Toeplitz algebras associated to endomorphisms of Ore semigroups

In this paper, we consider the Toeplitz algebra associated to actions of Ore semigroups on C⁎C⁎-algebras. In particular, we consider injective and surjective actions of such semigroups. We use the theory of groupoid dynamical systems to represent the Toeplitz algebra as a groupoid crossed product. We also discuss the K-theory of the Toeplitz algebra in some examples. For instance, we show that for the semigroup of positive matrices, the K-theory of the associated Toeplitz algebra vanishes.