Journal of Operator Theory

Volume 56, Issue 2, Fall 2006 pp. 357-376.

The completion of a

$C^*$ -algebra with a locally convex topology

Authors: Fabio Bagarello (1), Maria Fragoulopoulou (2), Atsushi Inoue (3) and Camillo Trapani (4)
Author institution: (1) Dipartimento di Metodi e Modelli Matematici, Facolt\`a di ingegneria, Universita di Palermo, Palermo, I-90128, Italy
(2) Department of Mathematics, University of Athens, Athens, 15784, Greece
(3) Department of Applied Mathematics, Fukuoka University, Fukuoka, 814-0180, Japan
(4) Dipartimento di Matematica ed Applicazioni, Universita di Palermo, Palermo, I-90123, Italy

Summary: There are examples of

$C^*$ -algebras

$\A$ that accept a locally convex

$*$ -topology

$\tau$ coarser than the given one, such that

$\widetilde{\A}[\tau]$ (the completion of

$\A$ with respect to

$\tau$ ) is a

$GB^*$ -algebra. The multiplication of

$\A[\tau]$ may be or not be jointly continuous. In the second case,

$\widetilde{\A}[\tau]$ may fail being a locally convex

$*$ -algebra, but it is a partial

$*$ -algebra. In both cases the structure and the representation theory of

$\widetilde{\A}[\tau]$ are investigated. If

$\overline{\A}_+^{\,\tau}$ denotes the

$\tau$ -closure of the positive cone

$\A_+$ of the given

$C^*$ -algebra

$\A$ , then the property

$\overline{\A}_+^{\,\tau} \cap (-\overline{\A}_+^{\,\tau})= \{0\}$ is decisive for the existence of certain faithful

$*$ -representations of the corresponding

$*$ -algebra

$\widetilde{\A}[\tau]$ .

Contents Full-Text PDF