Complete positivity, tensor products and C^*-nuclearity for inverse limits of C^*-algebras

Abstract

The paper aims at developing a theory of nuclear (in the topological algebraic sense) pro-C^*-algebras (which are inverse limits of C^*-algebras) by investigating completely positive maps and tensor products. By using the structure of matrix algebras over a pro-C^*-algebra, it is shown that a unital continuous linear map between pro-C^*-algebras A and B is completely positive iff by restriction, it defines a completely positive map between the C^*-algebras b(A) and b(B) consisting of all bounded elements of A and B. In the metrizable case, A and B are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C^*-topology α and the projective pro-C^*-topology υ on A ⊗ B are shown to be minimal and maximal pro-C^*-topologies; and α coincides with the topology of biequicontinous convergence iff either A or B is abelian. A nuclear pro-C^*-algebra A is one that satisfies, for any pro-C^*-algebra (or a C^*-algebra) B, any of the equivalent requirements; (i) α= υ on A ∃ B (ii) A is inverse limit of nuclear C^*-algebras (iii) there is only one admissible pro-C^*-topologyon A ⊗ B (iv) the bounded part b(A) of A is a nuclear C^*-algebra (v) any continuous complete state map A→ B^* can be approximated in simple weak^* convergence by certain finite rank complete state maps. This is used to investigate permanence properties of nuclear pro-C^*-algebras pertaining to subalgebras, quotients and projective and inductive limits. A nuclearity criterion for multiplier algebras (in particular, the multiplier algebra of Pedersen ideal of a C^*-algebra) is developed and the connection of this C^*-algebraic nuclearity with Grothendieck's linear topological nuclearity is examined. A б-C^*-algebra A is a nuclear space iff it is an inverse limit of finite dimensional C^*-algebras; and if abelian, then A is isomorphic to the algebra (pointwise operations) of all scalar sequences.