Finitely correlated representations of product systems of \(C^{*}\)-correspondences over \(\mathbb N^k\) (Q610776)

Let \(A\) be a \(C^*\)-algebra. A \(C^*\)-correspondence over \(A\) is a Hilbert \(C^*\)-module over \(A\) equipped with a left action of \(A\). A product system of \(C^*\)-correspondences over \(\mathbb N ^k\) can be thought of as a collection of \(k\) such \(C^*\)-correspondences over \(A\) together with isomorphisms between them satisfying natural intertwining relations (see \textit{B.\,Solel}, [J.~Funct.\ Anal.\ 235, No.\,2, 593--618 (2006; Zbl 1102.46036)]). This concept, originally introduced by \textit{N.\,J.\thinspace Fowler} in [Pac.\ J.\ Math.\ 204, No.\,2, 335--375 (2002; Zbl 1059.46034)], provides a convenient language for the analysis of certain classes of \(C^*\)-algebras generalising Cuntz and Cuntz-Pimsner algebras and for studying certain problems in the multi-dimensional dilation theory. In particular, one can investigate isometric dilations of (completely) contractive representations of product systems of \(C^*\)-correpondences. Natural examples can be constructed out of higher-rank graphs. The paper under review is concerned with the study of so-called finitely correlated representations of a \(C^*\)-correspondence, or more generally of product systems of \(C^*\)-correspondences. These are the representations which arise as minimal isometric dilations of finite-dimensional fully coisometric completely contractive representations. They generalise a class of representations of Cuntz algebras studied by \textit{K.\,R.\thinspace Davidson, D.\,W.\thinspace Kribs} and \textit{M.\,E.\thinspace Shpigel} in [Can.\ J.\ Math.\ 53, No.\,3, 506--545 (2001; Zbl 0973.47057)]. The author shows in particular that finitely correlated representations always admit a unique minimal cyclic coinvariant subspace and hence are equipped with a natural complete unitary invariant given by the corresponding compression. The proof for the product systems is based on a clever observation allowing the author to extend the result from the case of a single \(C^*\)-correspondence. Some further applications of the above results are given for a class of non-selfadjoint operator algebras associated to graphs satisfying the strong double-cycle property (i.e., such that each vertex can be connected to a vertex lying on two distinct minimal cycles).