[Paper Review] Product systems of graphs and the Toeplitz algebras of higher-rank graphs

Iain Raeburn, Aidan Sims|arXiv (Cornell University)|May 27, 2003

Advanced Operator Algebra ResearchMathematics20 references74 citations

TL;DR

This paper generalizes Cuntz-Krieger algebras and their Toeplitz analogues to higher-rank graphs using product systems of graphs over semigroups, particularly N^k. It constructs a product system of Hilbert bimodules and establishes a uniqueness theorem for the Toeplitz algebra of finitely aligned k-graphs, especially those with infinite emitters, by analyzing the diagonal expectation and proving faithfulness under a strengthened Cuntz-Krieger relation that accounts for non-row-finite structures.

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

Motivation & Objective

• To extend the theory of Cuntz-Krieger algebras and their Toeplitz analogues to higher-rank graphs that are not necessarily row-finite.
• To define universal C*-algebras for product systems of graphs using Toeplitz-Cuntz-Krieger families, ensuring compatibility with Nica-covariant representations.
• To identify the conditions under which the associated product system of Hilbert bimodules is compactly aligned, enabling application of Fowler's theory.
• To establish a sharp uniqueness theorem for the Toeplitz algebra of k-graphs with infinite emitters, overcoming limitations of prior results.

Proposed method

• Construct a product system of Hilbert bimodules X(E) from a product system E of directed graphs over semigroups, particularly N^k.
• Define Toeplitz-E-families and establish a one-to-one correspondence with Toeplitz representations of X(E).
• Characterize finitely aligned product systems E for which X(E) is compactly aligned, enabling use of Fowler's theory on Nica-covariant representations.
• Introduce a strengthened Cuntz-Krieger relation involving the positivity of products of projections, replacing the standard sum-of-projections condition.
• Analyze the expectation onto the diagonal in T*C(E) to prove faithfulness of representations under the new relation.
• Use the amenability of Z^k and the existence of long paths avoiding finite sets of edges to show that certain products of projections are nonzero.

Experimental results

Research questions

• RQ1 How can the Cuntz-Krieger algebra and its Toeplitz analogue be generalized to higher-rank graphs that are not row-finite?
• RQ2 What conditions on a product system of graphs ensure that the associated Hilbert bimodule is compactly aligned?
• RQ3 What is the correct generalization of the Cuntz-Krieger relation for non-row-finite k-graphs?
• RQ4 Can a uniqueness theorem for the Toeplitz algebra of a k-graph be established that is sharp even when the graph has infinitely many edges at each vertex?
• RQ5 How does the diagonal expectation in the Toeplitz algebra relate to the faithfulness of representations in the non-row-finite case?

Key findings

• The paper defines a universal Toeplitz algebra T*C(E) for finitely aligned product systems of graphs, which is generated by a Toeplitz-Cuntz-Krieger E-family.
• A new Cuntz-Krieger relation is identified: for each vertex v and finite sets G_m ⊂ s^{-1}_{e_m}(v), the product ∏_{m=1}^k (t_v - ∑_{λ∈G_m} t_λ t_λ^*) > 0.
• This strengthened relation is both necessary and sufficient for the faithfulness of representations in the case of k-graphs with |Λ^{e_i}(v)| = ∞ for all v and i.
• The main uniqueness theorem shows that for such k-graphs, any Cuntz-Krieger family with nonzero vertex projections generates a faithful representation of C*(E).
• The proof relies on constructing paths μ_k of degree ∑_{i=1}^k e_i that avoid specified finite sets of edges, ensuring that t_μ_k t_μ_k^* is nonzero and survives under the product of projections.
• The result establishes that T*C(E) = C*(E) when the k-graph has no sources and each vertex emits infinitely many edges of each degree, unifying the Toeplitz and Cuntz-Krieger algebras in this case.