AF- Operator Spaces And C * - Algebra Projection Operator Space Tensor Product

In this paper, we first define and study the direct limit of operator spaces, we show that the uniqueness theorems for the direct limit of operator spaces under the significance of completely isometric injection. Approximately finite dimensional (AF) C*-algebras was introduced by Bratteli in [37]. As a large class of C*-algebras, the research of the theory of AF-algebras is very important and highly non-trivial. For this, we define the approximately finite dimensional (AF) operator spaces, thus AF-algebras become a special case of AF-operator spaces. For any AF-operator space we prove that it is agree with a direct limit of some operator spaces, and besides it’s closed subspace and the corresponding quotient space both are AF-operator spaces. These results cover the corresponding conclusions in [36]. If given AF-operator spaces V and W, we also prove that the projective tensor product V (?) W and the injective tensor product V(?)W W of operator spaces are also AF-operator spaces. In addition, Takesaki has cleverly constructed and disscussed the infinite tensor product of C*-algebras in [43], we learn his method and use the direct limit of operator spaces to define the infinite projective tensor product and the infinite injective tensor product of operator spaces. We extend the embedding theorem of infinite tensor product of C*-algebras in [44] to the operator spaces. Finally we consider the operator space projective tensor product of C*-algebras. For C*-algebras V and W, we prove that the Banach*-algebra V(?)W preserves*-homomorphism and a characterization of the convergence property of dual space (V(?)W)*will be given. Further, we study the prime ideals of V(?)W and prove that it’s prime ideals are agree with primitive adeals.