC  U_t^n$ we see also
that any operator of the form $\zeta U_t$ admits arbitrary $m$-th roots
of the form $\eta U_{t/m}$.
Hence, for any given $t\in\RR$, we can construct a sequence
of group homomorphisms
$g_{t,m}: \{ n/m!:\ n\in\ZZ\}\to {\rm GL}({\bf Z})$
$(m=1,2,\ldots)$ with $g_{t,m+1}$ extending $g_{t,m}$
such that $g_{t,m}(1)=U_t$ and
$g_{t,m}(n/m!) \in \CC U_{t/m!}^n =\CC U_{nt/m!}$ for all $m,n$.
Thus, for any $t\!\in\!\RR$, there is a homomorphism $g_t:\RR\!\to\!{\rm GL}({\bf Z})$
along with a function
$\widetilde{\rho}_t : \QQ \!\to\! \CC$
such that
$$g_t(n/m!) = g_{t,m}(n/m!) = \widetilde{\rho}_t(n/m!) U_{nt/m!} \quad
(n=0,\pm1,\pm2,\ldots; \ m=1,2,\ldots).$$
Let us fix any Hamel basis ${\cal H}$ in $\RR$ and define
the map $g: \RR\to {\rm GL}({\bf Z})$ resp. the function
$\widetilde{\rho}:\RR\to\CC\setminus\{ 0\}$ as
$$\vtop{
\hbox{$g(q_1 t_1 +\cdots + q_n t_n) := g_{t_1}(q_1) \cdots  g_{t_n}(q_n), $}
\hbox{$\widetilde{\rho}(t_1 q_1 + \cdots+ t_n q_n) :=
\lambda_n(q_1 t_1,\ldots,q_n t_n)^{-1}
\widetilde{\rho}_{t_1}(q_1) \cdots \widetilde{\rho}_{t_n}(q_n)$}
}$$
for $q_1,\ldots,q_n\in\QQ$ and $t_1,\ldots,t_n\in{\cal H}$.
We complete the proof by observing that $g$
is a group homemorphism with $g(1)=U_1$ and satisfying the identities
$g(q_1 t_1 +\cdots+ q_n t_n) = \lambda_n(q_1 t_1,\ldots,q_n t_n)^{-1}
\widetilde{\rho}(q_1 t_1 +\cdots+ q_n t_n) U_{q_1 t_1}\cdots U_{q_n t_n}$.

\bigskip
\bigskip
\it Proof of Proposition 1.3.
\smallskip
\rm

By assumption, for any tuple
$\big[{\bf x}^{(j)}\big]\in\XXX{1}\times\cdots\XXX{N}$
the map
$t\mapsto U_{1,t}{\bf x}^{(1)}\otimes\cdots\otimes U_{N,t}{\bf x}^{(N)}$
is weakly continuous. An application of Corollary 2.7
establishes the existence of functions
$\rho_{k,[{\bf x}^{(j)}]}:\RR\to \CC\setminus\{0\}$ such that
all the maps
$t\mapsto \rho_{k,[{\bf x}^{(k)}]}(t) U_{k,t} x^{(k)}(t)$ are continuous.
Let us fix any family $0\ne {\bf x}_0^{(j)} \in\XXX{k}$ $(j=1,\ldots,N)$
and define $\rho_k(t):=\rho_{k,[{\bf x}_0^{(j)}]}$.
To complete the proof
that each $\big[ \rho_k(t)U_{k,t}:\ t\in\RR \big]$
is a strongly continuous family,
it suffices to see that,
given any index $\ell$ with vector ${\bf x}^{(\ell)}\in\XXX{\ell}$
which is linearly independent of ${\bf x}_0^{(\ell)}$,
the function
$t\mapsto U_{\ell,t} {\bf x}^{(\ell)}$ is continuous.
This fact follows immediately from
Lemma 2.8 applied with ${\bf Z}:=\XXX{\ell}$,
${\bf z}_1(t):=U_{\ell,t} {\bf x}_0^{(\ell)}(t)$,
${\bf z}_z(t):=U_{\ell,t} {\bf x}^{(\ell)}(t)$,
$\zeta_1(t):=\rho_\ell(t)$,
$\zeta_2(t):=\rho_{\ell,[{\bf y}^{(k)}]} (t)$,
$\mu(t):=\rho_{\ell,[{\bf u}^{(k)}]} (t)$
where
${\bf y}^{(k)} := (1-\delta_{k\ell}) {\bf x}_0^{(k)} + \delta_{k\ell} {\bf y}^{(\ell)}$
and
${\bf u}^{(k)} := (1-\delta_{k\ell}) {\bf x}_0^{(k)} + \delta_{k\ell} {\bf y}^{(\ell)}$
in terms of the Kronecker-delta $\delta_{k\ell}:=[1$ for $k=\ell$ and $0$ else$]$.

