Approximate amenability of Schatten classes, Lipschitz algebras and second duals of Fourier algebras.

*(English)*Zbl 1228.46043A 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.

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.

- {\(\bullet\)}
- The Schatten class operators \(\mathcal S^p(H),\) (\(1\leq p<\infty\)), where \(H\) is an infinite-dimensional Hilbert space.
- {\(\bullet\)}
- The Lipschitz algebra \(\text{lip}_\alpha(X, d)\), where \(X\) is an infinite compact metric space.
- {\(\bullet\)}
- A proper Segal subalgebra of either \(L^1({\mathbb R}^k)\) or \(L^1({\mathbb T}^k)\), \(k\in\mathbb N\).
- {\(\bullet\)}
- The weighted convolution algebra \(\l^1({\mathbb N}_{\min},\omega)\), where \(\omega(n)\rightarrow\infty\) as \(n\rightarrow\infty\).
- {\(\bullet\)}
- The convolution algebra \(\l^1(S)\), where \(S\) is a Brandt semigroup over an amenable group with infinite index.

Reviewer: Hamid Reza Ebrahimi Vishki (Mashhad)

##### MSC:

46H20 | Structure, classification of topological algebras |

43A20 | \(L^1\)-algebras on groups, semigroups, etc. |