Fully representable and $^*$-semisimple topological partial $^*$-algebras

J.-P. Antoine¹ ; G. Bellomonte² ; C. Trapani²

¹Institut de Recherche en Mathématique et Physique Université Catholique de Louvain B-1348 Louvain-la-Neuve, Belgium
²Dipartimento di Matematica e Informatica Università di Palermo I-90123 Palermo, Italy

Studia Mathematica, Tome 208 (2012) no. 2, pp. 167-194

Résumé

We continue our study of topological partial $^*$-algebras, focusing our attention on $^*$-semisimple partial $^*$-algebras, that is, those that possess a {multiplication core} and sufficiently many $^*$-representations. We discuss the respective roles of invariant positive sesquilinear (ips) forms and representable continuous linear functionals, and focus on the case where the two notions are completely interchangeable (fully representable partial $^*$-algebras) with the aim of characterizing a $^*$-semisimple partial $^*$-algebra. Finally we describe various notions of bounded elements in such a partial $^*$-algebra, in particular, those defined in terms of a positive cone (order bounded elements). The outcome is that, for an appropriate order relation, one recovers the $\mathcal M$-bounded elements introduced in previous works.

Détail

DOI : 10.4064/sm208-2-4

Keywords: continue study topological partial * algebras focusing attention * semisimple partial * algebras those possess multiplication core sufficiently many * representations discuss respective roles invariant positive sesquilinear ips forms representable continuous linear functionals focus where notions completely interchangeable fully representable partial * algebras characterizing * semisimple partial * algebra finally describe various notions bounded elements partial * algebra particular those defined terms positive cone order bounded elements outcome appropriate order relation recovers mathcal m bounded elements introduced previous works