Borel complexity and automorphisms of C*-algebras

Abstract

Abstract. We show that if A is Z, O2, O∞, a UHF algebra of infinite type, or the tensor product of a UHF algebra of infinite type and O∞, then the conjugation action Aut(A) Aut(A) is generically turbulent for the point-norm topology. We moreover prove that if A is either (i) a separable C*-algebra which is stable under tensoring with Z or K, or (ii) a separable II1 factor which is McDuff or a free product of II1 factors, then the approximately inner automorphisms of A are not classifiable by countable structures.

Polish groupoids and functorial complexity

“…For example the relation of isomorphism of simple separable C*-algebras has been shown to transcend countable structures in [14]; see also [45]. Similar results have been obtained for several other equivalence relations, such as affine homeomorphism of Choquet simplexes [14], conjugacy of unitary operators on the infinite dimensional separable Hilbert space [29], conjugacy of ergodic measure-preserving transformations of the Lebesgue space [16], conjugacy of homeomorphisms of the unit square [21], conjugacy of irreducible representations of non type I groups [20] or C*-algebras [11,30], conjugacy and unitary equivalence of automorphisms of classifiable simple separable C*-algebras [31,34], isometry of separable Banach spaces [37] and complete order isomorphism of separable operator systems. Furthermore the relations of isomorphism and Lipschitz isomorphisms of separable Banach spaces, topological isomorphism of (abelian) Polish groups, uniform homeomorphism of complete separable metric spaces [15], and the relation of completely bounded isomorphism of separable operator spaces [2] have been shown to be not classifiable by the orbits of a Polish group action (and in fact to have maximal complexity among analytic equivalence relations).…”

Non-classification of free Araki–Woods factors and $\tau$-invariants

“…Finally note that a similar non-classification theorem for conjugacy of a single automorphism (i.e. an action of Z) was obtained in [KLP08] for the hyperfinite II 1 factor R and in [KLP14] for free product factors, using a spectral invariant. Such spectral invariants are however not preserved under cocycle conjugacy.…”

Equivariant Kirchberg–Phillips-Type Absorption for Amenable Group Actions

“…Using again that D is strongly self-absorbing, the flip automorphism on D ⊗ D is homotopic to the identity map; see [13]. Applying this homotopy on each individual opposed pair of tensor factors in the product ( I D) ⊗ ( I D), we obtain a G-homotopy between the identity map and the flip automorphism with respect to the action β σ ⊗ β σ ; see [35,Lemma 2.6] for a related argument. Thus it follows from Theorem 6.4 that the flip automorphism on ( I D) ⊗ ( I D), β σ ⊗ β σ is approximately G-inner.…”