Approximate amenability is not bounded approximate amenability (Q465383)

A Banach algebra \(A\) is approximately amenable if every continuous derivation \(D\) from \(A\) into the dual \(X^*\) of any Banach \(A\)-bimodule \(X\) is approximately inner, i.e., there exists a net \((\text{ad}_{\xi_i})\) of inner derivations (where \(\text{ad}_{\xi_i}(a)=a\cdot\xi_i-\xi_i\cdot a\)) such that \(D(a)=\lim\text{ad}_{\xi_i}(a)\) for each \(a\in A\). In the case where the approximating net can be chosen to be bounded for each continuous derivation \(D: A\to X^*\), the algebra \(A\) is said to be boundedly approximately amenable. The authors give examples of Banach algebras which are approximately amenable, but not boundedly approximately amenable. This result complements the information provided by the same authors in [J. Funct. Anal. 262, No. 9, 3929--3945 (2012; Zbl 1273.46034)], where they gave examples of boundedly approximately amenable Banach algebras that are not boundedly approximately contractible.