\smallskip
It is a well-known consequence of $(2.2)$ and the Schur lemma that
the factorization of a non-vanishing tensor products of linear operators
is unique up to constant factors with product one.
Hence the semigroup property ${\bf U}(s+t)={\bf U}(s){\bf U}(t)$ entails
the existence of functions $\lambda_k : \RR^2\to\CC\setminus\{0\}$
such that
$U_{k,s+t} =\lambda_k(s,t)U_{k,t}U_{s,t}$ $s,t,\!\in\!\RR ;\ k \!=\! 1,\ldots,N)$.
Given any index $\ell\!\in\!\{1,\ldots,N\}$,
by applying Lemma 2.9 again with ${\bf Z}\!:=\!\XXX{\ell}$,
the operators $U_t \!:=\! U_{\ell,t}$ and
the function $\rho:=\rho_\ell$
constructed above,
we obtain the existence of a function
$\widetilde{\rho}_\ell: \RR\!\to\!\CC$ making the family
$\big[ \widetilde{\rho}_\ell U_{\ell,t}: t\!\in\!\RR\big]$ a one-parameter group.


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

\vskip10mm
%\bf 3. On Cartan factors of types 2,3; proof of Theorem 1.6
\centerline{\bf CARTAN FACTORS OF TYPES 2,3; PROOF OF THEOREM 1.6}
\rm

\bigskip
Throughout this section let
$\HHHx$ be an arbitrarily fixed complex Hilbert space.
We shall write
\DEL{
equipped with the
inner product $\langle . \vert . \rangle$, norm $\norm{.}$, \quad
}%ENDDEL
$\hhh^* :=[\HHHx \ni{\bf x}\mapsto \langle {\bf x}\vert \hhh\rangle]$
for the dual functionals,
${\rm Ball}(\HHHx):= \{ \hhh\in\HHHx:\ \norm{\hhh}<1 \}$ for the unit ball,
$\partial{\rm Ball}(\HHHx):= \{ \hhh\in\HHHx:\ \norm{\hhh}=1\}$ for the unit sphere
and
${\bf g}^* \land \hhh^* := {\bf g}^* \otimes \hhh^* - \hhh^* \otimes {\bf g}^*$
for the basic antisymmetric functionals, respectively, where
${\bf g}^* \otimes \hhh^* := [({\bf x},{\bf y})\mapsto {\bf g}^* ({\bf x}) \hhh^*({\bf y})]$.
The spaces ${\cal F}_2,{\cal F}_3$ in Theorem 1.6 are isomorphic copies of the Banach spaces
$\FFF_2,\FFF_3$ of all
symmetric resp. antisymmetric continuous bilinear functionals $\HHHx \times \HHHx \to \CC$.
Recall [11] that in both cases $k=2,3$, the surjective linear isometries of $\FFF_k$
are of the form
$[U\otimes U]\Phi = \big[ ({\bf x},{\bf y}) \mapsto \Phi(U{\bf x},U{\bf y})\big]$ $(\Phi\in\FFF_k)$.
Therefore the conclusion that, given any of the indices $k\in\{ 2,3\}$
with a one-parameter group
$\big[ {\bf U}(t) :\ t\in\RR\big]$ of the form ${\bf U}(t) = U_t\otimes U_t$
such that all the functions $t\mapsto \Phi(U_t{\bf x},U_t{\bf y})$
$(\Phi\in\FFF_k ;\ {\bf x},{\bf x}\in\HHHx)$ are continuous,
there are functions $\kappa,\widetilde{\kappa}:\RR\to\TT$ making
$\big[ \kappa(t)U_t:\ t\in \RR\big]$ resp.
$\big[ \widetilde{\kappa}(t)U_t:\ t\in \RR\big]$
a strongly continuous family resp. a one parameter group
is immediate from Propositions 3.3,3.5 and Corollary 3.4 below
even with $\kappa,\widetilde{\kappa}$ admitting only the values $\pm1$.
Since the results of [10, Sections 4,5] concern only a single underlying Hilbert space,
hence we can conclude Theorem 1.6 (without refering to [10, Section 6]).

