Topological vector spaces, compacta, and unions of subspaces

In the first two sections, we study when a σ-compact space can be covered by a point-finite family of compacta. The main result in this direction concerns topological vector spaces. Theorem 2.4 implies that if such a space L admits a countable point-finite cover by compacta, then L has a countable network. It follows that if f is a continuous mapping of a σ-compact locally compact space X onto a topological vector space L, and fibers of f are compact, then L is a σ-compact space with a countable network (Theorem 2.10). Therefore, certain σ-compact topological vector spaces do not have a stronger σ-compact locally compact topology.In the last, third section, we establish a result going in the orthogonal direction: if a compact Hausdorff space X is the union of two subspaces which are homeomorphic to topological vector spaces, then X is metrizable (Corollary 3.2).     

On spaces in which countably compact sets are closed, and hereditary properties

A space X is called C-closed if every countably compact subset of X is closed in X. We study the properties of C-closed spaces. Among other results, it is shown that countably compact C-closed spaces have countable tightness and under Martin's Axiom or 2^ω0<2^ω1, C-closed is equivalent to sequential for compact Hausdorff spaces. Furthermore, every hereditarily quasi-k Hausdorff space is Fréchet-Urysohn, which generalizes a theorem of Arhangel'sk in [4]. Also every hereditarily q-space is hereditarily of pointwise countable type and contains an open dense first countable subspace.     

On smooth, nonlinear surjections of banach spaces

It is shown that (1) every infinite-dimensional Banach space admits aC ¹ Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC ^∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces. Supported by an NSF Postdoctoral Fellowship in Mathematics.     

A full-invariant theorem and some applications

Let (X,τ₁) and (Y,τ₂) be two Hausdorff locally convex spaces with continuous duals X′ and Y′, respectively, L(X,Y) be the space of all continuous linear operators from X into Y, K(X,Y) be the space of all compact operators of L(X,Y). Let WOT and UOT be the weak operator topology and uniform operator topology on K(X,Y), respectively. In this paper, we characterize a full-invariant property of K(X,Y); that is, if the sequence space λ has the signed-weak gliding hump property, then each λ-multiplier WOT-convergent series ∑_iT_i in K(X,Y) must be λ-multiplier convergent with respect to all topologies between WOT and UOT if and only if each continuous linear operator T :(X,τ₁)→(λ^β,σ(λ^β,λ)) is compact. It follows from this result that the converse of Kalton's Orlicz–Pettis theorem is also true.     

A generalization of a theorem on biseparating maps

In J. Math. Anal. Appl. 12 (1995) 258–265, Araujo et al. proved that for any linear biseparating map  from C(X) onto C(Y), where X and Y are completely regular, there exist ω in C(Y) and an homeomorphism h from the realcompactification vX of X onto vY, such that

The compact version of this result was proved before by Jarosz in Bull. Canad. Math. Soc. 33 (1990) 139–144. In Contemp. Math., Vol. 253, 2000, pp. 125–144, Henriksen and Smith asked to what extent the result above can be generalized to a larger class of algebras. In the present paper, we give an answer to that question as follows. Let A and B be uniformly closed Φ-algebras. We first prove that every order bounded linear biseparating map from A onto B is automatically a weighted isomorphism, that is, there exist ω in B and a lattice and algebra isomorphism ψ between A and B such that

(a)=ωψ(a) for all aA.

We then assume that every universally σ-complete projection band in A is essentially one-dimensional. Under this extra condition and according to a result from Mem. Amer. Math. Soc. 143 (2000) 679 by Abramovich and Kitover, any linear biseparating map from A onto B is automatically order bounded and, by the above, a weighted isomorphism. It turns out that, indeed, the latter result is a generalization of the aforementioned theorem by Araujo et al. since we also prove that every universally σ-complete projection band in the uniformly closed Φ-algebra C(X) is essentially one-dimensional.     

A theorem of Riesz type with Pettis integrals in topological vector spaces

Let X be a locally convex Hausdorff space and let C₀(S,X) be the space of all continuous functions f:S→X, with compact support on the locally compact space S. In this paper we prove a Riesz representation theorem for a class of bounded operators T:C₀(S,X)→X, where the representing integrals are X-valued Pettis integrals with respect to bounded signed measures on S. Under the additional assumption that X is a locally convex space, having the convex compactness property, or either, X is a locally convex space whose dual X^′ is a barrelled space for an appropriate topology, we obtain a complete identification between all X-valued Pettis integrals on S and the bounded operators T:C₀(S,X)→X they represent. Finally we give two illustrations of the representation theorem proved, in the particular case when X is the topological dual of a locally convex space.     

