Journal of Operator Theory

Volume 63, Issue 2, Spring 2010 pp. 425-454.

Poisson transform for higher-rank graph algebras and its applications

Authors: Adam Skalski (1) and Joachim Zacharias (2)
Author institution: (1) Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF and Department of Mathematics, University of Lodz, ul. Banacha 22, 90-238 Lodz, Poland
(2) School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, U.K.

Summary: Higher-rank graph generalisations of the Popescu--Poisson transform are constructed, allowing us to develop a dilation theory for higher rank operator tuples. These dilations are joint dilations of the families of operators satisfying relations encoded by the graph structure which we call $\Lambda$-contractions or $\Lambda$-isometries. Besides commutant lifting results and characterisations of pure states on higher rank graph algebras several applications to the structure theory of non-selfadjoint graph operator algebras are presented generalising recent results in special cases.

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