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.     

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.     

Differential operators on Hopf algebras and some functorial properties

 Following the defintion of differential operators on noncommutative rings as given by Lunts and Rosenberg in [LR], we investigate the situation for Hopf algebras. We also prove some results for general rings, in order to understand the functorial properties of D. Received: 7 December 2001     

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.     

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.     

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.     

Completely continuous and weakly completely continuous abstract Segal algebras

Let $\mathcal{A}$ be a Banach algebra. It is obtained a necessary and sufficient condition for the complete continuity and also weak complete continuity of symmetric abstract Segal algebras with respect to $\mathcal{A}$ , under the condition of the existence of an approximate identity for $\mathcal{B}$ , bounded in $\mathcal{A}$ . In addition, a necessary condition for the weak complete continuity of $\mathcal{A}$ is given. Moreover, the applications of these results about some group algebras on locally compact groups are obtained.     

On existence of finite universal Korovkin sets in Segal algebras

We prove that there exists a finite universal Korovkin set w.r.t positive operators for the centre of a Segal algebra on a compact groupG if and only ifG is metrizable. As a consequence it follows that a Segal algebra on a compact abelian group admits a finite universal Korovkin set w.r.t. positive operators iff the group is metrizable.     

The Generalized Segal–Bargmann Transform and Special Functions

Analysis of function spaces and special functions are closely related to the representation theory of Lie groups. We explain here the connection between the Laguerre functions, the Laguerre polynomials, and the Meixner–Pollacyck polynomials on the one side, and highest weight representations of Hermitian Lie groups on the other side. The representation theory is used to derive differential equations and recursion relations satisfied by those special functions.     

Decomposition rank and absorbing extensions of type I algebras

We prove the following: Let A and B be separable C^*-algebras. Suppose that B is a type I C^*-algebra such that

(i)

B has only infinite dimensional irreducible *-representations, and

(ii)

B has finite decomposition rank.

If

0→B→C→A→0     

Tensor Banach algebras of projective type. I

We construct a tensor Banach functorT that establishes a correspondence between every projectively admissible Banach spaceE over a complete normed fieldK with a tensor Banach algebra of projective typeT .All-Union Scientific-Research Center for the Study of Surfaces and the Vacuum. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 17–29, April, 1992.     

Gromov's translation algebras, growth and amenability of operator algebras

Translation algebras of finitely generated *-algebras of bounded linear operators on a separable Hilbert space are introduced. Two equivalent forms of amenability for finitely generated *-algebras in terms of the existence of Følner sequences are introduced. These are related to the existence of traces on the associated translation algebra and, in the context of C^*-algebras, are related to weak-filterability and to the existence of hypertraces.