[Paper Review] The Rohlin property for automorphisms on simple C*-algebras

Huaxin Lin|arXiv (Cornell University)|Feb 23, 2006

Advanced Operator Algebra Research15 references9 citations

TL;DR

This paper establishes that if an automorphism α on a unital separable simple C*-algebra A with tracial rank zero satisfies the tracial Rokhlin property and its induced action on K₀(A) fixes a dense subgroup, then the crossed product A ⋊ₐℤ also has tracial rank zero. The result resolves a generalized version of Kishimoto's conjecture for AH-algebras with slow dimension growth and real rank zero.

ABSTRACT

We study a general Kishimoto's problem for automorphisms on simple C*-algebras with tracial rank zero. Let $A$ be a unital separable simple C*-algebra with tracial rank zero and let $α$ be an automorphism. Under the assumption that $α$ has certain Rokhlin property, we present a proof that $A times_α\Z$ has tracial rank zero. We also show that if the induced map $α_{*0}$ on $K_0(A)$ fixes a "dense" subgroup of $K_0(A)$ then the tracial Rokhlin property implies a stronger Rokhlin property. Consequently, the induced crossed product C*-algebras have tracial rank zero.

Motivation & Objective

• Investigate the preservation of tracial rank zero under crossed products by automorphisms on simple C*-algebras.
• Address a generalized version of Kishimoto's conjecture on crossed products of A𝕋-algebras with real rank zero.
• Establish conditions under which the crossed product A ⋊ₐℤ retains tracial rank zero when A has tracial rank zero.
• Provide sufficient conditions on the automorphism α involving its action on K₀(A) to ensure the tracial cyclic Rokhlin property.
• Extend results from the approximately inner case to broader classes of automorphisms using K-theory and tracial Rokhlin conditions.

Proposed method

• Use the tracial Rokhlin property as a generalization of classical Rokhlin towers for C*-algebra automorphisms.
• Apply the tracial cyclic Rokhlin property, defined via asymptotic morphisms and central sequences of projections.
• Utilize the condition that αʳ*₀ fixes a subgroup G ⊂ K₀(A) with dense image under ρ_A to imply stronger Rokhlin-type behavior.
• Leverage the Universal Coefficient Theorem (UCT) and amenability to ensure classification-theoretic properties.
• Construct central sequences of projections and partial isometries to model the dynamics of α and control the asymptotic behavior.
• Use sequential asymptotic morphisms and approximation techniques to control traces and K-theory in the crossed product.

Experimental results

Research questions

• RQ1 Under what conditions does the crossed product A ⋊ₐℤ of a unital separable simple C*-algebra A with TR(A) = 0 also have TR(A ⋊ₐℤ) = 0?
• RQ2 Can the tracial Rokhlin property be strengthened to the tracial cyclic Rokhlin property when αʳ*₀ fixes a dense subgroup of K₀(A)?
• RQ3 Does the condition that αʳ is homotopic to the identity in KL(A,A) imply that A ⋊ₐℤ has tracial rank zero?
• RQ4 How does the action of α on K₀(A) influence the structural properties of the crossed product algebra?
• RQ5 To what extent can Kishimoto’s conjecture be generalized beyond the approximately inner case?

Key findings

• If α has the tracial Rokhlin property and αʳ*₀ fixes a subgroup G ⊂ K₀(A) with ρ_A(G) dense in Aff(T(A)), then α satisfies the tracial cyclic Rokhlin property.
• Under these conditions, the crossed product A ⋊ₐℤ has tracial rank zero, i.e., TR(A ⋊ₐℤ) = 0.
• The result applies to unital separable simple amenable C*-algebras with TR(A) = 0 satisfying the UCT.
• For AH-algebras with slow dimension growth and real rank zero, if α has the tracial Rokhlin property and [αʳ] = [id_A] in KL(A,A), then A ⋊ₐℤ is again an AH-algebra with slow dimension growth and real rank zero.
• The proof relies on constructing central sequences of projections and partial isometries that model the dynamics of α and satisfy asymptotic commutation and trace control.
• The key technical step is replacing Lemma 6.1 of [22] with a new argument using the density of ρ_A(G) in Aff(T(A)) to produce projections with controlled K-theory and trace behavior.