C^∗-Segal Algebras with Order Unit, II

Abstract

We introduce the notion of a (noncommutative) C*-Segal algebra as a Banach algebra (A, · A ) which is a dense ideal in a C*-algebra (C, · C ), where · A is strictly stronger than · C on A. Several basic properties are investigated and, with the aid of the theory of multiplier modules, the structure of C*-Segal algebras with order unit is determined.

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

“…Let A be a Segal algebra in a C * -algebra C. The approximate ideal of A is the set E A := span • A {ab : a, b ∈ A}; this is a Banach subalgebra of A which is contained as a Segal algebra in C. Remark 1.5. Strictly speaking, this is not the definition of approximate ideal originally given in [5,12], but it coincides with the original one by [12, Lemmas 2.9 and 2.16].…”

“…Our main result is a characterization theorem for C * -Segal algebra valued function algebras with an order unitization. Together with results of [13], this yields a structure theorem for these algebras. As an Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.…”

“…Recently, a new approach to Segal algebras was introduced in [5,12], which allows a more detailed study of their structure and properties, particularly in the absence of an approximate identity ( [5] the commutative case and [12] the noncommutative case). The main new tools used in these papers are the notion of "approximate ideal" together with the theory of multiplier modules.…”

“…The primary motivation of this investigation was to gain a better understanding of the order structure of (not necessarily closed) ideals of C * -algebras. Indeed, the main results of [5,12] are structure theorems for self-adjoint C * -Segal algebras with an order unit whose norm coincides with the order unit norm on the self-adjoint part. These results were extended in [13] to C * -Segal algebras with an order unitization, i.e., to which an order unit can be added in a structure preserving way.…”

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

“…Theorems 5.7 and 5.21 suggest that the theory of Nachbin algebras can be extended to the non-commutative setting. In fact, first steps in this direction have already been taken in [20].…”