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We introduce new notions of approximate amenability for a Banach algebra A. A Banach algebra A is n-approximately weakly amenable, for n ∈ N, if every continuous derivation from A into the n-th dual space A<sup>(n)</sup> is approximately inner. First we examine the relation between m-approximately weak amenability and n-approximately weak amenability for distinct m,n ∈ N. Then we investigate (2n+1)-approximately weak amenability of module extension Banach algebras. Finally, we give an example of a Banach algebra that is 1-approximately weakly amenable but not 3-approximately weakly amenable.

Content uploaded by Taher Yazdanpanah

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All content in this area was uploaded by Taher Yazdanpanah

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... In [2,3,7,8,10] the authors studied the approximate amenability of certain Banach algebras having the property that every continuous derivation from them into a related dual Banach bimodule is approximately inner. Approximately inner and non-inner derivations arise naturally in the theory of operator algebras and abstract harmonic analysis. ...

The aim of this paper is to display sufficient conditions on Banach algebras
without bounded approximate identity such that every continuous derivation is approximately inner. We apply these results to prove weak amenability of Banach algebras and
approximate n-weak amenability of extension Banach algebras.

... A Banach algebra A is called ideally amenable if H 1 (A, I * ) = {0} for every closed two-sided ideal I in A. Ideal amenability of group algebras L 1 (G), L ∞ (G) and M (G), where G is a locally compact group, are studied in [12]. Many new results for ideal amenability of Banach algebras are given in [11,13,15,16,24,21], and [22]. Definition 1.1. ...

A Banach algebra A is said to be approximate ideally amenable if every
continuous derivation from A into I¤ is approximately inner for all two{sided ideal
I of A. In this paper, we study the approximate ideal amenability of several Banach
algebras. Moreover, we show that approximate ideal amenability is di®erent from
approximate amenability and approximate weak amenability.

In this paper we investigate n-approximately weak amenability of a Banach algebra A, and show that for n ≥ 2, n-approximately weak amenability passes from A" to A, where A" is the second dual of A equipped with the first Arens product. Also we prove that under certain condition n-approximately weak amenability inherits by closed subalgebras of A.

We start by discussing general necessary and sufficient conditions for a module extension Banach algebra to be n-weakly amenable, for n = 0, 1, 2, .... Then we investigate various special cases. All these case studies finally provide us with a way to construct an example of a weakly amenable Banach algebra which is not 3-weakly amenable. This answers an open question raised by H. G. Dales, F. Ghahramani and N. Grønbæk.

We consider when certain Banach sequence algebras A on the set N are approximately amenable. Some general results are obtained, and we resolve the special cases where A = p for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras p (ω).

In the present paper, it is shown that a left cancellative
semigroup S (not necessarily with identity) is left amenable
whenever the Banach algebra ℓ1(S) is approximately amenable. It is also proved that if S is a Brandt semigroup over a group
G with an index set I, then ℓ1(S) is approximately
amenable if and only if G is amenable. Moreover ℓ1(S) is amenable if and only if G is amenable and I is finite. For a
left cancellative foundation semigroup S with an identity such
that for every Ma(S)-measurable subset B of S
and s ∈ S the set sB is Ma(S)-measurable,
it is proved that if the measure algebra Ma(S) is approximately
amenable, then S is left amenable. Concrete examples are given
to show that the converse is negative.

We continue the investigation of notions of approximate amenability that were introduced in work of the second and third authors together with R.J. Loy. It is shown that every boundedly approximately contractible Banach algebra has a bounded approximate identity, and that the Fourier algebra of the free group on two generators is not operator approximately amenable. Further examples are obtained of ℓ1-semigroup algebras which are approximately amenable but not amenable; using these, we show that bounded approximate contractibility need not imply sequential approximate amenability. Results are also given for Segal algebras on locally compact groups, and algebras of p-pseudo-functions on discrete groups.