Category product densities and liftings

In this paper we investigate two main problems. One of them is the question on the existence of category liftings in the product of two topological spaces. We prove, that if X×Y is a Baire space, then, given (strong) category liftings ρ and σ on X and Y, respectively, there exists a (strong) category lifting π on the product space such that π is a product of ρ and σ and satisfies the following section property:

     

Various products of category densities and liftings

We extend earlier work [M.R. Burke, N.D. Macheras, K. Musia?, W. Strauss, Category product densities and liftings, Topology Appl. 153 (2006) 1164-1191] of the authors on the existence of category liftings in the product of two topological spaces X and Y such that X×Y is a Baire space. For given densities ρ, σ on X and Y, respectively, we introduce two ‘Fubini type’ products ρ⊙σ and ρ?σ on X×Y. We present a necessary and sufficient condition for ρ⊙σ to be a density. Provided (X,Y) and (Y,X) have the Kuratowski-Ulam property, we prove for given category liftings ρ, σ on the factors the existence of a category lifting π on the product, dominating the density ρ?σ and such that

     

Splitting of liftings in products of probability spaces II

For a probability measure R on a product of two probability spaces that is absolutely continuous with respect to the product measure we prove the existence of liftings subordinated to a regular conditional probability and the existence of a lifting for R with lifted sections which satisfies in addition a rectangle formula. These results improve essentially some of the results from the former work of the authors [W. Strauss, N.D. Macheras, K. Musia?, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389-2408], by weakening considerably the assumptions and by presenting more direct and shorter proofs. In comparison with [W. Strauss, N.D. Macheras, K. Musia?, Splitting of liftings in products of probability spaces, Ann. Probab. 32 (2004) 2389-2408] it is crucial for applications intended that we can now prescribe one of the factor liftings completely freely. We demonstrate the latter by applications to τ-additive measures, transfer of strong liftings, and stochastic processes.     

Representations of pointed Hopf algebras and their Drinfel'd quantum doubles

We study representations of nontrivial liftings of nilpotent type of quantum linear spaces and their Drinfel'd quantum doubles. We construct a family of Verma-type modules in both cases and prove a parametrization theorem for simple modules. We compute the Loewy and socle series of Verma modules under a mild restriction on the datum of a lifting. We find bases and dimensions of simple modules.     

Central intertwining lifting,maximum entropy and their permanence

The central intertwining lifting is used to establish a maximum principle for the commutant lifting theorem. This maximum principle is used to prove that the central intertwining lifting is also a maximal entropy solution for the commutant lifting theorem, when T is a unilateral shift of finite multiplicity. The maximum principle is based on the residual spaces for intertwining liftings, and is motivated by Robinson's minimum energy delay principle for outer functions. A permanence property for the central intertwining lifting is also given.     

A harmonic-type maximal principle in commutant lifting

In this note, we prove a harmonic-type maximal principle for the Schur parametrization of all intertwining liftings of an intertwining contraction in the commutant lifting theorem.     

Path connected components of the space of Volterra-type integral operators

We study the topological structure of the space of Volterra-type integral operators on Fock spaces endowed with the operator norm. We prove that the space has the same connected and path connected components which is the set of all those compact integral operators acting on the spaces. We also obtain a characterization of isolated points of the space of the operators and show that there exists no essentially isolated Volterra-type integral operator.     

Various products for Lebesgue densities

In Macheras and Strauss (Atti Sem Math Fis Univ Modena, L, pp 349–361, 2002) and Musial et al. (J Theor Probab 20:545–560, 2007) various products for primitive liftings in the factors of a product of probability spaces have been considered. In this paper we settle for the d-dimensional Lebesgue densities open problems from Macheras and Strauss (Atti Sem Math Fis Univ Modena, L, pp 349–361, 2002) and Musial et al. (J Theor Probab 20:545–560, 2007) by applying results relying on the metrical group structure of \mathbb R^d{{\mathbb R}^d}, if d ? \mathbb N{d\in{\mathbb N}}. In particular, a lifting problem from Musial et al. (Arch Math 83:467–480, 2004), Question 3.3, is decided to the negative for the Lebesgue densities. The relation of the Lebesgue density in the product space and the results of the products taken for the Lebesgue densities in the factors under order is discussed. The results can be carried over to densities and liftings dominating Lebesgue densities and to multiplicative and positive linear liftings on function spaces.     

