Collectionwise weak continuity duals

A function f: (X, τ) → (Y, σ) is weakly collectionwise continuous if for some C ? 2 ^X with τ ? C we have f ^?1(V) ∈ C for each V ∈ σ. In this case, f is said to be C-continuous. If also τ ? C* ? 2 ^X , C*-continuity is a dual to C-continuity if C?C* = τ and then the pair (C-continuity, C*-continuity) is a decomposition of continuity. In this paper, two natural topological methods are found for construction of a dual to any collectionwise weak continuity. Some known decompositions are improved.     

Über die multiplikative Funktionalgleichung für stochastische Matrizen

Letγ(n, R) andγ ^* (n, R) stand for the semigroups of all stochastic and all invertible stochasticn × n real matrices respectively, and letK be a commutative field. The following theorem is proved in the paper. Every functionf :γ(n, R) → K orf :γ ^* (n, R) → K satisfying the multiplicative functional equationf(X)f(Y) = f(XY) is of the formf(X) = ψ(detX), whereψ: 〈?1, 1〉 → K orψ: 〈?1, 0) ∪ (0, 1〉 → →K, respectively, is an arbitrary multiplicative function. The concept of pseudostochastic matrix \(\left( {\sum\limits_{j = 1}^n {a_{ij} = 1} } \right)\) and the canonical form of such a matrix derived in [3] are used.     

Spectral Synthesis and Topologies on Ideal Spaces for Banach*-Algebras

This paper continues the study of spectral synthesis and the topologies τ_∞ and τ_r on the ideal space of a Banach algebra, concentrating on the class of Banach ^*-algebras, and in particular on L¹-group algebras. It is shown that if a group G is a finite extension of an abelian group then τ_r is Hausdorff on the ideal space of L¹(G) if and only if L¹(G) has spectral synthesis, which in turn is equivalent to G being compact. The result is applied to nilpotent groups, [FD]⁻-groups, and Moore groups. An example is given of a non-compact, non-abelian group G for which L¹(G) has spectral synthesis. It is also shown that if G is a non-discrete group then τ_r is not Hausdorff on the ideal lattice of the Fourier algebra A(G).     

Embedding paratopological groups into topological products

We show that a Hausdorff paratopological group G admits a topological embedding as a subgroup into a topological product of Hausdorff first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the Hausdorff number of G is countable, i.e., for every neighbourhood U of the neutral element e of G there exists a countable family γ of neighbourhoods of e such that _?V∈γVV−1⊆U. Similarly, we prove that a regular paratopological group G can be topologically embedded as a subgroup into a topological product of regular first-countable (second-countable) paratopological groups if and only if G is ω-balanced (totally ω-narrow) and the index of regularity of G is countable.As a by-product, we show that a regular totally ω-narrow paratopological group with countable index of regularity is Tychonoff.     

Verallgemeinerungen eines Satzes von H. Steinhaus

The following result is due to H. Steinhaus [20]: “If A,B?R are sets of positive inner Lebesgue measure and if the function f: R x R→R is defined by f(x,y):=x+y (x,y?R), then the interior of f(A x B) is non void”. In this note there is proved, that the theorem of H. Steinhaus remains valid, if

1. R is replaced by certain topological measure spaces X, Y and a Hausdorff space Z,
2. f is a continuous function from an open set T?X x Y into Z and satisfies a special local (respectively global) solvability condition in T,
3. A?X is a set of positive outer measure, B?Y contains a set of positive measure and A x B?T.

     

Counting maximal topologies on countable groups and rings

Two non-discrete Hausdorff group topologies τ₁, τ₂ on a group G are called transversal if the least upper bound τ₁∨τ₂ of τ₁ and τ₂ is the discrete topology. We show that a countable group G admitting non-discrete Hausdorff group topologies admits c² pairwise transversal complete group topologies on G (so c² maximal group topologies). Moreover, every abelian group G admits ²|G|² pairwise transversal complete group topologies.     

Characterization and properties of extreme operators intoC(Y)

LetE be a real (or complex) Banach space,Y a compact Hausdorff space, andC(Y) the space of real (or complex) valued continuous functions onY. IfT is an extreme point in the unit ball of bounded linear operators fromE intoC(Y), then it is shown thatT ^* maps (the natural imbedding inC(Y) ^* of)Y into the weak ^*-closure of extS(E ^*), provided thatY is extremally disconnected, orE=C(X), whereX is a dispersed compact Hausdorff space.     

On the reconstruction problem for factorizable homeomorphism groups and foliated manifolds

For a group G of homeomorphisms of a regular topological space X and a subset U⊆X, set . We say that G is a factorizable group of homeomorphisms, if for every open cover U of X, _?U∈UG generates G.

