International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 79268, 34 pages

Unbounded C*-seminorms, biweights, and *-representations of partial *-algebras: A review

Camillo Trapani

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.


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.