## $^*$-Representations, seminorms and structure properties of normed quasi $^*$-algebras

### Volume 186 / 2008

Studia Mathematica 186 (2008), 47-75
MSC: Primary 46L08; Secondary 46L51, 47L60.
DOI: 10.4064/sm186-1-6

#### Abstract

The class of $^*$-representations of a normed quasi $^*$-algebra $({\frak X}, {\cal A}_{0})$ is investigated, mainly for its relationship with the structure of $({\frak X}, {\cal A}_{0})$. The starting point of this analysis is the construction of GNS-like $^*$-representations of a quasi $^*$-algebra $({\frak X}, {\cal A}_{0})$ defined by invariant positive sesquilinear forms. The family of bounded invariant positive sesquilinear forms defines some seminorms (in some cases, $C^*$-seminorms) that provide useful information on the structure of $({\frak X}, {\cal A}_{0})$ and on the continuity properties of its $^*$-representations.