Journal of Operator Theory

Volume 53, Issue 2, Spring 2005 pp. 399-429.

Product systems of graphs and the Toeplitz algebras of higher-rank graphs

Authors: Iain Raeburn (1) and Aidan Sims (2)
Author institution: (1) School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia
(2) School of Mathematical and Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia

Summary: There has recently been much interest in the

$C^*$ -algebras of directed graphs. Here we consider product systems

$E$ of directed graphs over semigroups and associated

$C^*$ -algebras

$C^*(E)$ and

$\mathcal{T}C^*(E)$ which generalise the higher-rank graph algebras of Kumjian-Pask and their Toeplitz analogues. We study these algebras by constructing from

$E$ a product system

$X(E)$ of Hilbert bimodules, and applying recent results of Fowler about the Toeplitz algebras of such systems. Fowler's hypotheses turn out to be very interesting graph-theoretically, and indicate new relations which will have to be added to the usual Cuntz-Krieger relations to obtain a satisfactory theory of Cuntz-Krieger algebras for product systems of graphs; our algebras

$C^*(E)$ and

$\mathcal{T}C^*(E)$ are universal for families of partial isometries satisfying these relations. Our main result is a uniqueness theorem for

$\mathcal{T}C^*(E)$ which has particularly interesting implications for the

$C^*$ -algebras of non-row-finite higher-rank graphs. This theorem is apparently beyond the reach of Fowler's theory, and our proof requires a detailed analysis of the expectation onto the diagonal in

$\mathcal{T}C^*(E)$ .

Contents Full-Text PDF