• Dinesh J Karia

      Articles written in Proceedings – Mathematical Sciences

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

      Subhash J Bhatt Dinesh J Karia

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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*-algebrasA andB is completely positive iff by restriction, it defines a completely positive map between the C*-algebrasb(A) andb(B) consisting of all bounded elements ofA andB. In the metrizable case,A andB are homeomorphically isomorphic iff they are matricially order isomorphic. The injective pro-C*-topology α and the projective pro-C*-topology v on A⊗B are shown to be minimal and maximal pro-C*-topologies; and α coincides with the topology of biequicontinous convergence iff eitherA orB is abelian. A nuclear pro-C*-algebraA is one that satisfies, for any pro-C*-algebra (or a C*-algebra)B, any of the equivalent requirements; (i) α =v onA ⊗B (ii)A is inverse limit of nuclear C*-algebras (iii) there is only one admissible pro-C*-topologyon A⊗B (iv) the bounded partb(A) ofA 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 σ-C*-algebraA is a nuclear space iff it is an inverse limit of finite dimensional C*-algebras; and if abelian, thenA is isomorphic to the algebra (pointwise operations) of all scalar sequences.

    • On a Class of Smooth Frechet Subalgebras of $C^\ast$-Algebras

      Subhash J Bhatt Dinesh J Karia Meetal M Shah

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      The paper contributes to understanding the differential structure in a $C^∗$-algebra. Refining the Banach $(D^∗_p)$-algebras investigated by Kissin and Shulman as noncommutative analogues of the algebra $C^p[a,b]$ of 𝑝-times continuously differentiable functions, we investigate a Frechet $(D^∗\infty)$-subalgebra $\mathcal{B}$ of a $C^∗$-algebra as a noncommutative analogue of the algebra $C^\infty[a,b]$ of smooth functions. Regularity properties like spectral invariance, closure under functional calculi and domain invariance of homomorphisms are derived expressing $\mathcal{B}$ as an inverse limit over 𝑛 of Banach $(D^∗_n)$-algebras. Several examples of such smooth algebras are exhibited.

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