Rank-One Isometries of Buildings and Quasi-Morphisms of Kac–Moody Groups

Given an irreducible non-spherical non-affine (possibly non-proper) building X, we give sufficient conditions for a group G < Aut(X) to admit an infinite-dimensional space of non-trivial quasi-morphisms. The result applies in particular to all irreducible (non-spherical and non-affine) Kac–Moody groups over integral domains. In particular, we obtain finitely presented simple groups of infinite commutator width, thereby answering a question of Valerii G. Bardakov [MK, Prob. 14.13]. Independently of these considerations, we also include a discussion of rank-one isometries of proper CAT(0) spaces from a rigidity viewpoint. In an appendix, we show that any homogeneous quasi-morphism of a locally compact group with integer values is continuous.     

Some cardinal invariants on the space

Let C_α(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X such that closed under finite unions. We define two properties (E1) and (E2) on the triple (α,X,Y) which yield new equalities and inequalities between some cardinal invariants on C_α(X,Y) and some cardinal invariants on the spaces X, Y such as: Theorem If Y is an equiconnected space with a base consisting of φ-convex sets, then for each fC(X,Y), χ(f,C_α(X,Y))=αa(X).w_e(f(X)).Corollary Let Y be a noncompact metric space and let the triple (α,X,Y) satisfy (E1). The following are equivalent:

(i) C_α(X,Y) is a first-countable space.

(ii) π-character of the space C_α(X,Y) is countable.

(iii) C_α(X,Y) is of pointwise countable type.

(iv) There exists a compact subset K of C_α(X,Y) such that π-character of K in the space C_α(X,Y) is countable.

(v) αa(X)₀.

(vi) C_α(X,Y) is metrizable.

(vii) C_α(X,Y) is a q-space.

(viii) There exists a sequence of nonempty open subset of C_α(X,Y) such that each sequence with g_nO_n for each nω, has a cluster point in C_α(X,Y).

Elementarity and Dimensions

An alternative proof of Fedorchuk’s recent result that dim X ≤ Dg X for compact Hausdorff spaces X is given. The problem is reduced to the metric case by using the Lowenheim-Skolem theorem.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 292–298.Original Russian Text Copyright © 2005 by K. P. Hart.     

A note on approximation of semigroups of contractions on Hilbert spaces

We show that every contractive C ₀-semigroup on a separable, infinite-dimensional Hilbert space X can be approximated by unitary C ₀-groups in the weak operator topology uniformly on compact subsets of ℝ₊. As a consequence we get a new characterization of a bounded H ^∞-calculus for the negatives of generators of bounded holomorphic semigroups. Applications of our results to the study of a topological structure of the set of (almost) weakly stable contractive C ₀-semigroups on X are also discussed. The author was partially supported by the Marie Curie “Transfer of Knowledge” programme, project “TODEQ”, and by a MNiSzW grant Nr. N201384834.     

Strong q-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces

The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space ${\Upsilon\colon\widehat X\to X}The main result is that, for any projective compact analytic subset Y of dimension q > 0 in a reduced complex space X, there is a neighborhood Ω of Y such that, for any covering space U\colon[^(X)]? X{\Upsilon\colon\widehat X\to X} in which [^(Y)] o U^-1(Y){\widehat Y\equiv\Upsilon^{-1}(Y)} has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components, there exists a C ^∞ exhaustion function j{\varphi} on [^(X)]{\widehat X} which is strongly q-convex on [^(W)]=U^-1(W){\widehat\Omega=\Upsilon^{-1}(\Omega)} outside a uniform neighborhood of the q-dimensional compact irreducible components of [^(Y)]{\widehat Y}.     

Approximative compactness and continuity of metric projector in Banach spaces and applications

First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T~ ,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.     

