Double multipliers andA*-algebras of the first kind

Abstract

LetAbe anA*-algebra and letdenote its auxiliary norm closure. The multiplier algebras of dualA*-algebras of the first kind have been studed by Tomiuk [12], [13] and Wong[15]. In this paper we study the double multiplier algebra ofA*-algebras of the first kind. In particular, we prove that, ifAis anA*-algebra of the first kind, then the double multiplier algebraM(A) ofAis *-isomorphic and (auxiliary norm) isometric to a subalgebra ofM(), extending in the process some results established by Tomiuk[12]. We also consider the embedding of the double multiplier algebra ofAin**, when the latter is regarded as an algebra with the Arens product, and, in particular, we show that, for an annihilator A*-algebra,M(A) is *-isomorphic and (auxiliary norm) isometric to**.

C∗-Segal Algebras with Order Unit, II

“…Despite many important examples in analysis, such as the Schatten classes for example, the general structure and properties of C*-Segal algebras is not well understood. The multiplier algebra and the bidual of self-adjoint C*-Segal algebras were described in [1,13] and, in the presence of an approximate identity, the form of the closed ideals of C*-Segal algebras was given in [6]. Commutative C*-Segal algebras were studied by Arhippainen and the first-named author in [5].…”

On dense ideals of C*-algebras and generalizations of the Gelfand–Naimark Theorem

“…A C * -Segal algebra is a Banach algebra which is a dense ideal in a C * -algebra; it need not be self-adjoint. The multiplier and the bidual algebras of selfadjoint C * -Segal algebras were described in [1,16,18] and, in the presence of an approximate identity, the form of the closed ideals of C * -Segal algebras was given in [7]. Otherwise, however, not much is known about the general structure of C * -Segal algebras.…”

Order structure, multipliers, and Gelfand representation of vector-valued function algebras

“…In particular, most results have relied on the additional assumption of an approximate identity. The multiplier algebra and the bidual of self-adjoint C * -Segal algebras were described in [2,14], and the form of the closed ideals of C * -Segal algebras with an approximate identity was given in [6].…”