\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\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}}



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












\bye