\bigskip

%\bf 3.1. Lemma.
\rm LEMMA 3.1.
\rm [10, 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 map with respect to the distance
${\rm dist}(\TT{\bf g} ,\TT\hhh) :=
\min_{\abs{\kappa}=\abs{\lambda}=1} \abs{\kappa {\bf g} - \lambda\hhh}$.
Then ${\bf F}(t) =\TT \hhh_t$ $(t\in\RR)$ for some
continuous function $t\mapsto \hhh_t \in \partial {\rm Ball}(\HHHx)$.


\bigskip

%\bf 3.2 Lemma.
\rm LEMMA 3.2.
\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
\it Proof. \rm
We can apply Lemma 3.1 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 a suitable function $\kappa:\RR\to\TT$.
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 3.3. Proposition.
\rm PROPOSITION 3.3.
\it
If $t\mapsto \phi({\bf e}_t,{\bf e}_t)$
with unit vectors ${\bf e}_t\in\HHHx$ is continuous for all $\Phi\in \FFF_2$
then there is a function $\sigma:\RR\to\{-1,1\}$
such that $t\mapsto \sigma(t) {\bf e}_t$ is continuous.

\bigskip
\it Proof. \rm Given any vector $\hhh\in\HHHx$, with the functional
$\Phi_\hhh ({\bf x},{\bf y}):= \hhh^* \otimes \hhh^* (\in \FFF_2)$
%$\big\langle {\bf x} \big\vert \hhh \big\rangle \big\langle {\bf y} \big\vert \hhh \big\rangle$
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 a
{\it universal subnet} [4] for it.
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 3.2 applies.


\bigskip
%\bf 3.4. Corollary.
\rm COROLLARY 3.4.
\it
Let $t\mapsto [{\bf e}_t,{\bf f}_t]$ be a map
$\RR\to\{${\rm orthonormed 2-frames}$\}$. If the map
$t\mapsto \Phi(\xi {\bf e}_t+\eta {\bf f}_t,\xi {\bf e}_t+\eta{\bf f}_t)$
is continuous for all $\Phi\in \FFF_2$ 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
\it Proof. \rm
By Proposition 3.3, 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 3.5 Proposition.
\rm PROPOSITION 3.5.
\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 $\Phi\in \FFF_3$. 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
\it Proof. \rm
Consider a convergent net $t_\nu\to t\in\RR$ and let $t_{\nu_\alpha} (\to t)$ be again a
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.

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

\vskip10mm

%\bf 4. Proof of Proposition 1.5
\centerline{\bf PROOF OF PROPOSITION 1.5}
\medskip
\rm
Let ${\bf U}:=\big[ U^t:\ t\in \RR\big]$ be a strongly continuous one-parameter
subgroup of $\HHHx$
such that
%for some constant $M\in[1,\infty)$ we have
$$M\norm{{\bf x}} \le \norm{U^t {\bf x}} \le M \norm{{\bf x}} \qquad
({\bf x}\in\HHHx,\ t\in\RR) \ .$$
Consider a Banach limit $L_{R\to\infty}$ 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
$$\liminf_{R\to\infty} f(R) \le L_{R\to \infty}f(R) \le \limsup_{R\to\infty} f(R) \quad
\hbox{whenever} \ \ {\rm range}(f) \subset\RR .
\leqno(4.1)$$
Since all the functions $t\mapsto U^t{\bf x}$ are continuous and bounded,
the operation
$$\sspr{{\bf x}}{{\bf y}} := L_{R\to\infty} \left(
{1\over 2R} \int_{-R}^R \spr{U^t{\bf x}}{U^t{\bf y}}\ dt \right) $$
is well-defined for every couple of vectors ${\bf x},{\bf y}\in\HHHx$.
The sesquilinearity of the product $\spr..$ along with the inequalities (4.1)
entail immediately that
$\sspr..$ is also a scalar product on $\HHHx$ with
$$M^{-2} \spr{{\bf x}}{{\bf x}}  \le \sspr{{\bf x}}{{\bf x}} \le M^{2} \spr{{\bf x}}{{\bf x}}
\qquad ({\bf x}\in\HHHx)\ .$$
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
$\left\vert \int_I \spr{U^t{\bf x}}{U^t{\bf y}}\ dt \right\vert \le M^2 s \norm{{\bf x}}^2$
whenever $I$ is an interval of length $\abs{s}$.
Therefore
$$
\sspr{U^s{\bf x}}{U^s{\bf y}} =
\vtop{
\hbox{$L_{R\to\infty} \Big(  {1\over 2R} \int_{-R}^R \spr{U^{t+s}{\bf x}}{U^{t+s}{\bf y}}\ dt \Big) =$}
\hbox{$= L_{R\to\infty} \Big(  {1\over 2R} \int_{-R+s}^{R+s} \spr{U^t{\bf x}}{U^t{\bf y}}\ dt \Big) =$}
\hbox{$= L_{R\to\infty} \Big(  {1\over 2R} \Big[\int_{-R}^R +\int_{-R+s}^{-R} + \int_R^{R+s} \Big]
\spr{U^t{\bf x}}{U^t{\bf y}}\ dt \Big) =$}
\hbox{$= L_{R\to\infty} \Big(  {1\over 2R} \int_{-R+s}^{R+s} \spr{U^t{\bf x}}{U^t{\bf y}}\ dt + O(R^{-1}) \Big) =$}
\hbox{$= L_{R\to\infty} \Big(  {1\over 2R} \int_{-R+s}^{R+s} \spr{U^t{\bf x}}{U^t{\bf y}}\ dt \Big) =
\sspr{{\bf x}}{{\bf y}} \ .$}
}$$



