Poisson Transform for Higher-rank Graph Algebras and Its Applications

Higher-rank graph generalisations of the Popescu-Poisson transform are constructed, allowing to obtain joint dilations of the families of operators satisfying certain commutation relations encoded by the graph structure. Several applications to the analysis of the structure of concrete graph operator algebras are presented. The starting point of classical dilation theory was the construction of a dilation of a contractive operator on a Hilbert space to a unitary by B. Sz.-Nagy in 1953 ([SzF]). Under a certain minimality condition this dilation is unique up to unitary equivalence. There exists also a minimal isometric dilation whose adjoint leaves the original Hilbert space invariant. In particular, this provides a quick proof of von Neumann’s celebrated inequality for Hilbert space contractions. There is a connection between Sz.-Nagy’s and Stinespring’s Theorems: a contraction defines a completely positive map and using this it turns out that the existence of minimal isometric dilations of contractions can be derived from Stinespring’s theorem. This idea links dilation theory of contractions to representations of the Toeplitz algebra and plays a key role in the more recent developments of dilation theory for tuples of operators. In 1989, G. Popescu proved in [Po1] a result analogous to Sz.-Nagy’s for a rowcontraction, that is a (finite or infinite) sequence of contractions (Ti) n i=1 on a Hilbert space H such that ∑n i=1 TiT ∗ i ≤ IH. Building on the earlier work of A.E.Frazho and J.Bunce he showed that each such object can be dilated to a row-isometry, which in turn provides a representation of the Cuntz algebra On. This may be regarded as a dilation of the completely positive map X 7→ ∑n i=1 TiXT ∗ i . In a series of further papers G. Popescu unveiled the whole array of connections between row-contractions, their dilations, operator algebras related to free semigroups on n-generators, completely positive maps on B(H) and their dilations to endomorphisms. This led him to develop a theory which can be justifiably called noncommutative (free) complex analysis (see for example [Po5] and references therein), with concrete operators on a full Fock space playing the role of certain classes of analytic functions; in particular, it allowed to establish a von Neumann inequality for row contractions. His most important tool is a noncommutative Poisson transform (introduced in [Po3] and later used in [BJKW]). 2000 Mathematics Subject Classification. Primary 47A20, Secondary 05C20, 46L05, 47A13, 47L75.