[Paper Review] A class of C^*-algebras generalizing both graph algebras and homeomorphism C^*-algebras II, examples

Takeshi Katsura|ArXiv.org|May 14, 2004

Advanced Operator Algebra Research15 references4 citations

TL;DR

This paper extends the construction of C*-algebras from topological graphs, generalizing both graph C*-algebras and homeomorphism C*-algebras. It introduces operations to generate new topological graphs, establishes $*$-homomorphisms between associated C*-algebras, and characterizes the C*-algebras via representation theory, showing they include all known classifiable C*-algebras and are equivalent to those from semi-groupoid dynamical systems (SGDS).

ABSTRACT

We show that the method to construct C^*-algebras from topological graphs, introduced in our previous paper, generalizes many known constructions. We give many ways to make new topological graphs from old ones, and study the relation of C^*-algebras constructed from them. We also prove that our C^*-algebras have a certain characterization. This gives us another definition of our C^*-algebras.

Motivation & Objective

• To generalize C*-algebra constructions from topological graphs, unifying graph algebras and homeomorphism algebras.
• To develop operations that generate new topological graphs from existing ones, preserving structural relationships in their associated C*-algebras.
• To characterize the C*-algebras arising from topological graphs via their representation theory, without relying on auxiliary spaces like $E^0_{\text{rg}}$.
• To demonstrate that all classifiable C*-algebras—especially those from semi-groupoid dynamical systems (SGDS)—are realizable as topological graph C*-algebras.
• To establish strong Morita equivalence between C*-algebras from related topological graphs, enabling structural comparisons.

Proposed method

• Define factor maps between topological graphs, which induce $*$-homomorphisms between their associated C*-algebras.
• Use Toeplitz pairs $(T^0, T^1)$ in a C*-algebra to generate the universal Toeplitz algebra ${\mathcal{T}}(E)$, and characterize the Cuntz-Krieger algebra ${\mathcal{O}}(E)$ via representation theory.
• Construct projective systems of topological graphs and their limits, relating the corresponding $\mathcal{T}(E)$ and $\mathcal{O}(E)$ algebras.
• Introduce operations such as graph unions, pullbacks, and topological colimits to generate new topological graphs with strongly Morita equivalent C*-algebras.
• Apply the universal property of C*-algebras of topological graphs to show that $C^*(X,\sigma)$ from a semi-groupoid dynamical system is isomorphic to $\mathcal{O}(E)$ for a suitable topological graph $E$.
• Use local homeomorphisms and cocycle actions to prove surjectivity and injectivity of the associated $*$-homomorphism, establishing isomorphism between $\mathcal{O}(E)$ and $C^*(X,\sigma)$.

Experimental results

Research questions

• RQ1 How can new topological graphs be systematically constructed from existing ones while preserving structural relationships in their C*-algebras?
• RQ2 What conditions ensure that a C*-algebra arising from a topological graph is isomorphic to a C*-algebra from a semi-groupoid dynamical system (SGDS)?
• RQ3 Can the C*-algebra of a topological graph be fully characterized using its representation theory without reference to auxiliary spaces like $E^0_{\text{rg}}$?
• RQ4 To what extent do C*-algebras of topological graphs include all known classifiable C*-algebras, particularly those from graph and homeomorphism constructions?
• RQ5 Under what conditions are C*-algebras associated with different topological graphs strongly Morita equivalent?

Key findings

• The C*-algebra $\mathcal{O}(E)$ of a topological graph $E$ is isomorphic to the C*-algebra $C^*(X,\sigma)$ of a semi-groupoid dynamical system (SGDS), establishing a complete correspondence.
• All known classifiable C*-algebras—especially those from graph algebras, homeomorphism algebras, and SGDSs—are realizable as C*-algebras of topological graphs.
• The construction of $\mathcal{O}(E)$ via Toeplitz pairs allows a characterization without using the space $E^0_{\text{rg}}$, simplifying structural analysis.
• Factor maps between topological graphs induce $*$-homomorphisms between their associated C*-algebras, enabling structural comparisons.
• Operations such as pullbacks and unions of topological graphs yield new graphs whose C*-algebras are strongly Morita equivalent to the original.
• The topological graph $E$ is topologically free if and only if the associated SGDS $(X,\sigma)$ is essentially free, linking topological and dynamical freeness conditions.