[Paper Review] On C*-algebras Associated with Sofic Shifts

Toke Meier Carlsen|ArXiv.org|Sep 14, 2000

Advanced Operator Algebra Research6 references19 citations

TL;DR

This paper establishes that for any sofic shift Λ, Matsumoto's C*-algebra 𝒪_Λ is isomorphic to the Cuntz-Krieger algebra associated with the left Krieger cover graph of Λ. The isomorphism is constructed via universal properties by defining mutually inverse *-homomorphisms between the two algebras, using the structure of the left Krieger cover and the generating partial isometries of 𝒪_Λ.

ABSTRACT

We show that for a sofic shift Lambda, Matsumoto's C*-algebra O_Lambda is isomorphic to the Cuntz-Krieger algebra of the left Krieger cover graph of Lambda.

Motivation & Objective

• To resolve the open question of whether Matsumoto's C*-algebra 𝒪_Λ for a sofic shift Λ is isomorphic to the Cuntz-Krieger algebra of its left Krieger cover graph.
• To establish a structural link between two C*-algebra constructions—Matsumoto's and Cuntz-Krieger's—on sofic shifts.
• To prove that the K₀ and K₁ invariants of both algebras coincide, and go further to construct a full *-isomorphism.
• To leverage universal properties of C*-algebras to define and verify inverse *-homomorphisms between the two algebras.
• To demonstrate that the algebraic structure of the left Krieger cover graph fully captures the C*-algebraic structure of the sofic shift.

Proposed method

• Construct a *-homomorphism ϕ from the Cuntz-Krieger algebra 𝒪_{B_Λ} to Matsumoto's algebra 𝒪_Λ by mapping generators s_e to S_{ℒ(e)}E_{r(e)}.
• Construct a dual *-homomorphism ψ from 𝒪_Λ to 𝒪_{B_Λ} by mapping S_i to ∑_{ℒ(e)=i} s_e and E_{r(e)} to s_e^*s_e.
• Use the universal property of Cuntz-Krieger algebras to ensure the existence of these homomorphisms based on the relations satisfied by the generators.
• Verify that the compositions ϕ∘ψ and ψ∘ϕ are identity maps on the respective algebras using the orthogonality and range projections of partial isometries.
• Leverage the left-resolving property of the left Krieger cover graph to ensure consistency in the multiplication and adjoint relations of the generators.
• Use the isomorphism between 𝒟_Λ and the algebra of characteristic functions on cylinder sets to relate the projections in 𝒪_Λ to measurable sets in the shift space.

Experimental results

Research questions

• RQ1 Is Matsumoto's C*-algebra 𝒪_Λ for a sofic shift Λ isomorphic to the Cuntz-Krieger algebra of its left Krieger cover graph?
• RQ2 Do the universal relations defining the Cuntz-Krieger algebra and Matsumoto's algebra align under the left Krieger cover structure?
• RQ3 Can the K₀ and K₁ invariants of the two algebras be lifted to a full *-isomorphism?
• RQ4 Does the left Krieger cover graph provide a complete algebraic model for the C*-algebra of a sofic shift?
• RQ5 Can the generating partial isometries of 𝒪_Λ be systematically lifted to the generators of the Cuntz-Krieger algebra via the left Krieger cover?

Key findings

• The C*-algebra 𝒪_Λ associated with a sofic shift Λ is isomorphic to the Cuntz-Krieger algebra 𝒪_{B_Λ} of its left Krieger cover graph.
• The isomorphism is explicitly constructed via two mutually inverse *-homomorphisms: ϕ: 𝒪_{B_Λ} → 𝒪_Λ and ψ: 𝒪_Λ → 𝒪_{B_Λ}.
• The map ψ sends S_i to ∑_{ℒ(e)=i} s_e and E_{r(e)} to s_e^*s_e, preserving the universal relations of the Cuntz-Krieger algebra.
• The map ϕ sends s_e to S_{ℒ(e)}E_{r(e)}, and the composition ϕ∘ψ acts as the identity on 𝒪_Λ due to orthogonality and projection relations.
• The composition ψ∘ϕ acts as the identity on 𝒪_{B_Λ}, confirming that ϕ and ψ are inverse isomorphisms.
• The proof relies on the left-resolving property of the left Krieger cover graph to ensure consistency in the multiplication and adjoint relations of the partial isometries.