International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 79268, 34 pages
doi:10.1155/IJMMS/2006/79268
Unbounded C*-seminorms, biweights, and *-representations of partial *-algebras: A review
Dipartimento di Matematica ed Applicazioni, Università di Palermo, Palermo 90123, Italy
Received 1 March 2006; Revised 30 June 2006; Accepted 18 July 2006
Copyright © 2006 Camillo Trapani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The notion of (unbounded) C*-seminorms plays a relevant role in the representation theory of *-algebras and partial *-algebras. A rather complete analysis of the case of *-algebras has given rise to a series of interesting concepts like that of semifinite
C*-seminorm and spectral C*-seminorm that give information on the properties of
*-representations of the given *-algebra A and also on the structure of the *-algebra itself, in particular when
A is endowed with a locally convex topology. Some of these results extend to partial *-algebras too. The state of the art on this topic is reviewed in this paper, where the possibility of constructing unbounded C*-seminorms from certain families of positive sesquilinear forms, called biweights, on a (partial)
*-algebra A is also discussed.