Unitary equivalence of automorphisms of separable C*-algebras

Abstract

Abstract. We prove that the automorphisms of any separable C*-algebra that does not have continuous trace are not classifiable by countable structures up to unitary equivalence. This implies a dichotomy for the Borel complexity of the relation of unitary equivalence of automorphisms of a separable unital C*-algebra: Such relation is either smooth or not even classifiable by countable structures. IntroductionIf A is a separable C*-algebra, the group Aut(A) of automorphisms of A is a Polish group with respect to the topology of pointwise norm convergence. An automorphism of A is called (multiplier) inner if it is induced by the action by conjugation of a unitary element of the multiplier algebra M (A) of A. Inner automorphisms form a Borel normal subgroup Inn(A) of the group of automorphisms of A. The relation of unitary equivalence of automorphisms of A is the coset equivalence relation on Aut(A) determined by Inn(A). (The reader can find more background on C*-algebras in Section 2.) The main result presented here asserts that if A does not have continuous trace, then it is not possible to effectively classify the automorphisms of A up to unitary equivalence using countable structures as invariants; in particular this rules out classification by K-theoretic invariants. In the course of the proof of the main result we will show that the existence of an outer derivation on a C*-algebra A is equivalent to a seemingly stronger statement, that we will refer to as Property AEP (see Definition 4.4), implying in particular the existence of an outer derivable automorphism of A.The notion of effective classification can be made precise by means of Borel reductions in the framework of descriptive set theory (the monographs [20] and [15] are standard references for this subject). If E and E ′ are equivalence relations on standard Borel spaces X and X ′ respectively, then a Borel reduction from E to F is a Borel function f : X → X ′ such that for every x,y ∈ X, xEy if and only if f (x) E ′ f (y). The Borel function f witnesses an effective classification of the objects of X up to E, with E ′ -equivalence classes of objects of X ′ as invariants. This framework captures the vast majority of concrete classification results in mathematics. (In [11] and [12] the computation of most of the invariants in the theory of C*-algebras is shown to be Borel.)2010 Mathematics Subject Classification. 03E15, 46L40, 46L57.

Borel complexity and automorphisms of C*-algebras

“…This consequence is proved using different methods in [25], in much greater generality (for separable C*-algebras which do not have continuous trace).…”

Borel equivalence relations in the space of bounded operators

“…Moreover, each of these is known to be not classifiable by countable structures; this is evident for c 0 /ℓ p by Example 3.2, and can be shown for [0,1] N /c 0 and [0, 1] N /ℓ p using Lemma 3.1 of [14]. In fact, we have the following, part (a) of which is essentially due to Oliver [16]:…”

Nonclassifiability of UHF $L^p$-operator algebras

“…A prototypical example of turbulent group action is the action of ℓ 1 on R N by translation. A standard argument allows one to deduce the following nonclassification criterion from turbulence of the action ℓ 1 R N and Hjorth's turbulence theorem [H2, Theorem 3.18]; see for example [L,Lemma 3.2 and Criterion 3.3].…”