Cb-frames for operator spaces

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⁎C⁎-algebra Cr⁎(F2), which is derived from the infinite convex decomposition of the biorthogonal system (λs,δs)s∈F2(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 Cr⁎(Γ) has a cb-frame. Finally, we show that, in contrast to Banach space case, there exists a separable operator space, which cannot be completely isomorphic to a subspace of an operator space with a cb-basis.