We show that every abelian Polish group is the topological factor-group of a closed subgroup of the full unitary group of a separable Hilbert space with the strong operator topology. It follows that all orbit equivalence relations induced by abelian Polish group actions are Borel reducible to some orbit equivalence relations induced by actions of the unitary group.

We prove a functional central limit theorem for stationary random sequences given by the transformations T ;! (x; y) = (2x; y + ! + x) mod 1 on the two-dimensional torus. This result is based on a functional central limit theorem for ergodic stationary martingale diierences with values in a separable Hilbert space of square integrable functions.

We shall show that every differential operator of 2-nd order in a real separable Hilbert space can be decomposed into a regular and an irregular operator. Then we shall characterize irregular operators and differential operators satisfying the maximum principle. Results obtained for the Lévy laplacian in [3] will be generalized for irregular differential operators satisfying the maximum principle.

The theory of linearized general relativity is quantized using the projection operator formalism, in which no gauge choices are made. The result of this exercise is the construction of a separable, reproducing kernel for the physical Hilbert space using coherent states. The work is then compared with legacy canonical quantum gravity results.

We consider the mixed states of the two-qubit system under the action of the group G = SU(2) × SU(2) of local unitary transformations. The Hilbert space of this system is the tensor product H = C ⊗ C. Let P be the algebra of real polynomial functions on the affine space of all hermitian operators of trace 1 on H. The polynomials f ∈ P which are invariant under G form the subalgebra P ⊆ P. We co...

We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of Wwopt = σ − cmax σ I, where cmax σ is a non-negative number and I is the identity matrix. We show the relation between the weakly optimal wit...

In this paper, we discuss elliptically contoured distribution for random variables defined on a separable Hilbert space. It is a generalization of the multivariate elliptically contoured distribution to distributions on infinite dimensional spaces. Some theoretical properties of the Hilbertian elliptically contoured distribution are discussed, examples on functional data are investigated to ill...

Let T be a quasidiagonal operator on a separable Hilbert space. It is shown that there exists a sequence of operators {Tn} such that dim(C∗(Tn)) < ∞ and ‖T − Tn‖ → 0 if and only if C ∗(T ) is exact. Dedicated to Yuriko Hoshiya and Isaac Seiji Brown.

We prove that every 2-local automorphism of the unitary group or the general linear group on a complex infinite-dimensional separable Hilbert space is an automorphism. Thus these types of transformations are completely determined by their local actions on the two-points subsets of the groups in question.

We study regularizing properties for transition semigroups related to Ornstein Uhlenbeck processes with values in a Banach space E which is continuously and densely embedded in a real and separable Hilbert space H . Namely we study conditions under which the transition semigroup maps continuous and bounded functions into differentiable functions. Via a Girsanov type theorem such properties exte...