2016/01/23 by Rui Liu, Liu, Rui, Zhong-Jin Ruan +1 · 1 citation
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.1601.06218
17 pages, to appear in JFA
arxiv created 2016/01/23 · arxiv updated 2016/01/26
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 Cr^*(F2), which is derived from the infinite convex decomposition of the biorthogonal system (λs, δs)s ∈ F2. 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 Γ is weakly amenable if and only if the reduced group C*-algebra C^*r(Γ) 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.