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 $\mathbf{X}=\left\{{\mathbf{X}}_p:p\in P\right\}$ 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 $C^{\ast }$-algebra ${\mathcal{N}\mathcal{T}}_{\mathbf{X}}$ 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.

24 pages; fixed typos, added paragraph at end of introduction