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

Camillo Trapani¹

¹ Dipartimento di Matematica ed Applicazioni Università di Palermo I-90123 Palermo, Italy

Studia Mathematica, Tome 186 (2008) no. 1, pp. 47-75

Résumé

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.

Détail

DOI : 10.4064/sm186-1-6

Keywords: class * representations normed quasi * algebra frak cal investigated mainly its relationship structure frak cal starting point analysis construction gns like * representations quasi * algebra frak cal defined invariant positive sesquilinear forms family bounded invariant positive sesquilinear forms defines seminorms cases * seminorms provide useful information structure frak cal continuity properties its * representations