Topological *-algebras with C*-enveloping algebras II

Universal C*-algebras C*(A) exist for certain topological *-algebras called algebras with a C*-enveloping algebra. A Frechet *-algebra A 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 pi, 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,alpha), any topological *-algebra B containing C-c(G,A) as a dense *-subalgebra and contained in the crossed product C*-algebra C*(G,A,alpha) satisfies E(B) = C*(G,A,alpha). If G = R, if B is an alpha -invariant dense Frechet *-subalgebra of A such that E(B) = A, and if the action alpha on B is m-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed product S(R,B,alpha) satisfies E(S(R,B,alpha)) = C*(R,A,alpha). When G is a Lie group. the C-infinity-elements C-infinity(A), the analytic elements C-omega(A) as well as the entire analytic elements C-omega(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-alpha is constructed satisfying A = C*-ind lim I-alpha; and the locally convex inductive limit ind lim I-alpha 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.