[Paper Review] AF-algebras and the tail-equivalence relation on Bratteli diagrams

R. Exel, Jean Renault|arXiv (Cornell University)|Jul 16, 2003

Advanced Operator Algebra Research1 references4 citations

TL;DR

This paper establishes a canonical isomorphism between the C*-algebra associated with the tail-equivalence relation on infinite paths of a Bratteli diagram and the AF-algebra defined by the same diagram. Using groupoid C*-algebra techniques and conditional expectations on path spaces, the authors show that the tail-equivalence relation generates an approximately proper equivalence relation whose associated C*-algebra is isomorphic to the AF-algebra via a canonical *-homomorphism, thereby linking dynamical systems on Bratteli diagrams to AF-algebra structure.

ABSTRACT

Given a Bratteli diagram D we consider the compact topological space formed by all infinite paths on D. Two such path are said to be tail-equivalent when they "have the same tail", i.e. when they eventually coincide. This equivalence relation is approximately proper and hence one may consider the C*-algebra associated to it according to a procedure recently introduced by the first named author and A. Lopes. The main result of this work is the proof that this algebra is isomorphic to the AF-algebra associated to the given Bratteli diagram.

Motivation & Objective

• To establish a structural link between the tail-equivalence relation on infinite paths of a Bratteli diagram and the AF-algebra it defines.
• To demonstrate that the C*-algebra arising from the approximately proper tail-equivalence relation is isomorphic to the AF-algebra of the Bratteli diagram.
• To provide a concrete realization of the groupoid C*-algebra associated with the tail-equivalence relation and relate it to the standard AF-algebra construction.

Proposed method

• Define the space Ω of infinite paths on a Bratteli diagram D using the product topology on ∏ₙ∈ℕ Eₙ, where Eₙ is the set of edges from Vₙ to Vₙ₊₁.
• Introduce the tail-equivalence relation ∼ on Ω, where two paths are equivalent if they eventually agree, and define the equivalence relations ∼ₙ for finite initial segments.
• Construct a conditional expectation Eₙ: C(Ω) → C(Ω; Rₙ) mapping functions to those constant on ∼ₙ-classes, using averaging over fibers of fixed tail.
• Realize the groupoid C*-algebra C*(G) for the equivalence relation R, where G is the groupoid of pairs (α,β) with α∼β, and identify C(Ω) as a subalgebra of C*(G).
• Define projections êₙ in the Toeplitz algebra T(R,E) and their images êₙ̂ in C*(G), showing they satisfy êₙ êₙ₊₁ = êₙ₊₁ and êₙ f êₙ = Eₙ(f) êₙ.
• Prove that the canonical *-homomorphism φ: T(R,E) → C*(G) factors through the quotient by the redundancy ideal, yielding an isomorphism ψ: C*(R,E) → C*(G).

Experimental results

Research questions

• RQ1 Is the C*-algebra associated with the tail-equivalence relation on a Bratteli diagram isomorphic to the AF-algebra defined by the same diagram?
• RQ2 How does the structure of the tail-equivalence relation relate to the standard inductive limit construction of AF-algebras?
• RQ3 Can the groupoid C*-algebra of the tail-equivalence relation be explicitly realized and compared to the standard AF-algebra?
• RQ4 What role do conditional expectations and projections play in relating the two C*-algebraic structures?
• RQ5 Does the redundancy ideal in the Toeplitz algebra vanish under the homomorphism to the groupoid C*-algebra?

Key findings

• The C*-algebra associated with the tail-equivalence relation on the path space Ω of a Bratteli diagram is isomorphic to the AF-algebra defined by the same diagram.
• The canonical *-homomorphism φ: T(R,E) → C*(G) from the Toeplitz algebra of the equivalence relation to the groupoid C*-algebra factors through the quotient by the redundancy ideal, yielding an isomorphism ψ: C*(R,E) → C*(G).
• The projections êₙ in T(R,E) map to projections êₙ̂ in C*(G) satisfying êₙ êₙ₊₁ = êₙ₊₁, reflecting the inductive limit structure of the AF-algebra.
• The conditional expectation Eₙ: C(Ω) → C(Ω; Rₙ) is realized as averaging over fibers of fixed tail, and its image corresponds to continuous functions on the quotient space Ω/Rₙ.
• The function j: C*(G) → C₀(G) is injective, and the support of j(k) for k in the image of the n-th projection lies within Rₙ, ensuring compatibility with the equivalence relation structure.
• The map ψ is an isomorphism, as shown by verifying that the kernel of φ contains the redundancy ideal and that the image generates the full groupoid C*-algebra.