  In this paper, we consider the Toeplitz algebra associated to actions of Ore
semigroups on $C^{*}$-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.
