Journal of Operator Theory

Volume 64, Issue 2, Fall 2010 pp. 349-376.

$C^*$-algebras associated to product systems of Hilbert bimodules

Authors: Aidan Sims (1) and Trent Yeend (2)
Author institution: (1) School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia
(2) School of Mathematical and Physical Sciences, University of Newcastle, NSW 2308, Australia

Summary: Let $(G,P)$ be a quasi-lattice ordered group and let $X$ be a compactly aligned product system over $P$ of Hilbert bimodules in the sense of Fowler. Under mild hypotheses we associate to $X$ a $C^*$-algebra which we call the Cuntz--Nica--Pimsner algebra of $X$. Our construction generalises a number of others: a sub-class of Fowler's Cuntz--Pimsner algebras for product systems of Hilbert bimodules; Katsura's formulation of Cuntz--Pimsner algebras of Hilbert bimodules; the $C^*$-algebras of finitely aligned higher-rank graphs; and Crisp and Laca's boundary quotients of Toeplitz algebras. We show that for a large class of product systems $X$, the universal representation of $X$ in its Cuntz--Nica--Pimsner algebra is isometric.

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