### Operator algebras and approximate diagonals

Pages 16-22

The author argues that two approaches dominate the study of amenable algebras, groups, modules, etc. They are the homological approach and the approach based on the invariance in respect of a particular group of transformations. In the latter case, an invariant mean serves as a convenient instrument. In particular, a mean is determined as a positive finitely additive measure which is identified using the algebra of all subsets of the group in question.In the first part of the article, the author introduces definitions of an inversely amenable module and an inversely amenable C* algebra. The criteria for the inverse amenability for C* algebras is formulated using virtual diagonals constructed with the help of means, which are invariant in respect of components of amenability in a certain space of limited functions. In the further part of the article, the author presents necessary and sufficient conditions of inverse amenability based on the existence of approximate diagonals. Unlike the standard approach applied to describe amenable Banach algebras, the above approach offers a set of invariant means that are more easily perceived by intuition.

DOI: 10.22227/1997-0935.2013.9.16-22

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