Large sublattices in subsets of Banach lattices

A classical problem (initially studied by N. Kalton and A. Wilansky) concerns finding closed infinite dimensional subspaces of X / Y, where Y is a subspace of a Banach space X. We study the Banach lattice analogue of this question. For a Banach lattice X, we prove that X / Y contains a closed infinite dimensional sublattice under the following conditions: either (i) Y is a closed infinite codimensional subspace of X, and X is either order continuous or a C(K) space, where K is a compact subset of \({\mathbb {R}}^n\); or (ii) Y is the range of a compact operator.     

An Analog of Titchmarsh’s Theorem for the Fourier–Walsh Transform

The space clos(X) of all nonempty closed subsets of an unbounded metric space X is considered. The space clos(X) is endowed with a metric in which a sequence of closed sets converges if and only if the distances from these sets to a fixed point θ are bounded and, for any r, the sequence of the unions of the given sets with the exterior balls of radius r centered at θ converges in the Hausdorff metric. The metric on clos(X) thus defined is not equivalent to the Hausdorff metric, whatever the initial metric space X. Conditions for a set to be closed, totally bounded, or compact in clos(X) are obtained; criteria for the bounded compactness and separability of clos(X) are given. The space of continuous maps from a compact space to clos(X) is considered; conditions for a set to be totally bounded in this space are found.     

On the Dimension of Preimages of Certain Paracompact Spaces

It is proved that if X is a normal space which admits a closed fiberwise strongly zero-dimensional continuous map onto a stratifiable space Y in a certain class (an S-space), then IndX = dimX. This equality also holds if Y is a paracompact σ-space and ind Y = 0. It is shown that any closed network of a closed interval or the real line is an S-network. A simple proof of the Kateˇ tov–Morita inequality for paracompact σ-spaces (and, hence, for stratifiable spaces) is given.     

The Kreps-Yan theorem for Banach ideal spaces

Consider a closed convex cone C in a Banach ideal space X on some measure space with σ-finite measure. We prove that the fulfilment of the conditions C∩X ₊ = {0} and C??X ₊ guarantees the existence of a strictly positive continuous functional on X whose restriction to C is nonpositive.     

Dimension scales of bicompacta

We introduce the notion of a (stable) dimension scale d-sc(X) of a space X, where d is a dimension invariant. A bicompactum X is called dimensionally unified if dim F = dim_G F for every closed F ? X and for an arbitrary abelian group G. We prove that there exist dimensionally unified bicompacta with every given stable scale dim-sc.     

Embeddings of quotient division algebras of rings of differential operators

Let k be an algebraically closed field of characteristic zero, let X and Y be smooth irreducible algebraic curves over k, and let D(X) and D(Y) denote respectively the quotient division rings of the ring of differential operators of X and Y. We show that if there is a k-algebra embedding of D(X) into D(Y), then the genus of X must be less than or equal to the genus of Y, answering a question of the first-named author and Smoktunowicz.     

Symmetric function kernels and sweeping of measures

This is a potential theoretic study of balayage (sweeping) of a positive Radon measure ω on a locally compact (Hausdorff) space X onto a closed, or, more generally, a quasiclosed set A ? X (that is, a set which can be approximated in outer capacity by closed sets). The setting is that of potentials with respect to a suitable symmetric function kernel G: X × X → [0,+∞]. We consider energy capacity, not as a set function, but as a functional, acting on positive numerical functions on X. The finiteness of the upper capacity of the function 1 _A Gω is sufficient for the possibility of the sweeping in question (1 _A denoting the indicator function of A and Gω the G-potential of ω).     

Expansive homoclinic classes of generic <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-vector fields

Let γ be a hyperbolic closed orbit of a C ¹ vector field X on a compact C ^∞ manifold M of dimension n ≥ 3, and let H _X(γ) be the homoclinic class of X containing γ. In this paper, we prove that C ¹-generically, if H _X(γ) is expansive and isolated, then it is hyperbolic.     

On Symmetric Spaces With Convergence in Measure on Reflexive Subspaces

A closed subspace H of a symmetric space X on [0, 1] is said to be strongly embedded in X if in H the convergence in X-norm is equivalent to the convergence in measure. We study symmetric spaces X with the property that all their reflexive subspaces are strongly embedded in X. We prove that it is the case for all spaces, which satisfy an analogue of the classical Dunford–Pettis theorem on relatively weakly compact subsets in L₁. At the same time the converse assertion fails for a broad class of separableMarcinkiewicz spaces.     

