Cb-frames for operator spaces

TL;DR

This paper introduces the concept of cb-frames for operator spaces, establishing their existence for certain algebras and characterizing when operator spaces possess cb-frames, linking to properties like weak amenability.

Contribution

It defines cb-frames for operator spaces, constructs a concrete example for reduced free group C*-algebra, and characterizes operator spaces with cb-frames in relation to approximation properties.

Findings

Existence of a concrete cb-frame for $C_r^*(F_2)$

Operator space has a cb-frame iff it has the completely bounded approximation property

A discrete group is weakly amenable iff its reduced group C*-algebra has a cb-frame

Abstract

In this paper, we introduce the concept of cb-frames for operator spaces. We show that there is a concrete cb-frame for the reduced free group C*-algebra $C_r^*(F_2)$, which is derived from the infinite convex decomposition of the biorthogonal system $(\lambda_s, \delta_s)_{s \in F_2}$. We show that, in general, a separable operator space X has a cb-frame if and only if it has the completely bounded approximation property if and only if it is completely isomorphic to a completely complemented subspace of an operator space with a cb-basis. Therefore, a discrete group $\Gamma$ is weakly amenable if and only if the reduced group C*-algebra $C^*_r(\Gamma)$ has a cb-frame. Finally, we show that, in contrast to Banach space case, there exists a separable operator space, which can not be completely isomorphic to a subspace of an operator space with a cb-basis.