Cyclic cohomology of Banach algebras of quivers and their inverse semigroups

In this paper, we consider the path semigroup ℓ1ℓ1-algebra for a quiver and the inverse semigroup ℓ1ℓ1-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>1n>1. We then determine the simplicial and cyclic cohomology of the inverse semigroup algebra. The work uses the Connes–Tzygan long exact sequence.