Topological algebras with C^*-enveloping algebras

Abstract

LetA be a complete topological ^*algebra which is an inverse limit of Banach^*algebras. The (unique) enveloping algebra E(A) of A, providing a solution of the universal problem for continuous representations of A into bounded Hilbert space operators, is known to be an inverse limit of C^*-algebras. It is shown that S(A) is a C^*-algebra iff A admits greatest continuous C^*-seminorm iff the continuous states (respectively, continuous extreme states) constitute an equicontinuous set. A Q-algebra (i.e., one whose quasiregular elements form an open set) A has C^*-enveloping algebra. There exists (i) a Frechet algebra with C^*-enveloping algebra that is not a Q-algebra under any topology and (ii) a non-Q spectrally bounded algebra with C^*-enveloping algebra.A hermitian algebra with C^*-enveloping algebra turns out to be a Q-algebra. The property of having C^*-enveloping algebra is preserved by projective tensor products and completed quotients, but not by taking closed subalgebras. Several examples of topological algebras with C^*-enveloping algebras are discussed. These include several pointwise algebras of functions including well-known test function spaces of distribution theory, abstract Segal algebras and concrete convolution algebras of harmonic analysis, certain algebras of analytic functions (with Hadamard product) and Kothe sequence algebras of infinite type. The enveloping C^*-algebra of a hermitian topological algebra with an orthogonal basis is isomorphic to the C^*-algebra c₀ of all null sequences.