## Pseudo-amenable and pseudo-contractible Banach algebras.(English)Zbl 1118.46046

The authors introduce two new properties related to amenability for a Banach algebra $$A$$. They call $$A$$ pseudo-amenable if there is a (not necessarily bounded) net $$(u_\alpha)$$ in $$A\widehat{\otimes}A$$ such that $$au_\alpha-u_\alpha a\to 0$$ and $$\pi(u_\alpha)a\to a$$ for each $$a\in A$$ (where $$\pi$$ is specified by $$\pi(a\otimes b)=ab$$). If, in fact, $$au_\alpha=u_\alpha a$$, then $$A$$ is said to be pseudo-contractible. These notions are compared with some previous notions of amenability such as amenability, contractibility, approximate amenability, and approximate contractibility. On the other hand, the authors exhibit several classes of Banach algebras (including group algebras, measure algebras, and Segal algebras for certain locally compact groups) satisfying the new amenability conditions. The paper concludes with a list of open questions.

### MSC:

 46H20 Structure, classification of topological algebras 43A20 $$L^1$$-algebras on groups, semigroups, etc. 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.) 47B47 Commutators, derivations, elementary operators, etc.
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