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

**Keywords:**$C^*$-algebra of directed graphs, Hilbert bimodules, Toeplitz algebras, product systems of graphs.

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