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

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.

34 pages; Introduction extended; to appear in the Journal of Functional Analysis

References in corpus (11)

• Free Semigroupoid Algebras
• Ideal Structure in Free Semigroupoid Algebras from Directed Graphs
• Representations of Higher Rank Graph Algebras
• Classifying Higher Rank Analytic Toeplitz Algebras
• Dilation Theory for Rank 2 Graph Algebras
• Poisson transform for higher-rank graph algebras and its applications
• Isometric dilations of non-commuting finite rank $n$-tuples
• E_0-dilation of strongly commuting CP_0-semigroups
• Partially Isometric Dilations of Noncommuting $N$-tuples of Operators
• Dilating covariant representations of the non-commutative disc algebras
• On isometric dilations of product systems of C*-correspondences and applications to families of contractions associated to higher-rank graphs