Approximate identities in approximate amenability (Q418709)

The theory of ``amenability of Banach algebras'' started with the seminal memoir of \textit{B. E. Johnson} [Cohomology in Banach algebras. Mem. Am. Math. Soc. 127 (1972; Zbl 0256.18014)] and has been an active field of research in the past four decades. The ``approximate versions'' of amenability for Banach algebras were introduced and studied by \textit{F. Ghahramani} and \textit{R. J. Loy} [J. Funct. Anal. 208, No. 1, 229--260 (2004; Zbl 1045.46029)].NEWLINENEWLINEIn this nice paper, among other things, the authors mainly answer some open questions on approximate amenability raised by the first author and some of his colleagues. In this respect, they show that: {\parindent=6mm \begin{itemize}\item[{\(\bullet\)}] In contrast to the situation for amenability, a boundedly approximately amenable Banach algebra need not have a bounded approximate identity. \item[{\(\bullet\)}] The two notions of bounded approximate amenability and bounded approximate contractibility are not the same. \item[{\(\bullet\)}] The direct sum of two approximately amenable Banach algebras need not be approximately amenable. \item[{\(\bullet\)}] A 1-codimensional closed ideal of a boundedly approximately amenable Banach algebra need not be approximately amenable.NEWLINENEWLINE\end{itemize}}