A strong generic ergodicity property of unitary and self-adjoint operators

Abstract

Abstract. Consider the conjugacy action of the unitary group of an infinite-dimensional separable Hilbert space on the unitary operators. A strong generic ergodicity property of this action is established, by showing that any conjugacy invariants assigned in a definable way to unitary operators, and taking as values countable structures up to isomorphism, generically trivialize. Similar results are proved for conjugacy of self-adjoint operators and for measure equivalence. The proofs make use of the theory of turbulence for continuous actions of Polish groups, developed by Hjorth. These methods are also used to give a new solution to a problem of Mauldin in measure theory, by showing that any analytic set of pairwise orthogonal measures on the Cantor space is orthogonal to a product measure. IntroductionTwo of the main classes of operators on the infinite-dimensional separable complex Hilbert space H, which are studied extensively in functional analysis and operator theory, are the unitary and self-adjoint operators. The unitary operators equipped with the strong (equivalently, the weak) operator topology form a Polish (i.e. completely metrizable separable) group U(H) under multiplication, while the self-adjoint operators with norm at most 1 form a Polish space S 1 (H), when equipped with the strong topology. The unitary group U(H) acts continuously on both U(H) and S 1 (H) by conjugation, and our main purpose in this paper is to study (unitary) conjugacy invariants of unitary and self-adjoint operators, i.e. functions f : U(H) → X and g : S 1 (H) → Y with the property that f (V UV −1 ) = f (U) and g(V SV −1 ) = g(S), whenever U , V are in U(H) and S is in S 1 (H). For example, any such f and g, for which f (U) and g(S) depend only on the spectrum or the point spectrum of U and S respectively, constitute conjugacy invariants.The spectral theorem for unitary operators provides a complete invariant under conjugacy, which consists of:

Conjugacy, orbit equivalence and classification of measure-preserving group actions

“…By Theorem 5.1 in [17], we have that ≈ Pc(2 N ) is a generically turbulent equivalence relation. Thus E 0 < B ≈ 2 N c and ≈ c is not classifiable by countable structures.…”

“…If X is uncountable then P c (X) is a dense G δ subset of P (X), cf. [17]. Absolute equivalence ≈ is a Borel equivalence relation in P (X), c.f.…”

“…Given this, the task becomes to produce many G actions that are nonconjugate and ergodic on H. To this end we prove a version of Foreman and Weiss anti-classification result in [8] (Theorem A) for extensions of actions. This is based on Theorem 5.1 of Kechris and Sofronidis in [17]:…”

Borel equivalence relations in the space of bounded operators

“…Theorem (Kechris-Sofronidis [13]). Unitary equivalence of self-adjoint (or unitary) operators is not classifiable by countable structures.…”

All automorphisms of the Calkin algebra are inner

“…On the other hand, the complete invariant for the former given by the spectral theorem is of much higher complexity. As a matter of fact, in [26] it was proved that the unitary equivalence of self-adjoint operators does not admit any effectively assigned complete invariants coded by countable structures.…”