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.

Derivations on the module extension Banach algebras

UDC 517.986 We correct some results presented in [M. Eshaghi Gordji, F. Habibian, A. Rejali, <em> Ideal amenability of module extension Banach algebras</em>, Int. J. Contemp. Math. Sci., <strong>2</strong>, No. 5, 213–219 (2007)] and, using the obtained consequences, we find necessary and sufficient conditions for the module extension to be -weakly amenable, where is a closed ideal of the Banach algebra and is a closed -submodule of the Banach -bimodule We apply this result to the module extension where are two Banach -bimodules.

Characterization of homological properties of θ-Lau product of Banach algebras

Let A and B be two Banach algebras and ?? ?(B): In this paper, we investigate biprojectivity and biflatness of ?-Lau product of Banach algebras A x? B. Indeed, we show that A x? B is biprojective if and only if A is contractible and B is biprojective. This generalizes some known results. Moreover, we characterize biflatness of ?-Lau product of Banach algebras under some conditions. As an application, we give an example of biflat Banach algebras A and X such that the generalized module extension Banach algebra X o A is not biflat. Finally, we characterize pseudo-contractibility of ?-Lau product of Banach algebras and give an affirmative answer to an open question.

Biprojectivity and biflatness of generalized module extension Banach algebras

We investigate biprojectivity and biflatness of generalized module extension Banach algebra A Z B, in which A and B are Banach algebras and B is an algebraic Banach A-bimodule, with multiplication: (a, b)?(a',b') = (aa', ab' + ba' + bb')

Generalized Malcev-Neumann Series Modules with the Beachy-Blair Condition

We introduce a new class of extension rings called the generalized Malcev-Neumann series ring R((S;σ;τ)) with coefficients in a ring R and exponents in a strictly ordered monoid S which extends the usual construction of Malcev-Neumann series rings. Ouyang et al. in 2014 introduced the modules with the Beachy-Blair condition as follows: A right R-module satisfies the right Beachy-Blair condition if each of its faithful submodules is cofaithful. In this paper, we study the relationship between the right Beachy-Blair condition of a right R-module MR and its Malcev-Neumann series module extension MSR((S;σ;τ)).

Canonical transformations for fermions in superanalysis

Canonical transformations (Bogoliubov transformations) for fermions with an infinite number of degrees of freedom are studied within a calculus of superanalysis. A continuous representation of the orthogonal group is constructed on a Grassmann module extension of the Fock space. The pull-back of these operators to the Fock space yields a unitary ray representation of the group that implements the Bogoliubov transformations.