Correlations of functions of normal variables

Let (X₁, X₂,…, X_k, Y₁, Y₂,…, Y_k) be multivariate normal and define a matrix C by C_ij = cov(X_i, Y_j). If (i) (X₁,…, X_k) = (Y₁,…, Y_k) and (ii) C is symmetric positive definite, then 0 < varf(X₁,…, X_k) < ∞ corr(f(X₁,…, X_k),f(Y₁,…, Y_k)) > 0. Condition (i) is necessary for the conclusion. The sufficiency of (i) and (ii) follows from an infinite-dimensional version, which can also be applied to a pair of jointly normal Brownian motions.     

Lineability of sets of nowhere analytic functions

Although the set of nowhere analytic functions on [0,1] is clearly not a linear space, we show that the family of such functions in the space of C^∞-smooth functions contains, except for zero, a dense linear submanifold. The result is even obtained for the smaller class of functions having Pringsheim singularities everywhere. Moreover, in spite of the fact that the space of differentiable functions on [0,1] contains no closed infinite-dimensional manifold in C([0,1]), we prove that the space of real C^∞-smooth functions on (0,1) does contain such a manifold in C((0,1)).     

Completeness properties of the generalized compact-open topology on partial functions with closed domains

The primary goal of the paper is to investigate the Baire property and weak α-favorability for the generalized compact-open topology τ_C on the space of continuous partial functions f :A→Y with a closed domain AX. Various sufficient and necessary conditions are given. It is shown, e.g., that is weakly α-favorable (and hence a Baire space), if X is a locally compact paracompact space and Y is a regular space having a completely metrizable dense subspace. As corollaries we get sufficient conditions for Baireness and weak α-favorability of the graph topology of Brandi and Ceppitelli introduced for applications in differential equations, as well as of the Fell hyperspace topology. The relationship between τ_C, the compact-open and Fell topologies, respectively is studied; moreover, a topological game is introduced and studied in order to facilitate the exposition of the above results.     

On algebraic K-theory of real algebraic varieties with circle action

Assume that X is a compact connected orientable nonsingular real algebraic variety with an algebraic free S¹-action so that the quotient Y=X/S¹ is also a real algebraic variety. If π : X→Y is the quotient map then the induced map between reduced algebraic K-groups, tensored with ,

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is onto, where , denoting the ring of entire rational (regular) functions on the real algebraic variety X, extending partially the Bochnak–Kucharz result that

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for any real algebraic variety X. As an application we will show that for a compact connected Lie group G .     

Universal spaces for the subdifferentials of sublinear operators with values in the spaces of continuous functions

Given a continuous sublinear operator P: V → C(X) from a Hausdorff separable locally convex space V to the Banach space C(X) of continuous functions on a compact set X we prove that the subdifferential ∂P at zero is operator-affinely homeomorphic to the compact subdifferential ∂ ^c Q, i.e., the subdifferential consisting only of compact linear operators, of some compact sublinear operator Q: ł² → C(X) from a separable Hilbert space ł², where the spaces of operators are endowed with the pointwise convergence topology. From the topological viewpoint, this means that the space L ^c (ł², C(X)) of compact linear operators with the pointwise convergence topology is universal with respect to the embedding of the subdifferentials of sublinear operators of the class under consideration.     

Composition Operators and Vector-valued BMOA

Analytic composition operators are studied on X-valued versions of BMOA, the space of analytic functions on the unit disk that have bounded mean oscillation on the unit circle, where X is a complex Banach space. It is shown that if X is reflexive and C _φ is compact on BMOA, then C _φ is weakly compact on the X-valued space BMOA _C (X) defined in terms of Carleson measures. A related function-theoretic characterization is given of the compact composition operators on BMOA.     

Remarkable Classes of Unital AM-Spaces

We define and investigate two classes of unital Banach AM-spaces, the elements of which are the sums of continuous functions and discrete functions. Neither class is almost Dedekind σ-complete, although one has the Cantor property. One class has the rather rare property of having a sequentially order continuous norm and we deduce that any C(K) space can be embedded as a sublattice of a C(X) space with a sequentially order continuous norm. Finally we identify the order continuous and sequentially order continuous duals of spaces in these classes, which promise to be a rich source of further examples.     

Convergence of best approximations in a smooth Banach space

Let X be a reflexive, strictly convex Banach space such that both X and X^* have Fréchet differentiable norms, and let {C_n} be a sequence of non-empty closed convex subsets of X. We prove that the sequence of best approximations {p(x ¦ C_n)} of any x ε X converges if and only if lim C_n exists and is not empty. We also discuss measurability of closed convex set valued functions.