NONCOMMUTATIVE DISC ALGEBRAS FOR SEMIGROUPS

We study noncommutative disc algebras associated to the free product o f discrete subsemigroups of R+. These algebras are associated to gener alized Cuntz algebras, which are shown to be simple and purely infinit e. The nonself-adjoint subalgebras determine the semigroup up to isomo rphism. Moreover, we establish a dilation theorem for contractive repr esentations of these semigroups which yields a variant of the von Neum ann inequality. These methods are applied to establish a solution to t he truncated moment problem in this context.