Bhatt, S. J. (2001) Topological ^{*}algebras with C^{*}enveloping algebras II Proceedings of the Indian Academy of Sciences  Mathematical Sciences, 111 (1). pp. 6594. ISSN 02534142

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Official URL: http://www.ias.ac.in/mathsci/vol111/feb2001/06509...
Related URL: http://dx.doi.org/10.1007/BF02829541
Abstract
Universal C^{*}algebras C^{*}(A) exist for certain topological ^{*}algebras called algebras with a C^{*}enveloping algebra. A Frechet ^{*}algebraA has a C^{*}enveloping algebra if and only if every operator representation of A maps A into bounded operators. This is proved by showing that every unbounded operator representation Λ, continuous in the uniform topology, of a topological ^{*}algebra A, which is an inverse limit of Banach ^{*}algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping proC^{*}algebra E(A) of A. Given a C^{*}dynamical system (G,A,α), any topological ^{*}algebra B containing C_{c }(G,A) as a dense C^{*}subalgebra and contained in the crossed product C^{*}algebra C^{*}(G,A,α) satisfies E(B) =C^{*}(G,A,α;). If G = R, if B is an α;invariant dense Frechet ^{*}subalgebra of A such that E(B) =A, and if the action α; on B is mtempered, smooth and by continuous ^{*}automorphisms: then the smooth Schwartz crossed productS (R,B,α;) satisfies E(S(R,B,α;)) =C^{*}(R,A,α;). WhenG is a Lie group, the C^{∞}elements C^{∞}(A), the analytic elements C^{ω}(A) as well as the entire analytic elements C^{eω}(A) carry natural topologies making them algebras with a C^{*}enveloping algebra. Given a nonunital C^{*}algebra A, an inductive system of ideals I_{α;} is constructed satisfying A =C^{*}ind lim I_{α;}; and the locally convex inductive limit ind lim I_{α;} is an mconvex algebra with the C^{*}enveloping algebra A and containing the Pedersen ideal K _{A} of A. Given generators G with weakly Banach admissible relations R, we construct universal topological^{*}algebra A(G, R) and show that it has a C^{*}enveloping algebra if and only if (G, R) is C^{*}admissible.
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Keywords:  Frechet ^{*}algebra; Topological ^{*}algebra; C^{*}enveloping Algebra; Unbounded Operator Representation; O^{*}algebra ;smooth Frechet Algebra Crossed Product; Pedersen Ideal of a C^{*}algebra; Groupoidcalgebra; Universal Algebra on Generators with Relations 
ID Code:  59681 
Deposited On:  07 Sep 2011 05:17 
Last Modified:  18 May 2016 10:09 
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