Topological ^*-algebras with C^*-enveloping algebras II

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 pro-C^*-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 m-tempered, 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 non-unital 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 m-convex 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.