Approximate weak amenability of Banach algebras

Abstract

In this paper we deal with four generalized notions of amenability which are called approximate, approximate weak, approximate cyclic and approximate n-weak amenability. The first two were introduced and studied by Ghahramani and Loy in [9]. We introduce the third and fourth ones and we show by means of some examples, their distinction with their classic analogs.Our main result is that under some mild conditions on a given Banach algebra A, if its second dual A * * is (2n − 1)-weakly [respectively approximately/ approximately weakly/ approximately n-weakly] amenable, then so is A. Also if A is approximately (n + 2)-weakly amenable, then it is approximately n-weakly amenable. Moreover we show the relationship between approximate trace extension property and approximate weak [respectively cyclic] amenability. This answers question 9.1 of [9] for approximate weak and cyclic amenability.

A Generalization of Cyclic Amenability of Banach Algebras

“…We need the following result which has been proven in [7,Proposition 4.6]. Recall that a Banach algebra A is said to be a dual Banach algebra if there is a closed submodule A * of A * such that A = (A * ) * .…”

Arens Regularity and Amenability of Lau Product of Banach Algebras Defined by a Banach Algebra Morphism

“…The purpose of the present paper is to determine the Gel'fand space of A × T B which turns out to be nontrivial even though A × T B need not be commutative and to discuss the Arens regularity as well as the amenability of A × T B. These topics are central to the general theory of Banach algebras [3]; and are of current relevance [3,5,6]. Arens [1,2] showed that the given product on a Banach algebra A induces two canonical products on the second dual A of A; and A is Arens regular if these two products coincide.…”

More on cyclic amenability of the Lau product of Banach algebras defined by a Banach algebra morphism

“…A Banach algebra A is called approximately cyclic amenable, if every continuous cyclic derivation D : A → A * is approximately inner. The concepts of approximate cyclic amenability was introduced and studied in [6]; see also [14].…”

“…In the same paper and the subsequent one [8], the authors showed the distinction between each of these concepts and the corresponding classical notions and investigated properties of algebras in each of these new classes. Motivated by this notions, Esslamzadeh and Shojaee [6] defined the concept of approximate cyclic amenability for Banach algebras and investigated the hereditary properties for this new notion. Shojaee and Bodaghi in [14,Theorem 2.3] showed that for Banach algebras A and B, if direct product A⊕ B with ℓ 1 -norm is approximately cyclic amenable, then so are A and B.…”