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

## TL;DR

This paper proves a uniqueness theorem for Nica-Toeplitz algebras associated with compactly aligned product systems, clarifying conditions under which these algebras are uniquely determined by their representations.

## Contribution

It establishes a new uniqueness theorem for Nica-Toeplitz algebras of compactly aligned product systems, extending previous results in the theory of $C^*$-algebras.

## Key findings

- Proves a uniqueness theorem for Nica-Toeplitz algebras
- Clarifies conditions for algebraic uniqueness
- Extends understanding of product system $C^*$-algebras

## 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 $C^*$-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.