Product systems of graphs and the Toeplitz algebras of higher-rank graphs | NOVA. The University of Newcastle's Digital Repository

By Iain Raeburn, Aidan Sims. 2003

78 citations recorded by the source corpus.

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 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 re- cent 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 T C � (E) are universal for families of partial isometries satisfying these relations. Our main result is a uniqueness theorem for T C � (E) which has particularly in- teresting 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 T C � (E).

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/d772546b66794e8052f573c06692665e692025e9

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Trent Yeend
2. Aidan Sims
3. Ben Steinberg
4. R. Exel
5. Iain Raeburn
6. David Pask
7. K. Davidson
8. L. O. Clark

In Geometry and Topology

1. Trent Yeend
2. Aidan Sims
3. Ben Steinberg

In Mathematical Physics

1. Trent Yeend
2. Aidan Sims
3. Iain Raeburn
4. Ben Steinberg
5. R. Exel
6. David Pask
7. K. Davidson
8. L. O. Clark

Citations in the corpus

• C*-algebras associated to product systems of hilbert bimodules (2007, methodology): “Recall from [28] that for a nonempty subset G of F , MCE(G) denotes the set {λ ∈ Λ : d(λ) = ∨d(G), λ extends μ for all μ ∈ G}.”
• Groupoid models for the C*-algebras of topological higher-rank graphs (2006, unknown): “13] that C(GΛ) ∼= T C(Λ) and C(GΛ) ∼= C(Λ), where T C(Λ) and C(Λ) are defined and studied in [24] and [26], respectively.”