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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...

On a generalization of W*-modules

David P. Blecher, Jon E. Kraus (2010)

Banach Center Publications

a recent paper of the first author and Kashyap, a new class of Banach modules over dual operator algebras is introduced. These generalize the W*-modules (that is, Hilbert C*-modules over a von Neumann algebra which satisfy an analogue of the Riesz representation theorem for Hilbert spaces), which in turn generalize Hilbert spaces. In the present paper, we describe these modules, giving some motivation, and we prove several new results about them.

On a result of Peetre about interpolation of operator spaces

Fernando Cobos, Teresa Signes (2000)

Publicacions Matemàtiques

We establish interpolation formulæ for operator spaces that are components of a given quasi-normed operator ideal. Sometimes we assume that one of the couples involved is quasi-linearizable, some other times we assume injectivity or surjectivity in the ideal. We also show the necessity of these suppositions.

On an inclusion between operator ideals

Manuel A. Fugarolas (2011)

Czechoslovak Mathematical Journal

Let 1 ≤ q < p < ∞ and 1 / r : = 1 / p max ( q / 2 , 1 ) . We prove that ℒ r , p ( c ) , the ideal of operators of Geľfand type l r , p , is contained in the ideal Π p , q of ( p , q ) -absolutely summing operators. For q > 2 this generalizes a result of G. Bennett given for operators on a Hilbert space.

On bounded approximation properties of Banach spaces

Eve Oja (2010)

Banach Center Publications

This survey features some recent developments concerning the bounded approximation property in Banach spaces. As a central theme, we discuss the weak bounded approximation property and the approximation property which is bounded for a Banach operator ideal. We also include an overview around the related long-standing open problem: Is the approximation property of a dual Banach space always metric?

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