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On the trivial and nontrivial cohomology with inner symmetry groups of some classes of operator algebras. (English) Zbl 1185.46049

The authors prove vanishing and non-vanishing results for certain specialized cohomology theories (e.g., dihedral and cyclic) applied to Banach algebras. For example, they prove that the reflexive and dihedral cohomology of a stable \(C^*\)-algebra vanishes while the dihedral cohomology of a commutative Banach algebra \(A\) having an involution and satisfying \(\mathrm{codim}A^2 > 1\) does not.

MSC:

46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46H05 General theory of topological algebras
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