The cyclic and simplicial cohomology of the Cuntz semigroup algebra

The main objective of this paper is to determine the simplicial and cyclic cohomology groups of the Cuntz semigroup algebra ℓ1(Sm). We also determine the simplicial and cyclic cohomology of the tensor algebra of a Banach space, a class which includes the algebra on the free semigroup on m-generators ℓ1(FSm). In order to do so, we first establish some general results which can be used when studying simplicial and cyclic cohomology of Banach algebras in general. We then turn our attention to ℓ1(Sm), showing that the cyclic cohomology groups of degree n vanish when n is odd and are one-dimensional when n is even (n⩾2). Using the Connes–Tzygan exact sequence, these results are used to show that the simplicial cohomology groups of degree n vanish for n⩾1. A similar strategy is used for the tensor algebra of a Banach space.