Approximate amenability and a variant of continuous Hochschild cohomology (Q1732090)

The coboundaries in Hochschild cohomology of topological modules over a topological algebra do not always have closed range. Defining the separated cohomology spaces by taking closures of these coboundary ranges leads to possibly smaller cohomology, still highly relevant for approximate versions of amenability, projectivity, etc. A similar approach was recently proposed by \textit{A. Pourabbas} and \textit{A. Shirinkalam} [Quaest. Math. 40, No. 1, 107--118 (2017; Zbl 1430.46053)]. Particular interest for this separated cohomology spaces lies in their natural connection with pioneering works of \textit{B. E. Johnson} on Banach algebra cohomology [Am. J. Math. 94, 685--698 (1972; Zbl 0246.46040)]. Algebras of Lipschitz functions provide some insight into the new concepts.