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Erratum/Addendum to the paper "Some seminorms on quasi *-algebras" (Studia Math. 158 (2003), 99-115)

100%

Trapani C.

Studia Mathematica

2004

tom 160

nr 1

101

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

100%

Trapani C.

tom 186

nr 1

47-75

The class of *-representations of a normed quasi *-algebra (𝔛,𝓐₀) is investigated, mainly for its relationship with the structure of (𝔛,𝓐₀). The starting point of this analysis is the construction of GNS-like *-representations of a quasi *-algebra (𝔛,𝓐₀) 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 (𝔛,𝓐₀) and on the continuity properties of its *-representations.

Bounded elements and spectrum in Banach quasi *-algebras

100%

Trapani C.

Studia Mathematica

2006

tom 172

nr 3

249-273

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

Some seminorms on quasi *-algebras

100%

Trapani C.

tom 158

nr 2

99-115

Different types of seminorms on a quasi *-algebra (𝔄,𝔄₀) are constructed from a suitable family ℱ of sesquilinear forms on 𝔄. Two particular classes, extended C*-seminorms and CQ*-seminorms, are studied in some detail. A necessary and sufficient condition for the admissibility of a sesquilinear form in terms of extended C*-seminorms on (𝔄,𝔄₀) is given.

Locally convex quasi C*-algebras and noncommutative integration

63%

Trapani C. , Triolo S.

Studia Mathematica

2015

tom 228

nr 1

33-45

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.

Erratum/addendum to the paper: "Quasi *-algebras and generalized inductive limits of C*-algebras" (Studia Math. 202 (2011), 165-190)

63%

Bellomonte G. , Trapani C.

tom 217

nr 1

95-96

Bounded elements in certain topological partial *-algebras

51%

Antoine J.-P. , Trapani C. , Tschinke F.

nr 3

223-251

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, some remarks are made concerning representations in terms of so-called partial GC*-algebras of operators.

Quasi *-algebras of measurable operators

51%

Bagarello F. , Trapani C. , Triolo S.

Studia Mathematica

2006

tom 172

nr 3

289-305

Non-commutative $L^{p}$-spaces are shown to constitute examples of a class of Banach quasi *-algebras called CQ*-algebras. For p ≥ 2 they are also proved to possess a sufficient family of bounded positive sesquilinear forms with certain invariance properties. CQ*-algebras of measurable operators over a finite von Neumann algebra are also constructed and it is proven that any abstract CQ*-algebra (𝔛,𝔄₀) with a sufficient family of bounded positive tracial sesquilinear forms can be represented as a CQ*-algebra of this type.