*-representations, seminorms and structure properties of normed quasi *-algebras

The class of *-representations of a normed quasi *-algebra (x, u(0)) is investigated, mainly for its relationship with the structure of (x, u(0)). The starting point of this analysis is the construction of GNS-like *-representations of a quasi *-algebra (x, u(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 (x, u(0)) and on the continuity properties of its *-representations.