We introduce the concept of (completely bounded) frames for Banach and operator spaces, show the connection with the (completely) bounded approximation property and complemented embedding, and give the duality theorems for frames and associated basis in Banach spaces. We also prove that a separable operator space has the completely bounded approximation property if and only if it has a completely bounded frame if and only if it is completely isomorphic to a completely complemented subspace of an operator space with a completely bounded basis. We give 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 canonical biorthogonal system. Finally, we show that, in contrast to Banach space case, the Oikhberg-Ricard Hilbertian operator space can not be completely isomorphic to a subspace of an operator space with a cb-basis. Also, we will mention some recent progress (joint with Zhong-Jin Ruan).

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