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:
The purpose of this paper is to show that for any positive integer n, there exists no algorithm which decides for each non-cooperative n-person game in strategic form with partially computable payoff functions whether it has a pure Nash equilibrium or not. Copyright Springer-Verlag Berlin/Heidelberg 2004Undecidable problems, Non-cooperative games.,
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