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

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

2010-06-23

arxiv.org/abs/1006.4571v2

Mathematics

Operator Algebras

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

Scientific paper

10.1016/j.jfa.2010.10.004

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.

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