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Cyclic cohomology of Banach algebras of quivers and their inverse semigroups. (English) Zbl 1327.46074

Summary: In this paper, we consider the path semigroup \(\ell^1\)-algebra for a quiver and the inverse semigroup \(\ell^1\)-algebra of a quiver, the latter of which can be used in the construction of Cuntz-Krieger algebras. The main objectives of the paper are to determine the simplicial and cyclic cohomology groups of these algebras. First, we determine the simplicial and cyclic cohomology of the path algebra of the quiver, showing the simplicial cohomology groups of dimension \(n\) vanish for \(n>1\). We then determine the simplicial and cyclic cohomology of the inverse semigroup algebra. The work uses the Connes-Tzygan long exact sequence.

MSC:

46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
46J05 General theory of commutative topological algebras
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