Wold Decomposition for Representations of Product Systems of C-correspondences

Higher-rank versions of Wold decomposition are shown to hold for doubly commuting isometric representations of product systems of C∗correspondences over N 0 , generalising the classical result for a doubly commuting pair of isometries due to M. S lociński. Certain decompositions are also obtained for the general, not necessarily doubly commuting, case and several corollaries and examples are provided. Possibilities of extending isometric representations to fully coisometric ones are discussed. The classical notion of Wold decomposition refers to the unique decomposition of a Hilbert space isometry into a part which is unitary and a part which is isomorphic to a unilateral shift. For a simple proof and several applications of this result we refer to the classical monograph [SzF]. In analogy with the famous dilation problem for tuples of contractions, it is natural to ask whether some version of Wold decomposition is available for a tuple of commuting isometries. Indeed, M. S lociński established in [S lo] such a decomposition for a doubly commuting pair of isometries. This result (and its generalisations) was later used in [BCL] to provide models for tuples of commuting isometries and analyse the structure of C-algebras they generate. Another example of the analysis of the structure of a pair of commuting isometries, also of relevance to our work here, may be found in [Pop]. In recent years there has been an increased interest in Wold decompositions for objects of a different type. It originated from the work of G. Popescu, who in [Pope] established a result of this kind for a row contraction. Various related ideas were extended to an impressive degree in the series of papers of P.Muhly and B. Solel, who developed the theory of tensor algebras over C-correspondences. In particular in [MS2] they proved the existence of a Wold decomposition for an isometric representation of a C-correspondence over a C-algebra A. Another, more concrete, example of such a decomposition may be found in [JuK]. In this paper we establish a higher-rank version of the main result of [MS2]. The corresponding crucial concept of a product system of a C-correspondence over N0 was introduced in [Fow]; the notion has been exploited in the recent work by B. Solel on dilations of commuting completely positive maps ([So1−2]). Here we prove that every doubly commuting isometric representation of a product system of C-correspondences over N0 decomposes uniquely into a combination of fully coisometric and induced parts (in the classical terminology they correspond respectively to a unitary and a shift part). It turns out that for isometric representations which are not doubly commuting it is still possible to characterise maximal pieces of the Permanent address of the first named author: Department of Mathematics, University of Lódź, ul. Banacha 22, 90-238 Lódź, Poland. Research supported by the EPSRC grant no. RIS 24893 2000 Mathematics Subject Classification. Primary 47A45, Secondary 46L08.