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

TL;DR

This paper extends the theory of higher-rank graph $C^*$-algebras by introducing product systems of graphs, constructing associated Toeplitz algebras, and proving a key uniqueness theorem with implications for non-row-finite cases.

Contribution

It develops a new framework for $C^*$-algebras of product systems of graphs, generalizing higher-rank graph algebras and establishing a crucial uniqueness theorem.

Findings

Construction of $C^*(E)$ and $ ilde{C}^*(E)$ as universal algebras

Identification of new graph-theoretic relations for Cuntz-Krieger algebras

A uniqueness theorem with implications for non-row-finite higher-rank graphs

Abstract

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