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

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.