Topological algebras withC*-enveloping algebras

Abstract

LetA be a complete topological *algebra which is an inverse limit of Banach *algebras. The (unique) enveloping algebraE(A) ofA, providing a solution of the universal problem for continuous representations ofA into bounded Hilbert space operators, is known to be an inverse limit ofC*-algebras. It is shown thatS(A) is aC*-algebra iffA admits greatest continuousC*-seminorm iff the continuous states (respectively, continuous extreme states) constitute an equicontinuous set. AQ-algebra (i.e., one whose quasiregular elements form an open set)A hasC*-enveloping algebra. There exists (i) a Frechet algebra with C*-enveloping algebra that is not aQ-algebra under any topology and (ii) a non-Q spectrally bounded algebra withC*-enveloping algebra.A hermitian algebra withC*-enveloping algebra turns out to be aQ-algebra. The property of havingC*-enveloping algebra is preserved by projective tensor products and completed quotients, but not by taking closed subalgebras. Several examples of topological algebras withC*-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 envelopingC*-algebra of a hermitian topological algebra with an orthogonal basis is isomorphic to theC*-algebrac0 of all null sequences.