Theorem I. Let G, H be factorizable groups of homeomorphisms of X and Y respectively, and suppose that G, H do not have fixed points. Let φ be an isomorphism between G and H. Then there is a homeomorphism τ between X and Y such thatφ(g)=τ○g○τ−1for everyg∈G.     

Universal G-spaces for proper actions of locally compact groups

We establish the existence of universal G-spaces for proper actions of locally compact groups on Tychonoff spaces. A typical result sounds as follows: for each infinite cardinal number τ every locally compact, non-compact, σ-compact group G of weight w(G)?τ, can act properly on Rτ?{0} such that Rτ?{0} contains a G-homeomorphic copy of every Tychonoff proper G-space of weight ?τ. The metric cones Cone(G/H) with H⊂G a compact subgroup such that G/H is a manifold, are the main building blocks in our approach. As a byproduct we prove that the cardinality of the set of all conjugacy classes of such subgroups H⊂G does not exceed the weight of G.     

Closed structures on categories of topological spaces

The category of all topological spaces and continuous maps and its full subcategory of all T_o-spaces admit (up to isomorphism) precisely one structure of symmetric monoidal closed category (see [2]). In this paper we shall prove the same result for any epireflective subcategory of the category of topological spaces (particularly e.g. for the categories of Hausdorff spaces, regular spaces, Tychonoff spaces).     

Locally minimal topological groups 1

The aim of this paper is to go deeper into the study of local minimality and its connection to some naturally related properties. A Hausdorff topological group (G,τ) is called locally minimal if there exists a neighborhood U of 0 in τ such that U fails to be a neighborhood of zero in any Hausdorff group topology on G which is strictly coarser than τ. Examples of locally minimal groups are all subgroups of Banach-Lie groups, all locally compact groups and all minimal groups. Motivated by the fact that locally compact NSS groups are Lie groups, we study the connection between local minimality and the NSS property, establishing that under certain conditions, locally minimal NSS groups are metrizable. A symmetric subset of an abelian group containing zero is said to be a GTG set if it generates a group topology in an analogous way as convex and symmetric subsets are unit balls for pseudonorms on a vector space. We consider topological groups which have a neighborhood basis at zero consisting of GTG sets. Examples of these locally GTG groups are: locally pseudoconvex spaces, groups uniformly free from small subgroups (UFSS groups) and locally compact abelian groups. The precise relation between these classes of groups is obtained: a topological abelian group is UFSS if and only if it is locally minimal, locally GTG and NSS. We develop a universal construction of GTG sets in arbitrary non-discrete metric abelian groups, that generates a strictly finer non-discrete UFSS topology and we characterize the metrizable abelian groups admitting a strictly finer non-discrete UFSS group topology. Unlike the minimal topologies, the locally minimal ones are always available on “large” groups. To support this line, we prove that a bounded abelian group G admits a non-discrete locally minimal and locally GTG group topology iff |G|?c.     

A characterization of linearly semisimple groups

Let G = SpecA be an affine K-group scheme and à = {w ∈ A*: dim _K A*·w < ∞, dim _K w· A* < ∞}. Let 〈?,?〉: A* × Ã → K, 〈w, \(\tilde w\)〉:=tr(w~w), be the trace form. We prove that G is linearly reductive if and only if the trace form is non-degenerate on A*.     

Certain classes of sets in Banach spaces and a topological characterization of operators of type RN

We study properties of bounded sets in Banach spaces, connected with the concept of equimeasurability introduced by A. Grothendieck. We introduce corresponding ideals of operators and find characterizations of them in terms of continuity of operators in certain topologies. The following result (Corollary 9) follows from the basic theorems: Let T be a continuous linear operator from a Banach space X to a Banach space Y. The following assertions are equivalent:

1. T is an operator of type RN;
2. for any Banach space Z, for any number p, p > 0, and any p-absolutely summing operator U:Z → X the operator TU is approximately p-Radonifying;
3. for any Banach space Z and any absolutely summing operator U:Z → X the operator TU is approximately 1-Radonifying.

We note that the implication I)?2), is apparently new even if the operator T is weakly compact.     