Induced liftings, exchange rings and semi-perfect algebras

Several important classes of rings can be characterized in terms of liftings of idempotents with respect to various ideals: classical examples are semi-perfect rings, semi-regular rings and exchange rings. We begin with a study of some extensions of the concept of idempotent lifting and prove the generalizations of some classical lifting theorems. Then we describe the method of induced liftings, which allows us to transfer liftings from a ring to its subrings. Using this method we are able to show that under certain assumptions a subring of an exchange ring is also an exchange ring, and to prove that a finite algebra over a commutative local ring is semi-perfect, provided it can be suitably represented in an exchange ring.     

Quasi-multipliers of operator spaces

We use the injective envelope to study quasi-multipliers of operator spaces. We prove that all representable operator algebra products that an operator space can be endowed with are induced by quasi-multipliers. We obtain generalizations of the Banach-Stone theorem.     

A harmonic-type maximal principle in the three chains completion problem

In this note, we prove a harmonic-type maximal principle for the Schur parametrization of all contractive interpolants in the three chains completion problem (see [4]), which is analogous to the maximal principle proven in [2] in case of the Schur parametrization of all contractive intertwining liftings in the commutant lifting theorem.     

Stochastic Jacobi fields and vector fields induced by varying area on path spaces

Summary. We study two classes of vector fields on the path space over a closed manifold with a Wiener Riemannian measure. By adopting the viewpoint of Yang-Mills field theory, we study a vector field defined by varying a metric connection. We prove that the vector field obtained in this way satisfies a Jacobi field equation which is different from that of classical one by taking in account that a Brownian motion is invariant under the orthogonal group action, so that it is a geometric vector field on the space of continuous paths, and induces a quasi-invariant solution flow on the path space. The second object of this paper is vector fields obtained by varying area. Here we follow the idea that a continuous semimartingale is indeed a rough path consisting of not only the path in the classical sense, but also its Lévy area. We prove that the vector field obtained by parallel translating a curve in the initial tangent space via a connection is just the vector field generated by translating the path along a direction in the Cameron-Martin space in the Malliavin calculus sense, and at the same time changing its Lévy area in an appropriate way. This leads to a new derivation of the integration by parts formula on the path space. Received: 8 August 1996 / In revised form: 8 January 1997     

Lifting the 3-dimensional invariant of 2-plane fields on 3-manifolds

Homotopy classes of plane fields on 3-manifolds have been classified using a 2-dimensional invariant Γ and a 3-dimensional invariant θ by R. Gompf. Under regular covering maps, Γ lifts in the natural way. The lifting property of θ remained unresolved. In this paper, we present the lifting property of θ together with applications to Lens spaces. The applications help in specifying the liftings of the contact structures of the Lens space L(p,1) when lifted to S³.     

Fubini Type Products for Densities and Liftings

In our former paper (Fund. Math. 166, 281–303, 2000) we discussed densities and liftings in the product of two probability spaces with good section properties analogous to that for measures and measurable sets in the Fubini Theorem. In the present paper we investigate the following more delicate problem: Let (Ω,Σ,μ) and (Θ,T,ν) be two probability spaces endowed with densities υ and τ, respectively. Can we define a density on the product space by means of a Fubini type formula \((\upsilon\odot\tau)(E)=\{(\omega,\theta):\omega\in\upsilon(\{\bar {\omega}:\theta\in\tau(E_{\bar{\omega}}\})\}\), for E measurable in the product, and the same for liftings instead of densities? We single out classes of marginal densities υ and τ which admit a positive solution in case of densities, where we have sometimes to replace the Fubini type product by its upper hull, which we call box product. For liftings the answer is in general negative, but our analysis of the above problem leads to a new method, which allows us to find a positive solution. In this way we solved one of the main problems of Musia?, Strauss and Macheras (Fund. Math. 166, 281–303, 2000).     

Maximally movable Riemannian spaces with torsion

We prove that, among all Riemannian spaces of constant curvature, only three-dimensional spaces have torsion which is invariant under the group of motions. The torsion tensor in these spaces is covariantly constant and determines the torsion form. The ratio of the integral of this form over a bounded domain to its volume is a constant determining the torsion of the space. We introduce the notions of volume torsion and scalar torsion.     

Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations

We prove that the admissibility of any pair of vector-valued Schäffer function spaces (satisfying a very general technical condition) implies the existence of a “no past” exponential dichotomy for an exponentially bounded, strongly continuous cocycle (over a semiflow). Roughly speaking the class of Schäffer function spaces consists in all function spaces which are invariant under the right-shift and therefore our approach addresses most of the possible pairs of admissible spaces. Complete characterizations for the exponential dichotomy of cocycles are also obtained. Moreover, we involve a concept of a “no past” exponential dichotomy for cocycles weaker than the classical concept defined by Sacker and Sell (1994) in [23]. Our definition of exponential dichotomy follows partially the definition given by Chow and Leiva (1996) in [4] in the sense that we allow the unstable subspace to have infinite dimension. The main difference is that we do not assume a priori that the cocycle is invertible on the unstable space (actually we do not even assume that the unstable space is invariant under the cocycle). Thus we generalize some known results due to O. Perron (1930) [14], J. Daleckij and M. Krein (1974) [7], J.L. Massera and J.J. Schäffer (1966) [11], N. van Minh, F. Räbiger and R. Schnaubelt (1998) [26].     

Luzin measurability of Carathéodory type mappings

A topological space X is said to have the Scorza-Dragoni property if the following property holds: For every metric space Y and every Radon measure space (T,μ), any Carathéodory function is Luzin measurable, i.e., given ε>0, there is a compact set K in T with μ(T?K)?ε such that the mapping is continuous. We present a selection of spaces without the Scorza-Dragoni property, among which there are first countable hereditarily separable and hereditarily Lindelöf compact spaces, separable Moore spaces and even countable k-spaces. In the positive direction, it is shown that every space which is an ℵ₀-space and kR-space has the Scorza-Dragoni property. We also prove that every separately continuous mapping , where Y is a metric space, is Luzin measurable, provided the space X is strongly functionally generated by a countable collection of its bounded subsets. If Martin's Axiom is assumed then all metric spaces of density less than c, and all pseudocompact spaces of cardinality less than c, have the Scorza-Dragoni property with respect to every separable Radon measure μ. Finally, the class of countable spaces with the Scorza-Dragoni property is closely examined.     

The Knaster-Kuratowski-Mazurkiewicz theorem and abstract convexities

The Knaster-Kuratowski-Mazurkiewicz covering theorem (KKM), is the basic ingredient in the proofs of many so-called “intersection” theorems and related fixed point theorems (including the famous Brouwer fixed point theorem). The KKM theorem was extended from Rn to Hausdorff linear spaces by Ky Fan. There has subsequently been a plethora of attempts at extending the KKM type results to arbitrary topological spaces. Virtually all these involve the introduction of some sort of abstract convexity structure for a topological space, among others we could mention H-spaces and G-spaces. We have introduced a new abstract convexity structure that generalizes the concept of a metric space with a convex structure, introduced by E. Michael in [E. Michael, Convex structures and continuous selections, Canad. J. Math. 11 (1959) 556-575] and called a topological space endowed with this structure an M-space. In an article by Shie Park and Hoonjoo Kim [S. Park, H. Kim, Coincidence theorems for admissible multifunctions on generalized convex spaces, J. Math. Anal. Appl. 197 (1996) 173-187], the concepts of G-spaces and metric spaces with Michael's convex structure, were mentioned together but no kind of relationship was shown. In this article, we prove that G-spaces and M-spaces are close related. We also introduce here the concept of an L-space, which is inspired in the MC-spaces of J.V. Llinares [J.V. Llinares, Unified treatment of the problem of existence of maximal elements in binary relations: A characterization, J. Math. Econom. 29 (1998) 285-302], and establish relationships between the convexities of these spaces with the spaces previously mentioned.     

Compact convergence of σ-fields and relaxed conditional expectation

The collection of sub-σ-fields of a Borel measure space when endowed with the topology of strong convergence is in general not a compact space. The paper offers a completion of this space which makes it compact. The elements which are added to the space are called relaxed σ-fields. A notion of relaxed conditional expectation with respect to a relaxed σ-field is identified. The relaxed conditional expectation is a probability measure-valued map. It is shown that the conditional expectation operator is continuous on the completion of the space. Other properties of conditional expectation are lifted to and interpreted in the relaxed framework. Received: 22 February 1999 / Revised version: 23 October 2000 / Published online: 8 May 2001