Classification of homomorphisms and dynamical systems
Transactions of the American Mathematical Society
Let A be a unital simple C * -algebra, with tracial rank zero and let X be a compact metric space. Suppose that h 1 , h 2 : C(X) → A are two unital monomorphisms. We show that h 1 and h 2 are approximately unitarily equivalent if and only if for every f ∈ C(X) and every trace τ of A. Inspired by a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let X be a compact metric space and let α, β : X → X be two minimal homeomorphisms. Using the
... . Using the above-mentioned result, we show that two dynamical systems are approximately conjugate in that sense if and only if a K-theoretical condition is satisfied. In the case that X is the Cantor set, this notion coincides with the strong orbit equivalence of Giordano, Putnam and Skau, and the K-theoretical condition is equivalent to saying that the associate crossed product C * -algebras are isomorphic. Another application of the above-mentioned result is given for C * -dynamical systems related to a problem of Kishimoto. Let A be a unital simple AHalgebra with no dimension growth and with real rank zero, and let α ∈ Aut(A). We prove that if α r fixes a large subgroup of K 0 (A) and has the tracial Rokhlin property, then A α Z is again a unital simple AH-algebra with no dimension growth and with real rank zero. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use CLASSIFICATION OF HOMOMORPHISMS AND DYNAMICAL SYSTEMS 861 K-conjugate if and only if C(X) α Z ∼ = C(X) β Z. In this paper, we define a C * -version of approximate flip conjugacy and use the classification of monomorphisms from C(X) into a unital simple C * -algebra of tracial rank zero to give a K-theoretical condition for two minimal dynamical systems being approximate flip conjugate in that sense. We present another related application of the above-mentioned classification of monomorphisms from C(X). We study C * -dynamical systems (A, α), where A is a unital simple C * -algebra with tracial rank zero and α ∈ Aut(A) which satisfies a certain Rokhlin property. The Rokhlin property in ergodic theory was first adopted into operator algebras in the context of von Neumann algebras by A. Connes (). It was adopted by Herman and Ocneanu (), then by M. Rørdam (), A. Kishimoto and more recently by M. Isumi () in a much more general context of C * -algebras. Kishimoto has studied the problem when a crossed product of a simple AT-algebra A of real rank zero by an automorphism α ∈ Aut(A) is again an AT-algebra of real rank zero. A more general question is when A α Z is a unital simple AH-algebra with real rank zero if A is a unital simple AH-algebra. Given the classification theorem for simple separable amenable C * -algebra with tracial rank zero, a similar question is under what condition A α Z has tracial rank zero. In order to make reasonable sense, one has to assume that α is sufficiently outer. As proposed by A. Kishimoto (), the right description of "sufficiently outer" is that α has a Rokhlin property. One version of Rokhlin property was introduced in  called "tracial Rokhlin property" which is closely related to, but slightly weaker than, the so-called approximate Rokhlin property used in  (see Definition 3.12, below). It was shown in  that the tracial Rokhlin property occurs quite often. Tracial cyclic Rokhlin property was introduced in  which is a strong Rokhlin property (see Definition 3.13 below). For example, if α has the tracial cyclic Rokhlin property, then α * 0 fixes a large subset of K 0 (A). It is shown in  and  that if A has tracial rank zero and α has the tracial cyclic Rokhlin property, then A α Z has tracial Rokhlin property. Thus one may apply classification theorem in  to these simple crossed products. This leads to the question of when an automorphism α has tracial cyclic Rokhlin property. We will apply the results of classification of monomorphisms from C(X) into a unital simple C * -algebra with tracial rank zero to this problem. Among other things, we will show that, if α r * 0 | G = id G for some subgroup G ⊂ K 0 (A) for which ρ A (G) = ρ A (K 0 (A)) and α has tracial Rokhlin property, then α has tracial cyclic Rokhlin property. This result implies that, under the same condition, A α Z is a unital simple AH-algebra with no dimension growth and with real rank zero if A is. This solves the generalized version of the Kishimoto problem. The paper is organized as follows. In Section 2, we list some conventions that will be used in this paper. In Section 3, we present the main results. We first give (in Subsection 2.1) the classification of monomorphisms from C(X) into a unital simple C * -algebra with tracial topological rank zero. We then give two applications of the theorem, one for minimal dynamical systems (Subsection 2.2) and the other for the C * -dynamical systems and the Rokhlin property (Subsection 2.3). In Section 4, we give the proof of the classification theorem mentioned above and also give proofs of several approximate version of it. In Section 5, we present the proof of the theorems presented in Subsection 3.2. Finally, in Section 6, we give the proof of the main results in Subsection 2.3.