Journal of the
Korean Mathematical Society JKMS

Given a $C^*$-dynamical system $(\Cal A, G, \alpha)$ with a locally compact group $G$, two kinds of $C^*$-algebras are made from it, called the full $C^*$-crossed product and the reduced $C^*$-crossed product. In this paper, we extend the theory of the classical $C^*$-crossed product to the $C^*$-dynamical system $(\Cal A, M, \alpha)$ with a left-cancellative semigroup $M$ with unit. We construct a new $C^*$-algebra $ \Cal A \rtimes_{\alpha r} M$, the reduced crossed product of $A$ by the semigroup $M$ under the action $\alpha$ and investigate some properties of $\Cal A \rtimes_{\alpha r}M$.

Keywords: reduced crossed products by semigroups, regular isometric representation

MSC numbers: 46L55, 47O03