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

Bhatt, Subhash J. ; Karia, Dinesh J. (1991) Complete positivity, tensor products and C*-nuclearity for inverse limits of C*-algebras Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 101 (3). pp. 149-167. ISSN 0253-4142

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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.

Item Type:Article
Source:Copyright of this article belongs to Indian Academy of Sciences101.
Keywords:Inverse Limits of C* Algebras; completely Positive Maps; Tensor Products; Nuclear C*-and Nuclear Pro-c*-algebras; Multiplier Algebras; Nuclear Space
ID Code:59689
Deposited On:07 Sep 2011 04:47
Last Modified:18 May 2016 10:10

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