Minimal operator space structures on $C^*$ algebras

Question

Let $\Omega$ be a $\sigma$-finite measure space with measure $\mu.$ Then the natural operator space structure on $L_\infty(\Omega)$ is defined to be the operator space structure induced by the embedding of $L_\infty(\Omega)$ into some $B(\mathcal H)$ as a $C^*$ algebra. Is this the minimal operator space structure on $L_\infty(\Omega)$? What about the case when we replace $L_\infty(\Omega)$ by a general tracial von Neumann algebra?

Answer 1

An operator space carries the minimal operator space structure of and only if it is completely isometric to a subspace of a commutative $C^\ast$-algebra (Effros, Ruan. Operator Spaces, Proposition 3.3.1). The proof of the relevant direcion relies on the fact that for a commutative $C^\ast$-algebra $A$, the Gelfand transform $A\to C(\operatorname{ex}B_{A^\ast})$ is an $\ast$-isomorphism and the inclusion $C(\operatorname{ex}B_{A^\ast})\hookrightarrow C(B_{A^\ast})$ is a complete isometry since the convex hull of $\operatorname{ex}B_{A^\ast}$ is weak$^\ast$ dense in $B_{A^\ast}$.

For a general tracial von Neumann algebra this is no longer true. In fact, every bounded linear map from an operator space into a minimal operator space is completely bounded with the same norm. Since the transpose map on $M_k(\mathbb{C})$ for $k\geq 2$ has strictly larger cb-norm then norm, it follows that $M_k(\mathbb{C})$ does not carry the minimal operator space structure.