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

Abstract

There has recently been much interest in the $C^{\ast }$-algebras of directed graphs. Here we consider product systems $E$ of directed graphs over semigroups and associated $C^{\ast }$-algebras $C^{\ast }(E)$ and $\mathcal{T}{C}^{\ast }(E)$ 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 $C^{\ast }(E)$ and $\mathcal{T}{C}^{\ast }(E)$ are universal for families of partial isometries satisfying these relations. Our main result is a uniqueness theorem for $\mathcal{T}{C}^{\ast }(E)$ which has particularly interesting implications for the $C^{\ast }$-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 $\mathcal{T}{C}^{\ast }(E)$.

References in corpus (1)

• Higher rank graph C*-algebras

Cited by in corpus (30)

• C*-algebras associated to product systems of Hilbert bimodules
• Groupoid models for the C*-algebras of topological higher-rank graphs
• On ${\mathrm{C}}^{\ast }$-algebras associated to product systems
• von Neuman algebras of strongly connected higher-rank graphs
• Topological higher-rank graphs and the $C^{\ast }$-algebras of topological 1-graphs
• Topological aperiodicity for product systems over semigroups of Ore type
• Poisson transform for higher-rank graph algebras and its applications
• Gauge-invariant ideals in the C*-algebras of finitely aligned higher-rank graphs
• $K$-theory and homotopies of 2-cocycles on higher-rank graphs
• Regular dilations of representations of product systems
• Iterating the Cuntz-Nica-Pimsner construction for compactly aligned product systems
• Topological spaces associated to higher-rank graphs
• Finitely Correlated Representations of Product Systems of $C^{\ast }$-Correspondences over ${\mathbb{N}}^k$
• Product-system models for twisted $C^{\ast }$-algebras of topological higher-rank graphs
• Cuntz-Krieger type uniqueness theorem for topological higher-rank graph C*-algebras
• C*-algebras of commuting endomorphisms
• Cuntz's $ax+b$-semigroup C*-algebra over $\mathbb{N}$ and product system C*-algebras
• KMS states on the Toeplitz algebras of higher-rank graphs
• Realising the Toeplitz algebra of a higher-rank graph as a Cuntz-Krieger algebra
• Graphs of $C^{\ast }$-correspondences and Fell bundles
• Spatial realisations of KMS states on the C*-algebras of higher-rank graphs
• The $H^{\infty }$ algebras of higher rank graphs
• The C*-envelope of the tensor algebra of a directed graph
• KMS states on $C^{\ast }$-algebras associated to a family of $\ast$-commuting local homeomorphisms
• Co-universal C*-algebras associated to generalised graphs
• KMS states on $C^{\ast }$-algebras associated to higher-rank graphs
• The interplay between $k$-graphs and the Yang-Baxter equation
• On freely generated semigraph $C^{\ast }$-algebras
• Co-universal C*-algebras associated to aperiodic k-graphs
• Zappa-Szép actions of groups on product systems