Abelianization of the <Emphasis Type="Italic">F</Emphasis>-divided fundamental group scheme

Let (X , x ₀) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x ₀) produces a homomorphism from the abelianization of the F-divided fundamental group scheme of X to the F-divided fundamental group of the Albanese variety of X. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.     

Strange products of projections

Let H be an infinite-dimensional Hilbert space. We show that there exist three orthogonal projections X ₁,X ₂,X ₃ onto closed subspaces of H such that for every 0 ≠ z ₀ ∈ H there exist k ₁, k ₂, · · · ∈ {1, 2, 3} so that the sequence of iterates defined by z _n = X _kn z _n ?1 does not converge in norm.     

A categorical approach to absolute closure

Notions of strongly and absolutely closed objects with respect to a closure operator X on an arbitrary category X and with respect to a subcategory Y are introduced. This yields two Galois connections between closure operators on a given category X and subclasses of X, whose fixed points are studied. A relationship with some compactness notions is shown and examples are provided.     

Functional envelope of Cantor spaces

Given a topological dynamical system(X, T), where X is a compact metric space and T a continuous selfmap of X. Denote by S(X) the space of all continuous selfmaps of X with the compactopen topology. The functional envelope of(X, T) is the system(S(X), FT), where FT is defined by FT(?) = T ? ? for any ? ∈ S(X). We show that(1) If(Σ, T) is respectively weakly mixing, strongly mixing, diagonally transitive, then so is its functional envelope, where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(2) If(S(Σ), F_σ) is transitive then it is Devaney chaos, where(Σ, σ) is a subshift of finite type;(3) If(Σ, T) has shadowing property, then(SU(Σ), FT) has shadowing property,where Σ is any closed subset of a Cantor set and T a selfmap of Σ;(4) If(X, T) is sensitive, where X is an interval or any closed subset of a Cantor set and T : X → X is continuous, then(SU(X), FT) is sensitive;(5) If Σ is a closed subset of a Cantor set with infinite points and T : Σ→Σ is positively expansive then the entropy ent U(FT) of the functional envelope of(Σ, T) is infinity.     

On a Variety Related to the Commuting Variety of a Reductive Lie Algebra

For a reductive Lie algbera over an algbraically closed field of charasteristic zero, we consider a Borel subgroup B of its adjoint group, a Cartan subalgebra contained in the Lie algebra of B and the closure X of its orbit under B in the Grassmannian. The variety X plays an important role in the study of the commuting variety. In this note, we prove that X is Gorenstein with rational singularities.     

Variations on a theorem of Arhangel’skii and Pytkeev

We identify continuous real-valued functions on a Tychonoff space X with their (closed) graphs thus allowing for C(X) to naturally inherit the lower Vietoris topology from the ambient hyperspace. We then calculate a bitopological version of tightness using the weak Lindelöf numbers of finite powers of X. We also characterize bitopological versions of countable fan and strong fan tightness of the point-open topology with respect to the lower Vietoris topology on C(X) in terms of suitable covering properties of the powers X ⁿ formulated using the language of S ₁ and S _fin selection principles.     

Monge–Ampère equations in big cohomology classes

We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure μ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that μ can be written in a unique way as the top-degree self-intersection in the non-pluripolar sense of a closed positive current in α. We then extend Kolodziedj’s approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if μ has L ^1+ε -density with respect to Lebesgue measure. If μ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem, we finally explain how to construct singular Kähler–Einstein volume forms with minimal singularities on varieties of general type.     

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori, Ⅱ

We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S~2×S~1 or RP~3#RP~3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni(2014) in an earlier paper.     

Relative symplectic caps, 4-genus and fibered knots

We prove relative versions of the symplectic capping theorem and sufficiency of Giroux’s criterion for Stein fillability and use these to study the 4-genus of knots. More precisely, suppose we have a symplectic 4-manifold X with convex boundary and a symplectic surface Σ in X such that ?Σ is a transverse knot in ?X. In this paper, we prove that there is a closed symplectic 4-manifold Y with a closed symplectic surface S such that (X,Σ) embeds into (Y,S) symplectically. As a consequence we obtain a relative version of the symplectic Thom conjecture. We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in \(\mathbb {S}^{3} \). Further, we give a criterion for quasipositive fibered knots to be strongly quasipositive.     

On the stability of tangent bundle on double coverings

Let Y be a smooth projective surface defined over an algebraically closed field k with char k≠2, and let π : X →Y be a double covering branched along a smooth divisor. We show that y_X is stable with respect to π~*H if the tangent bundle y_Y is semi-stable with respect to some ample line bundle H on Y.