Finitely Correlated Representations of Product Systems of $C^*$-Correspondences over $\mathbb{N}^k$

TL;DR

This paper investigates minimal dilations of finite-dimensional, fully coisometric isometric representations of product systems over b^k, establishing uniqueness of a key invariant subspace and exploring algebraic properties related to graph algebras.

Contribution

It introduces the concept of a unique minimal cyclic coinvariant subspace for these representations and demonstrates its invariance and algebraic significance.

Findings

Existence of a unique minimal cyclic coinvariant subspace.

The compression to this subspace is a complete unitary invariant.

The generated algebra contains the projection onto this subspace, including free semigroup and higher-rank graph algebras.

Abstract

We study isometric representations of product systems of correspondences over the semigroup $\mathbb{N}^k$ which are minimal dilations of finite dimensional, fully coisometric representations. We show the existence of a unique minimal cyclic coinvariant subspace for all such representations. The compression of the representation to this subspace is shown to be complete unitary invariant. For a certain class of graph algebras the nonself-adjoint \textsc{wot}-closed algebra generated by these representations is shown to contain the projection onto the minimal cyclic coinvariant subspace. This class includes free semigroup algebras. This result extends to a class of higher-rank graph algebras which includes higher-rank graphs with a single vertex.