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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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