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. 149167. ISSN 02534142

PDF
 Publisher Version
2MB 
Official URL: http://www.ias.ac.in/j_archive/mathsci/101/3/1491...
Related URL: http://dx.doi.org/10.1007/BF02836797
Abstract
The paper aims at developing a theory of nuclear (in the topological algebraic sense) proC^{*}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 proC^{*}algebra, it is shown that a unital continuous linear map between proC^{*}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 proC^{*}topology α and the projective proC^{*}topology υ on A ⊗ B are shown to be minimal and maximal proC^{*}topologies; and α coincides with the topology of biequicontinous convergence iff either A or B is abelian. A nuclear proC^{*}algebra A is one that satisfies, for any proC^{*}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 proC^{*}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 proC^{*}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 бC^{*}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 Proc^{*}algebras; Multiplier Algebras; Nuclear Space 
ID Code:  59689 
Deposited On:  07 Sep 2011 04:47 
Last Modified:  18 May 2016 10:10 
Repository Staff Only: item control page