A uniqueness theorem for the Nica--Toeplitz algebra of a compactly aligned product system

Abstract

Fowler introduced the notion of a product system: a collection of Hilbert bimodules $\mathbf {X}=\{\mathbf {X}_p:p\in P\}$ indexed by a semigroup $P$, endowed with a multiplication implementing isomorphisms $\mathbf {X}_p\otimes _A \mathbf {X}_q\cong \mathbf {X}_{pq}$. When $P$ is quasi-lattice ordered, Fowler showed how to associate a $C^*$-algebra $\mathcal {NT}_\mathbf {X}$ to $\mathbf {X}$, generated by a universal representation satisfying some covariance condition. In this article we prove a uniqueness theorem for these so called Nica–Toeplitz algebras.