Fixed point theory and product,sub-, super-spaces

In this paper, we definen-segmentwise metric spaces and then we prove the following results:

1. (i)|Let (X, d) be ann-segmentwise metric space. ThenX ⁿ has the fixed point property with respect to uniformly continuous bounded functions if and only if, for any continuous functionF: C ^*(X) → C^*(X) and for anyn-tuple of distinct points x₁, x₂, ?, x_n ∈X, there exists anh ∈C ^*(X) such that $$F(h)(x_1 ) = h(x_1 ),i = 1,2,...,n;$$ whereC ^*(X) has either the uniform topology or the subspace product (Tychonoff) topology \((C^ * (X) \subseteq X^X )\) .
2. LetX _i (i = 1, 2, ?) be countably compact Hausdorff spaces such thatX ₁ × ? × X_n has the fixed point property for alln ∈N Then the product spaceX ₁ × X₂ × ? has the fixed point property. We shall also discuss several problems in the Fixed Point Theory and give examples if necessary. Among these examples, we have:
3. There exists a connected metric spaceX which can be decomposed as a disjoint union of a closed setA and an open setB such thatA andB have the fixed point property andX does not have.
4. There exists a locally compact metrizable spaceX which has the fixed point property but its one-point compactificationX ⁺ does not have the fixed point property.

Other relevant results and examples will be presented in this paper.     

On operator monotone and operator convex functions

We establish monotonicity and convexity criteria for a continuous function f: R⁺ → R with respect to any C*-algebra. We obtain an estimate for the measure of noncompactness of the difference of products of the elements of a W*-algebra. We also give a commutativity criterion for a positive τ-measurable operator and a positive operator from a von Neumann algebra.     

Remarks on absolutely star countable spaces

We prove the following statements: (1) every Tychonoff linked-Lindelöf (centered-Lindelöf, star countable) space can be represented as a closed subspace in a Tychonoff pseudocompact absolutely star countable space; (2) every Hausdorff (regular, Tychonoff) linked-Lindelöf space can be represented as a closed G _δ-subspace in a Hausdorff (regular, Tychonoff) absolutely star countable space; (3) there exists a pseudocompact absolutely star countable Tychonoff space having a regular closed subspace which is not star countable (hence not absolutely star countable); (4) assuming $2^{\aleph _0 } = 2^{\aleph _1 }$ , there exists an absolutely star countable normal space having a regular closed subspace which is not star countable (hence not absolutely star countable).     

Parabolic Singular Integrals in Probability Theory

The final aim of this work is to prove the Central Limit Theorem described in the motivations given below. The key for that is a Resolvant estimate, of the type of Theorem 1.1 in [21], adapted for the Parabolic Green function G(X, Y) which is the heat diffusion kernel in some domain Ω in time-space: i.e. we must estimate ${\int_{\Omega}\nabla_{Y}G(X, Y)\nabla_{Y}^{2}G(Y,Z)\;dY}The final aim of this work is to prove the Central Limit Theorem described in the motivations given below. The key for that is a Resolvant estimate, of the type of Theorem 1.1 in [21], adapted for the Parabolic Green function G(X, Y) which is the heat diffusion kernel in some domain Ω in time-space: i.e. we must estimate ò_W?_YG(X, Y)?_Y²G(Y,Z)  dY{\int_{\Omega}\nabla_{Y}G(X, Y)\nabla_{Y}^{2}G(Y,Z)\;dY}. Exactly as the estimate in [21] is based on [10] our estimate here is based on the main Theorem of this paper. This main theorem refers to rough singular integrals on the Gaussian potential on ∂Ω.     

Gysin maps and cycle classes for Hodge cohomology

Iff:X→Y is a projective morphism between regular varieties over a field, we construct Gysin maps $$f_ * :H^i \left( {X,\Omega _{X/Z}^j } \right) \to H_{f(x)}^{i + d} \left( {X,\Omega _{Y/Z}^j } \right)$$ for the Hodge cohomology groups, whered-dimY-dimX. These Gysin maps have the expected properties, and in particular may be used to construct a cycle class map $$Cl_X :CH^i \left( {X,S} \right) \to H^i \left( {X,\Omega _{X/Z}^i } \right)$$ whereX is quasi-projective over a field,S is the singular locus, andCH ⁱ(X, S) is the relative Chow group of codimension-i cycles modulo rational equivalence. Simple properties of this cycle map easily imply the infinite dimensionality theorem for the Chow group of zero cycles of a normal projective varietyX overC with \(H^n \left( {X,\mathcal{O}_X } \right) \ne 0\) , wheren=dimX. One also recovers examples of Nori of affinen-dimensional varieties which support indecomposable vector bundles of rankn.     

自反代数的环自同构和环反自同构

设A为Banach空间X中一自反代数使得在LatA中O ≠0且X_≠X,则A的每一环自同构￠（环反自同构φ）具有形式￠（A）=TAT^-1(φ（A）=TA^*T^-1),其中T：X→X（T：X^*→X）或为一有界线性双射算子或为一有界共轭线性性双射算子。特别地,￠和φ都是连续的。