International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 16, Article ID 79268, 34 pages
doi:10.1155/IJMMS/2006/79268

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

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.