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

**Keywords:**$GB^*$-algebra, unbounded $C^*$-seminorm, partial $*$-algebra.

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