Trace and integrable operators affiliated with a semifinite von Neumann algebra

New properties of the space of integrable (with respect to the faithful normal semifinite trace) operators affiliated with a semifinite von Neumann algebra are found. A trace inequality for a pair of projections in the von Neumann algebra is obtained, which characterizes trace in the class of all positive normal functionals on this algebra. A new property of a measurable idempotent are determined. A useful factorization of such an operator is obtained; it is used to prove the nonnegativity of the trace of an integrable idempotent. It is shown that if the difference of two measurable idempotents is a positive operator, then this difference is a projection. It is proved that a semihyponormal measurable idempotent is a projection. It is also shown that a hyponormal measurable tripotent is the difference of two orthogonal projections.     

*-algebras of unbounded operators affiliated with a von Neumann algebra

In the paper, the *-algebras of measurable operators, locally measurable operators, and τ-measurable operators associated with a von Neumann algebra M are considered. Conditions under which some of these algebras coincide are given. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 183–197.     

Paranormal measurable operators affiliated with a semifinite von Neumann algebra. II

Positivity - Let $${{\mathcal {M}}}$$ be a von Neumann algebra of operators on a Hilbert space $${\mathcal {H}}$$ and $$\tau $$ be a faithful normal semifinite trace on $$\mathcal {M}$$ . Let...     

Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

Let M be a type I von Neumann algebra with the center Z and let LS(M) be the algebra of all locally measurable operators affiliated with M. We prove that every Z-linear derivation on LS(M) is inner. In particular, all Z-linear derivations on the algebras of measurable and respectively totally measurable operators are spatial and implemented by elements of LS(M). The text was submitted by the authors in English.     

Classification of cocycles over ergodic automorphisms with values in the Lorentz group and recurrence of cocycles

It is proved that any SO₀(1, d)-valued cocycle over an ergodic (probability) measurepreserving automorphism is cohomologous to a cocycle having one of three special forms; the recurrence property of such cocycles is also studied.     

Group inverses of certain positive operators

Positive operators on certain polyhedral cones with the property that the group inverse of the operator is equal to some power of the operator are characterized. The special case of the Moore-Penrose inverse is also considered.     

Recurrence of matrix cocycles

This paper considers measurable cocycles with values in the subgroup of SL(2, ℂ) of diagonal and skew-diagonal matrices over an ergodic transformation preserving the probability measure. We prove the recurrence of such cocycles under certain conditions as well as the equivalence of two definitions of the recurrence.     

Ergodicity of Rokhlin cocycles

We develop a general study of ergodic properties of extensions of measure preserving dynamical systems. These extensions are given by cocycles (called here Rokhlin cocycles) taking values in the group of automorphisms of a measure space which represents the fibers. We use two different approaches in order to study ergodic properties of such extensions. The first approach is based on properties of mildly mixing group actions and the notion of complementary algebra. The second approach is based on spectral theory of unitary representations of locally compact Abelian groups and the theory of cocycles taking values in such groups. Finally, we examine the structure of self-joinings of extensions. We partially answer a question of Rudolph on lifting mixing (and multiple mixing) property to extensions and answer negatively a question of Robinson on lifting Bernoulli property. We also shed new light on some earlier results of Glasner and Weiss on the class of automorphisms disjoint from all weakly mixing transformations. Answering a question asked by Thouvenot we establish a relative version of the Foiaş—Stratila theorem on Gaussian—Kronecker dynamical systems. Research partially supported by KBN grant 2 P03A 002 14 (1998).     

New ring of a class of Bessel integral operators

In this paper, we introduce a new ring of ponderation functions which allowed us to describe and classify a large class of ring of Bessel integral operators. The main result is to give an explicit form of some ideals of such a ring of operators.     

Minimal cocycles with the scaling property and substitutions

‘Fractal’ functions are formulated as a minimal cocycle on a topological dynamics which admits nontrivial scaling transformations. In this paper, it is proved that if in addition it admits a continuous family of scaling transformations, then itscapacity is not ino(N ²). We define minimal cocycles with nontrivial scaling transformations coming from substitutions on a finite alphabet which are proved to have capacityO(N), so that they admit only a discrete family of scaling transformations. We also construct one which has capacityO(N ²) and admit a continuous family of scaling transformations. Supported by C.N.R.S.(U.R.A 225); Partially supported by the Japan Society for the Promotion of Science.     

Asymptotic stability of discrete cocycles

We introduce the notion of asymptotic stability of sequences of multifunctions associated with discrete cocycles. Some sufficient conditions for existence of attracting sets are given. The use of the topological (Kuratowski's) limits, as less complicated as commonly used Hausdorff metric, let us to weaken many standard assumptions. We show that in considered case existence of attractor is a property of a cocycle mapping itself and does not depend on properties of a parameter nor a state space. The obtained results generalize earlier on iterated function systems and can be applied for non-autonomous as well as random dynamical systems.     

Almost reduction and perturbation of matrix cocycles

In this note, we show that if all Lyapunov exponents of a matrix cocycle vanish, then it can be perturbed to become cohomologous to a cocycle taking values in the orthogonal group. This extends a result of Avila, Bochi and Damanik to general base dynamics and arbitrary dimension. We actually prove a fibered version of this result, and apply it to study the existence of dominated splittings into conformal subbundles for general matrix cocycles.