1 Introduction
The theory of operator spaces is a natural noncommutative quantisation of Banach space theory. Many problems in operator spaces are naturally motivated from both Banach space theory and operator algebra theory. Some properties such as local reflexivity, exactness, nuclearity and injectivity have been intensively studied (see [Reference Effros, Junge and Ruan5, Reference Effros, Ozawa and Ruan6, Reference Pisier10]). In particular, for any operator space V,
$$ \begin{align*}V\;\mbox{is nuclear}\Rightarrow V\;\mbox{is exact}\Rightarrow V\;\mbox{is locally reflexive}.\end{align*} $$
The first implication was proved in [Reference Pisier10] and the second in [Reference Effros, Ozawa and Ruan6]. In [Reference Effros, Ozawa and Ruan6], Effros et al. showed that an operator space V is nuclear if and only if V is locally reflexive and
$V^{**}$
is injective. As pointed out in [Reference Effros, Ozawa and Ruan6], local reflexivity is an essential condition in this result since Kirchberg [Reference Kirchberg8] had constructed a separable nonnuclear operator space V for which
$V^{**}=\prod _{n=1}^{+\infty } M_{n}$
. Turning to
$C^{*}$
-algebra theory, using Conne’s deep work in [Reference Connes3], Choi and Effros proved the following result in [Reference Choi and Effros1, Reference Choi and Effros2]:
$$ \begin{align*}\mbox{A}\: C^{*} \mbox{-algerba}\: \mathcal{A}\: \mbox{is nuclear}\Leftrightarrow \mbox{its second dual}\: \mathcal{A}^{**}\: \mbox{is injective}.\end{align*} $$
In [Reference Dong and Ruan4], Dong and Ruan showed that an operator space V is exact if and only if V is locally reflexive and
$V^{**}$
is weak
$^*$
exact.
In [Reference Effros, Junge and Ruan5], Effros et al. used the technique of mapping spaces to prove the most surprising result in the theory of operator spaces: the dual
$\mathcal {A}^{*}$
of any
$C^{*}$
-algebra
$\mathcal {A}$
is locally reflexive. In light of the fact that
$C^{*}$
-algebras need not be locally reflexive, it was thought that the same would be true for their dual operator spaces. It therefore came as quite a surprise to find that all such dual spaces, as well as all von Neumann algebraic preduals, are locally reflexive. In this short paper, we introduce the notion of weakly local reflexivity in operator space theory. We prove that any dual operator space is weakly locally reflexive.
2 Weakly local reflexivity
We first recall the definition of local reflexivity in operator space theory (see [Reference Effros and Ruan7]).
Definition 2.1. Suppose that V is an operator space. We say that V is locally reflexive if for any finite dimensional operator space L, every complete contraction
$\varphi : L\rightarrow V^{**}$
is the point-weak
$^{*}$
limit of a net of complete contractions
$\varphi _{\alpha }: L\rightarrow V$
.
Definition 2.2. We say that a dual operator space
$V^{*}$
is weakly locally reflexive if for any finite dimensional operator space L and every complete contraction
$\varphi : L\rightarrow V^{***}$
, there exists a net of complete contractions
$\varphi _{\alpha }: L\rightarrow V^{*}$
such that
$$ \begin{align*}\langle\varphi_{\alpha}(x), f\rangle\longrightarrow \langle\varphi(x), f\rangle\quad \mbox{for all } x\in L, f\in V.\end{align*} $$
It is well known that
$\mathcal {B(H)}$
is not locally reflexive for any infinite dimensional Hilbert space
$\mathcal {H}$
. However, the following result implies that
$\mathcal {B(H)}$
is weakly locally reflexive.
Theorem 2.3. Any dual operator space
$V^{*}$
is weakly locally reflexive.
Proof. For any finite dimensional subspaces
$E\subseteq V^{***}$
and
$F\subseteq V\subseteq V^{**}$
, it follows from [Reference Effros and Ruan7, Lemma 14.3.4] that for each
$n\in \mathbb {N}$
, we can find a mapping
$\psi ^{(n)}: E\rightarrow V^{*}$
such that
$\|(\psi ^{(n)})_{n}\|<1+1/n$
and
$$ \begin{align*}\langle\psi^{(n)}(x), f\rangle=\langle x, f\rangle\quad \mbox{for all } x\in E, f\in F.\end{align*} $$
Thus,
$\{\psi ^{(n)}\}$
is a sequence in the closed ball of radius 2 of
$B(E, V^{*})=(E\stackrel {\gamma }{\otimes } V)^{*}$
. From Alaoglu’s theorem and [Reference Paulsen9, Lemma 7.2], we may choose a limit point
$\psi : E\rightarrow V^{*}$
of the sequence
$\{\psi ^{(n)}\}$
in the point-weak
$^{*}$
topology. If
$r\leq n$
, then
$$ \begin{align*}\|(\psi^{(n)})_{r}\| \leq \|(\psi^{(n)})_{n}\| \leq 1+{1\over n}\end{align*} $$
and thus
$\|\psi _{r}\| \leq 1$
. It follows that
$\|\psi \|_{cb}\leq 1$
. Furthermore,
$$ \begin{align*}\langle\psi(x), f\rangle=\langle x, f\rangle\quad \mbox{for all } x\in E, f\in F.\end{align*} $$
Now for any finite dimensional operator space L and every complete contraction
$\varphi : L\rightarrow V^{***}$
, we fix
$E=\varphi (L)\subseteq V^{***}$
. For any finite dimensional subspaces
$F\subseteq V$
, it follows from the above proof that there exist complete contractions
$\psi _{F}: E\rightarrow V^{*}$
such that
$$ \begin{align*}\langle\psi_{F}(y), f\rangle=\langle y, f\rangle\quad \mbox{for all } y\in E=\varphi(L), f\in F.\end{align*} $$
Thus, the net
$\psi _{F}\circ \varphi : L\rightarrow V^{*}$
satisfies
$$ \begin{align*}\langle\psi_{F}\circ\varphi(x), f\rangle\longrightarrow \langle\varphi(x), f\rangle\quad \mbox{for all } x\in L, f\in V,\end{align*} $$
with
$\|\psi _{F}\circ \varphi \|_{cb}\leq 1$
. This implies that the dual operator space
$V^{*}$
is weakly locally reflexive.
