Noncommutative Disc Algebras for Semigroups

We study noncommutative disc algebras associated to the free product of discrete subsemigroups of R +. These algebras are associated to generalized Cuntz algebras, which are shown to be simple and purely innnite. The nonself-adjoint subalgebras determine the semigroup up to isomorphism. Moreover, we establish a dilation theorem for contractive representations of these semigroups which yields a variant of the von Neumann inequality. These methods are applied to establish a solution to the truncated moment problem in this context. The starting point of this work is an old result of Douglas 14] establishing that any properly isometric representation of the cone G + of a discrete subgroup of the real line generates the same C*-algebra, and that the ordered semigroup may be recovered from this algebra. We establish that the nonself-adjoint algebra generated by such a representation is a function algebra, and that two such algebras are isomorphic if and only if the semigroups are order isomorphic. Mlak 18] establishes a dilation theorem for contractive representations of these semigroups. This yields a variant of the von Neumann inequality. These results suggest thinking of these algebras as generalized disc algebras, and in fact the computation of their maximal ideal space and Shilov boundary make this connection even more compelling. we are lead to consider non-commutative disc algebras associated to the free product of ordered semigroups. We need to rst establish some facts about generalized Cuntz C*-algebras associated to these semigroups. The original Cuntz algebras 8] are associated to the free product of n copies of N. A larger class based on the free product of n copies of a countable dense subsemigroup of R + were investigated by Dinh 12, 13] motivated by the work on semigroups of endomorphisms of B(H), especially 4]. The fact that both of these authors use n copies 1991 Mathematics Subject Classiication. 47D25. Draft 06/2/97.