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bigskip
\bf Acknowledgements.
%
\rm
The author was supported by the projects \
T\'AMOP-4.2.2.A-11/1/KONV-2012-0060 \
Impulse Lasers for Use in Materials Science and Biophotonics \
and \
T\'AMOP-4.2.2.A-11/1/KONV-2012-0073 \
Telemedicine Oriented Research in the Fields of Mathematics, Informatics and Medical Sciences
of the European Union and co-financed by the European Social Fund.


\bigskip
\bigskip
\rm


%\bf References
\centerline{\rm REFERENCES}
\rm


\begin{description}

\rm

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

\smallskip
\item{[2]} F. Botelho, J. Jamison, \it
Projections on tensor products of Banach spaces, \rm
Arch. Math. {\bf 90} (2008), 341–352.

\smallskip
\item{[3]} Y. Friedman, B. Russo, \it
The Gelfand-Naimark theorem for JB*-triples, \rm
Duke Math. J. {\bf 53} (1986), 139-148.

\smallskip
\item{[4]} N.R. Howes, \it
Modern analysis, \rm
Springer-Verlag, New York, 1995.

\smallskip
\item{[5]} R. Iordanescu, \it
Jordan structures in analysis, geometry and physics, \rm
Editura Academiei Romane, Bucharest, 2009.

\smallskip
\item{[6]} J.-M. Isisdro, L.L. Stach\'o, \it
Holomorphic Automorphism Groups in Banach Spaces, \rm
North Holland Math. Studies 105,
North Holland Publ. Co., Oxford--New York--Amsterdam, 1985.

\smallskip
\item{[7]} W. Kaup, \it
A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, \rm
Math. Z., {\bf 83} (1983), 503--529.

\smallskip
\item{[8]}
F. Riest - B. Sz.-Nagy, \it
Vorlesungen \"uber Funktionalanalysis, \rm
VEB Deutscher Verlag der Wissenschaften, Berlin, 1968.

\smallskip
\item{[9]} R.A. Ryan, \it
Introduction to tensor products of Banach spaces, \rm
Springer Monographs in Mathematics,
Springer-Verlag, London-Berlin, 2002.

\smallskip
\item{[10]} L.L. Stach\'o, \it
On strongly continuous one-parameter groups of automorphisms of multilinear functionals, \rm
J. Math. Anal. Appl. {\bf 363} (2010), 419–430.


\smallskip
\item{[11]} H. Upmeier, \it
Symmetric Banach manifolds and Jordan C*-algebras, \rm
North-Holland Math. Studies 104,
North-Holland Publ. Co., Oxford--New York--Amsterdam, 1985.

\smallskip
\item{[12]}
K. Yosida, \it
Functional Analysis, \rm Classics Math.,
Springer-Verlag, Berlin, 1995.


\end{description}



\vskip10mm

\hfill
\vtop{\Smallit
\hbox{L.L. STACH\'O}
\hbox{Bolyai Institute}
\hbox{University of Szeged}
\hbox{Aradi V\'ertan\'uk tere 1}
\hbox{H-6720 Szeged}
\hbox{Hungary}
\hbox{stacho@math.u-szeged.hu}
}

\end{document}






\bye
