Discrete product systems and their C∗-algebras

Abstract

Continuous product systems, or product systems over R +, were recently introduced by Arveson to develope an index theory for continuous semigroups of ∗-endomorphisms of B(H) . In this paper, we continue his work by studying discrete product systems, or product systems over G + where G is a countable dense subgroup of R. We give examples of discrete product systems and compute their automorphism groups. To each discrete product system E over G +, we associate a separable C ∗-algebra O E(G) whose non-degenerate ∗-representations correspond bijectively to representations of E. This family of C ∗-algebras may be regarded as analogues of Cuntz algebras or CAR algebras. In some degenerate cases, these C ∗-algebras are the generalized Toeplitz algebras. We show that O E(G) is simple and study its gauge automorphisms, quasi-free automorphisms, and Fock state.