On Isometric Dilations of Product Systems of C-correspondences and Applications to Families of Contractions Associated to Higher-rank Graphs

Let E be a product system of C-correspondences over Nr 0 . Some sufficient conditions for the existence of a not necessarily regular isometric dilation of a completely contractive representation of E are established and difference between regular and -regular dilations discussed. It is in particular shown that a minimal isometric dilation is -regular if and only if it is doubly commuting. The case of product systems associated with higher-rank graphs is analysed in detail. Classical multi-dimensional dilation theory ([SzF]) for Hilbert space operators is concerned with dilating tuples of contractions to tuples of isometries or unitaries, preserving as many properties of the original family as possible. In particular if the tuple with which we start consists of mutually commuting operators, it is desirable to obtain a commuting dilation. Celebrated examples of S. Parrott, N.Varopoulos and others show that a joint dilation of three or more commuting contractions to commuting isometries need not exist. In general it is difficult to decide whether a given commuting tuple has a commuting isometric dilation. On the other hand the existence of so-called regular or -regular dilations (i.e. dilations satisfying additional conditions with respect to products of the original contractions and their adjoints, see for example [Tim]) can be detected via simple conditions corresponding to positive-definiteness of certain operator-valued functions associated with the initial tuple. In a recent paper [SZ2] together with J. Zacharias we considered dilations of Λcontractions, that is tuples of operators satisfying commutation relations encoded by a (higher rank) graph Λ. It has now become clear that using the constructions provided by I.Raeburn and A. Sims in [RaS] some of the results of [SZ2] can be viewed as statements on completely contractive representations of the canonical product system of C-correspondences associated to Λ. Product systems of Ccorrespondences were first defined in [Fow] as generalisations of product systems of Hilbert spaces and quickly proved to provide a natural framework for extensions of the classical multi-dimensional dilation theory to more complicated objects (see [So2] and references therein). The questions about the existence of a joint dilation of a family of contractions satisfying certain commutativity relations to an analogous family of isometries translates here into a question on the existence of an isometric dilation of a (completely) contractive representation of a given product system. Permanent address of the author: Department of Mathematics, University of Lódź, ul. Banacha 22, 90-238 Lódź, Poland. 2000 Mathematics Subject Classification. Primary 47A20, Secondary 05C20, 46L08, 47A13.