Weak module amenability for the second dual of a Banach algebra

In this paper we study the weak module amenability of Banach algebras which are Banach modules over another Banach algebra with compatible actions. We show that for every module derivation D : A ↦ ( A/J_A )∗ if D∗∗(A∗∗) ⊆ WAP (A/J_A ), then weak module amenability of A∗∗ implies that of A. Also we prove that under certain conditions for the module derivation D, if A∗∗ is weak module amenable then A is also weak module amenable.

THE BOCHNER–SCHOENBERG-EBERLEIN PROPERTY OF EXTENSIONS OF BANACH ALGEBRAS AND BANACH MODULES

Let A be a Banach algebra and let X be a Banach A-bimodule. We consider the Banach algebra ${A\oplus _1 X}$ , where A is a commutative Banach algebra. We investigate the Bochner–Schoenberg–Eberlein (BSE) property and the BSE module property on $A\oplus _1 X$ . We show that the module extension Banach algebra $A\oplus _1 X$ is a BSE Banach algebra if and only if A is a BSE Banach algebra and $X=\{0\}$ . Furthermore, we consider $A\oplus _1 X$ as a Banach $A\oplus _1 X$ -module and characterise the BSE module property on $A\oplus _1 X$ . We show that $A\oplus _1 X$ is a BSE Banach $A\oplus _1 X$ -module if and only if A and X are BSE Banach A-modules.

The research of some classes of almost periodic at infinity functions

The article under consideration is devoted to continuous almost periodic at infinity functions defined on the whole real axis and with their values in a complex Banach space. We consider different subspaces of functions vanishing at infinity, not necessarily tending to zero at infinity. We introduce the notions of slowly varying and almost periodic at infinity functions with respect to these subspaces. For almost periodic at infinity functions (with respect to a subspace) we give four different definitions. The first definition (approximating) is based on the approximation theorem. In the classical version, for almost periodic functions, they are represented as uniform closures of trigonometric polynomials. In our case, the Fourier coefficients are slowly varying at infinity functions. The second definition, which is an analogue of G. Bohr’s definition of an almost periodic function, is based on the concept of an ε-period. The third definition meets S. Bochner’s criterion for the almost periodicity of functions. The fourth definition is given in terms of factor space. With the help of the results of the theory of almost periodic vectors in Banach modules those four definitions are proved to be equivalent. In addition, it was proved that the introduced spaces of slowly varying and almost periodic at infinity functions with respect to different subspaces of functions vanishing at infinity coincide with the spaces of ordinary slowly varying and almost periodic at infinity functions, respectively. The feasibility of consideration of these functions is due to the fact that the solutions of some important classes of differential and difference equations are almost periodic at infinity. We consider differential equations whose right-hand side is a function vanishing at infinity and obtain necessary and sufficient conditions for their bounded solutions to be almost periodic at infinity functions. We also study an asymptotic representation of the solutions.

Effective descent morphisms for Banach modules

It is proved that a norm-decreasing homomorphism of commutative Banach algebras is an effective descent morphism for Banach modules if and only if it is a weak retract.