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Bounded elements and spectrum in Banach quasi *-algebras

Camillo Trapani (2006)

Studia Mathematica

A normal Banach quasi *-algebra (,) has a distinguished Banach *-algebra b consisting of bounded elements of . The latter *-algebra is shown to coincide with the set of elements of having finite spectral radius. If the family () of bounded invariant positive sesquilinear forms on contains sufficiently many elements then the Banach *-algebra of bounded elements can be characterized via a C*-seminorm defined by the elements of ().

Bounded elements in certain topological partial *-algebras

Jean-Pierre Antoine, Camillo Trapani, Francesco Tschinke (2011)

Studia Mathematica

We continue our study of topological partial *-algebras, focusing on the interplay between various partial multiplications. The special case of partial *-algebras of operators is examined first, in particular the link between strong and weak multiplications, on one hand, and invariant positive sesquilinear (ips) forms, on the other. Then the analysis is extended to abstract topological partial *-algebras, emphasizing the crucial role played by appropriate bounded elements, called ℳ-bounded. Finally,...

C*-seminorms on partial *-algebras: an overview

Camillo Trapani (2005)

Banach Center Publications

The main facts about unbounded C*-seminorms on partial *-algebras are reviewed and the link with the representation theory is discussed. In particular, starting from the more familiar case of *-algebras, we examine C*-seminorms that are defined from suitable families of positive linear or sesquilinear forms, mimicking the construction of the Gelfand seminorm for Banach *-algebras. The admissibility of these forms in terms of the (unbounded) C*-seminorms they generate is characterized.

Fully representable and *-semisimple topological partial *-algebras

J.-P. Antoine, G. Bellomonte, C. Trapani (2012)

Studia Mathematica

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...

Locally convex quasi C*-algebras and noncommutative integration

Camillo Trapani, Salvatore Triolo (2015)

Studia Mathematica

We continue the analysis undertaken in a series of previous papers on structures arising as completions of C*-algebras under topologies coarser that their norm topology and we focus our attention on the so-called locally convex quasi C*-algebras. We show, in particular, that any strongly *-semisimple locally convex quasi C*-algebra (𝔛,𝔄₀) can be represented in a class of noncommutative local L²-spaces.

Old and new results on Allan's GB*-algebras

Maria Fragoulopoulou, Atsushi Inoue, Klaus-Detlef Kürsten (2010)

Banach Center Publications

This is an expository paper on the importance and applications of GB*-algebras in the theory of unbounded operators, which is closely related to quantum field theory and quantum mechanics. After recalling the definition and the main examples of GB*-algebras we exhibit their most important properties. Then, through concrete examples we are led to a question concerning the structure of the completion of a given C*-algebra 𝓐₀[||·||₀], under a locally convex *-algebra topology τ, making the multiplication...

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