[Paper Review] Uniformly continuous orbit equivalence of Markov shifts and gauge actions on Cuntz-Krieger algebras

Kengo Matsumoto|arXiv (Cornell University)|Jul 19, 2016

Advanced Operator Algebra Research13 references3 citations

TL;DR

This paper introduces uniformly continuous orbit equivalence (UCOE) as a refinement of continuous orbit equivalence (COE) for one-sided topological Markov shifts, linking it to gauge actions on Cuntz–Krieger algebras. It establishes that UCOE implies strong continuous orbit equivalence (SCOE), and provides counterexamples showing the converse does not hold, clarifying the hierarchy of orbit equivalence relations in symbolic dynamics and C*-algebra classification.

ABSTRACT

We introduce a notion of uniformly continuous orbit equivalence as a subequivalence relation of continuous orbit equivalence of one-sided topological Markov shifts. It is described in terms of gauge actions on the associated Cuntz-Krieger algebras and continuous full groups of the Markov shifts.

Motivation & Objective

• To define and study uniformly continuous orbit equivalence (UCOE) as a subequivalence relation of continuous orbit equivalence (COE) in one-sided topological Markov shifts.
• To characterize UCOE in terms of gauge actions on Cuntz–Krieger algebras and continuous full groups of the shift spaces.
• To clarify the relationship between UCOE, SCOE, COE, and other orbit equivalence relations, and to examine the implications for C*-algebra isomorphisms.
• To provide counterexamples showing that UCOE does not imply COE or SCOE in reverse, and to analyze the role of AF algebras and $K_0$-groups in distinguishing these classes.
• To investigate the extent to which orbit equivalence relations correspond to isomorphisms of Cuntz–Krieger algebras and their fixed-point subalgebras.

Proposed method

• Introduces uniformly continuous orbit equivalence via continuous functions $k_1, l_1: X_A o bZ_+$ satisfying $\sigma_B^{k_1(x)}(h(\sigma_A(x))) = \sigma_B^{l_1(x)}(h(x))$ for all $x \in X_A$, with uniform continuity in the cocycle functions.
• Relates UCOE to the gauge action $\rho_t^A$ on the Cuntz–Krieger algebra $\cO_A$, particularly through the fixed-point subalgebra $\cF_A$ and its $K_0$-group.
• Uses the continuous full group $\Gamma_A$ and its AF-subgroup $\Gamma_A^{\text{AF}}$ to characterize homeomorphisms preserving finite-dimensional subalgebras.
• Applies the homomorphism $\Psi_h: C(X_B, \bbZ) \to C(X_A, \bbZ)$ induced by the orbit equivalence to compare cohomology classes in $H^A = C(X_A, \bbZ)/\{g - g \circ \sigma_A\}$.
• Employs the dimension group $K_0(\cF_A)$ and the class $[1] \in K_0(\cF_A)$ to distinguish non-isomorphic AF algebras, showing that $K_0(\cF_{B_2}) \not\cong K_0(\cF_{A_2})$.
• Constructs explicit counterexamples using matrices $A_2, A_4, B_2, F_2, B_3, C_3$ to demonstrate non-equivalence in the orbit equivalence hierarchy.

Experimental results

Research questions

• RQ1 Does uniformly continuous orbit equivalence (UCOE) imply strong continuous orbit equivalence (SCOE) in one-sided topological Markov shifts?
• RQ2 Can UCOE be characterized algebraically via the gauge action on Cuntz–Krieger algebras and their fixed-point subalgebras?
• RQ3 Is the converse of UCOE implying SCOE always true, or are there counterexamples?
• RQ4 How do the $K_0$-groups of the AF fixed-point algebras $\cF_A$ and $\cF_B$ distinguish between UCOE and COE classes?
• RQ5 To what extent does UCOE correspond to isomorphisms of Cuntz–Krieger algebras and their diagonal subalgebras $\cD_A$?

Key findings

• UCOE implies SCOE, as shown by the fact that if $c_1 = 1_{X_A}$ and $c_2 = 1_{X_B}$, then the cocycle functions are constant, leading to strong orbit equivalence.
• There exist examples where $\cO_{A_2} \cong \cO_{A_4}$ but $\cO_{A_2} \not\cong \cO_{A_4}$ under UCOE, showing that UCOE is strictly stronger than COE.
• The AF algebra $\cF_{B_2}$ has $K_0(\cF_{B_2}) \cong \bbZ^3$ with $[1] = 3$, while $\cF_{A_2}$ has $K_0(\cF_{A_2}) \cong \bbZ[1/2]$ with $[1] = 1$, so $\cF_{B_2} \not\cong \cF_{A_2}$, implying $\text{UCOE} \not\Rightarrow \text{COE}$ in reverse.
• The two-sided shifts $\bar{X}_{A_2}$ and $\bar{X}_{F_2}$ are not topologically conjugate, yet $\cO_{A_2} \cong \cO_{F_2}$ and $\det(\text{id} - A_2) = \det(\text{id} - F_2)$, so $\text{COE}$ holds but $\text{SCOE}$ does not.
• The AF algebra $\cF_{B_3}$ is not isomorphic to $\cF_{C_3}$, despite their two-sided shifts being conjugate, showing that $\text{COE} \not\Rightarrow \text{UCOE}$ in reverse.
• The converse of the implication from eventually one-sided conjugate to two-sided conjugate remains an open question, as noted in the paper.