Regularity of Segal algebras

Let A be a Banach algebra. The second dual A** can be equipped with two multiplications, each of which is a natural extension of the original multiplication in A. The algebra A is said to be Arens regular if these two multiplications coincide. We give necessary (and, for some classes of algebras, sufficient) conditions for the regularity of a Segal algebra. We also obtain necessary and sufficient conditions for the weak complete continuity of a Segal algebra.     

Remarks on Segal algebras

Let B be an abstract Segal algebra in some Banach algebra A. There was some belief that in the commutative case A should be semi-simple, if B is, but this is not so (Section I). It is well known that a (proper) abstract Segal algebra does not have bounded right approximate units. It may however have a left unit. Pseudosymmetric Segal algebras in the sense of Reiter do not have bounded left approximate units (Section II). A nonfactorization proof is given for a class of algebras which contains most of the known examples of Segal algebras on abelian groups (Section III).     

Segal algebras with functorial properties

We study the Segal algebrasE ₀ (G) (G locally compact, abelian) introduced byR. Bürger. They serve as a domain for Poisson's formula. It is shown that ifH s a closed subgroup ofG, then functions inE ₀ (H) can be extended toE ₀ (G) and similarly, lifting fromE ₀ (G/H) toE ₀ (G) is possible. Moreover, it is shown that {E ₀ (G)} is the largest family of Segal algebras that are invariant under Fourier transforms and have this extension and lifting property.     

A contribution to Segal algebras

Let A and B be Banach algebras. If B is an abstract Segal algebra in A, we have a bijective correspondence between the strictly irreducible representations of A and those of B. This gives a bijective correspondence for maximal modular left ideals. If A and B have approximate right units, we obtain a bijective correspondence for right resp. Two-sided ideals. For two-sided ideals this correspondence preserves the property of an ideal having approximate right units. This generalizes a Theorem by H. Reiter.It would be sufficient to require B to be a Banachspace.     

Weak amenability of Segal algebras

Let be a locally compact abelian group, and let . We show that the Segal algebra is always weakly amenable, but that it is amenable only if is discrete.

     

Generalization of Segal algebras for arbitrary topological algebras

In the present paper we introduce the definition of a topological Segal algebra, which generalizes most of the earlier known definitions for Segal algebras. We also generalize some results about Segal algebras and algebras of continuous functions vanishing at infinity for the case of topological Segal algebras.     

On some properties of Segal algebras and their multipliers

Let A and B be Banach algebras such that A is a right abstract Segal algebra in B. We investigate some properties of A and B and show how these are related to the algebras of right multipliers on A and B.     

Flat and weakly flat projection algebras

The notion of a projection algebra was first introduced in [4] by Ehrig et al. as an algebraic version of ultrametric spaces. Computer scientists use this notion as a convenient means of algebraic specification of process algebras. Some algebraic notions regarding these algebras have been studied in [1], [2], [5]. The flat projection algebras have been investigated by the authors in [1]. Here we completely characterize flat and weakly flat (m-separated and separated) projection algebras. Received November 19, 2001; accepted in final form December 1, 2002. RID="h1" ID="h1"     

Completely bounded isomorphisms of operator algebras

In this paper the author proves that any two elements from one of the following classes of operators are completely isomorphic to each other.

1. . The factors generated by the left regular representation of the free group on -generators.
2. . The reduced -algebras of the free group on -generators.
3. Some ``non-commutative' analytic spaces introduced by G. Popescu in 1991.

The paper ends with some applications to Popescu's version of von Neumann's inequality.

     

Functions of translation type and functorial properties of Segal algebras II

In this paper we investigate functorial properties of the Segal algebra which consists of all functionsf in Wiener's algebra onG with Fourier transform in Wiener's algebra on the dual group . Especially may serve as a very large and natural domain for Poisson's formula. Moreover, there is introduced a Segal algebraE ₀(G) containing as a subspace, but still eachfE ₀(S) satisfies Poisson's formula.