Fully representable and $^*$-semisimple topological partial $^*$-algebras
Volume 208 / 2012
Abstract
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.
