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

Volume 208 / 2012

J.-P. Antoine, G. Bellomonte, C. Trapani Studia Mathematica 208 (2012), 167-194 MSC: 08A55, 46K05, 46K10, 47L60. DOI: 10.4064/sm208-2-4

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.

Authors

  • J.-P. AntoineInstitut de Recherche en Mathématique et Physique
    Université Catholique de Louvain
    B-1348 Louvain-la-Neuve, Belgium
    e-mail
  • G. BellomonteDipartimento di Matematica e Informatica
    Università di Palermo
    I-90123 Palermo, Italy
    e-mail
  • C. TrapaniDipartimento di Matematica e Informatica
    Università di Palermo
    I-90123 Palermo, Italy
    e-mail

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