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Approximate and pseudo-amenability of various classes of Banach algebras. (English) Zbl 1179.46040
A Banach algebra $$A$$ is called approximately amenable if, for every Banach $$A$$-bimodule $$E$$ and every bounded derivation $$D: A \to E^*$$, there is a – not necessarily bounded – net $$(\varphi_\alpha )_\alpha$$ in $$E^*$$ such that $$Da= \lim_\alpha (a\cdot \varphi_\alpha- \varphi_\alpha \cdot a)$$ for each $$a\in A$$. (If we require the net $$(\varphi_\alpha)_\alpha$$ to be bounded, we obtain the usual Banach algebraic amenability in the sense of B. E. Johnson [“Cohomology in Banach algebras”, Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014)].)
Let $$\widehat{\otimes}$$ denote the projective tensor product. We call a Banach algebra $$A$$ pseudo-amenable if there is a net $$(d_\alpha)_\alpha$$ in $$A \widehat{\otimes} A$$ such that
$a\cdot d_\alpha- d_\alpha\cdot a\to 0 \qquad (a\in A)$
and
$a m(d_\alpha)\to a \qquad (a \in A),$
where $$m: A\widehat{\otimes} A\to A$$ is the map induced by multiplication in $$A$$. (If $$(d_\alpha )_\alpha$$ is bounded, we obtain again B. E. Johnson’s concept of amenability [“Approximate diagonals and cohomology of certain annihilator Banach algebras”, Am. J. Math. 94, 685–698 (1972; Zbl 0246.46040)].)
These and other generalized notions of amenability for Banach algebras were introduced by the second and third named author of the paper under review and R. J. Loy in a series of papers that started with [“Generalized notions of amenability”, J. Funct. Anal. 208, No. 1, 229–260 (2004; Zbl 1045.46029)].
The present paper is yet another installment in that series. The authors study various generalized notions of amenability, relate them to each other, and investigate examples, such as Fourier algebras, $$\ell^1$$-algebras of semigroups, Segal algebras on locally compact groups, and the algebras $$\text{PF}_p(\Gamma)$$ of $$p$$-pseudofunctions for discrete groups $$\Gamma$$. Among the noteworthy results of the paper are that the Fourier algebra of the free group in two generators is not (operator) approximately amenable and that for $$\text{PF}_p(\Gamma)$$ amenability, pseudo-amenability, and approximate amenability are all equivalent to the amenability of $$\Gamma$$.

##### MSC:
 46H05 General theory of topological algebras 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 43A20 $$L^1$$-algebras on groups, semigroups, etc. 43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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