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Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras. (English) Zbl 1228.46043
A Banach algebra \(A\) is said to be approximately amenable if every bounded derivation \(D\) from \(A\) into any dual Banach \(A\)-module \(X^*\) is approximately inner (i.e., \(D\) is the strong-operator topology limit of a net of inner derivations) for all Banach \(A\)-bimodules \(X.\) If the involved net of inner derivations can always be taken to be bounded, then \(A\) is said to be boundedly approximately amenable.
In the paper under review, the authors give a nice applicable criterion for non-approximate amenability of a Banach algebra. Indeed, they show that, if a Banach algebra \(A\) contains a SUM (separated, unbounded, multiplier-bounded) configuration, then \(A\) is not approximately amenable. This criterion is applied to show that neither of the following algebras are approximately amenable.
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The Schatten class operators \(\mathcal S^p(H),\) (\(1\leq p<\infty\)), where \(H\) is an infinite-dimensional Hilbert space.
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The Lipschitz algebra \(\text{lip}_\alpha(X, d)\), where \(X\) is an infinite compact metric space.
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A proper Segal subalgebra of either \(L^1({\mathbb R}^k)\) or \(L^1({\mathbb T}^k)\), \(k\in\mathbb N\).
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The weighted convolution algebra \(\l^1({\mathbb N}_{\min},\omega)\), where \(\omega(n)\rightarrow\infty\) as \(n\rightarrow\infty\).
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The convolution algebra \(\l^1(S)\), where \(S\) is a Brandt semigroup over an amenable group with infinite index.
Using different methods, the authors examine bounded approximate amenability of the bidual of a commutative Banach algebra \(A\) and they show that, if \(A^{**}\) (equipped with each Arens product) is boundedly approximately amenable, then \(A\) has a bounded approximate identity. Applying this result for the Fourier algebra \(A(G)\) on a locally compact group \(G,\) it is proved that \(A(G)^{**}\) is boundedly approximately amenable if and only if \(G\) is finite. Some results related to approximate amenability of the projective tensor product of Banach algebras are also given.

MSC:
46H20 Structure, classification of topological algebras
43A20 \(L^1\)-algebras on groups, semigroups, etc.
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