ON THE SETS OF GATEAUX NON-DIFFERENTIABILITY OF LIPSCHITZ ISOMORPHISM BETWEEN BANACH SPACES＊t

We prove that for every Lipschitz isomorphism f from a separable Hilbert space H to a Banach space Y with Radon-Nikodym property, there is a bounded surjective ...     

Linear maps preserving zero products on nest subalgebras of von Neumann algebras

In this paper, it is proved that every surjective linear map preserving identity and zero products in both directions between two nest subalgebras with non-trivial nests of any factor von Neumann algebra is an isomorphism; and that every surjective weakly continuous linear map preserving identity and zero Jordan products in both directions between two nest subalgebras with non-trivial nests of any factor von Neumann algebra is either an isomorphism or an anti-isomorphism.     

A NOTE ON WEAKLY PRIMITIVE RINGS

It is well known that,for a subring of a full linear ring over a vector spaec,2-foldtransitive implies k-fold transitive for every natual integer k,and a primitive ring withminimal oneside ideal is a two side nonsingular ring and every isomorphism can be inducedby a semi-linear one to one transformation.This paper generalizes these results to weaklyprimitive rings.     

Towards a Ryll‐Nardzewski‐type theorem for weakly oligomorphic structures

A structure is called weakly oligomorphic if its endomorphism monoid has only finitely many invariant relations of every arity. The goal of this paper is to show that the notions of homomorphism‐homogeneity, and weak oligomorphy are not only completely analogous to the classical notions of homogeneity and oligomorphy, but are actually closely related. We first prove a Fraïssé‐type theorem for homomorphism‐homogeneous relational structures. We then show that the countable models of the theories of countable weakly oligomorphic structures are mutually homomorphism‐equivalent (we call first order theories with this property weakly ω‐categorical). Furthermore we show that every weakly oligomorphic homomorphism‐homogeneous structure contains (up to isomorphism) a unique homogeneous, homomorphism‐homogeneous core, to which it is homomorphism‐equivalent. As a consequence we obtain that every countable weakly oligomorphic structure is homomorphism‐equivalent to a finite or ω‐categorical structure. As a corollary we obtain a characterization of positive existential theories of weakly oligomorphic structures as the positive existential parts of ω‐categorical theories.     

Transfinite Diameter with Respect to the Neumann Function

In this paper, we study holomorphically isomorphic Abelian groups, i.e., Abelian groups with isomorphic holomorphs. We also study a generalization of the concept of holomorphic isomorphism, namely, almost holomorphic isomorphism, which is deeply connected with normal Abelian subgroups of holomorphs of Abelian groups. Torsion-free Abelian groups that are determined by their holomorphs are highlighted from different classes. In particular, it has been found that any homogeneous separable group can be determined by its holomorph in the class of all Abelian groups.     

Linearization of multipolynomials and applications

We prove that every multipolynomial between Banach spaces is the composition of a canonical multipolynomial with a linear operator, and that this correspondence establishes an isometric isomorphism between the spaces of multipolynomials and linear operators. Applications to composition ideals of multipolynomials and to multipolynomials that are of finite rank, approximable, compact, and weakly compact are provided.     

次可分与拟可分的Banach空间

Ｘ是Ｂａｎａｃｈ空间，对Ｘ的每一弱可分且弱紧的子集Ｋ，若（ｋ，ω）可度量，则称Ｘ是次可分的。如果对任意的可分，则称Ｘ是拟可分的。本文证明了这两类新引入的空间比可分空间与弱紧生成的Ｂａｎａｃｈ空间都要广泛并且保留了可分空间的许多重要性质。     

Finite separable metacyclic 2-groups

A finite separable metacyclic 2-group G can be written as the semidirect product of a cyclic group with another cyclic group. Necessary and sufficient conditions are given for when all other split decompositions of G result in, up to isomorphism, this same semidirect product representation for G.     

Isomorphisms of adjoint Chevalley groups over integral domains

It is shown that every automorphism of an adjoint Chevalley group over an integral domain containing the rational number field is uniquely expressible as the product of a ring automorphism, a graph automorphism and an inner automorphism while every isomorphism between simple adjoint Chevalley groups can be expressed uniquely as the product of a ring isomorphism, a graph isomorphism and an inner automorphism. The isomorphisms between the elementary subgroups are also found having analogous expressions.

     

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.     

Factoring weakly compact operators

The main result is that every weakly compact operator between Banach spaces factors through a reflexive Banach space. Applications of the result and technique of proof include new results (e.g., separable conjugate spaces embed isomorphically in spaces with boundedly complete bases; convex weakly compact sets are affinely homeomorphic to sets in a reflexive space) and simple proofs of known results (e.g., there is a reflexive space failing the Banach-Saks property; if X is separable, then $\text{X =}\text{Z}{}^7\text{Z}$ for some Z; there is a separable space which does not contain l₁ whose dual is nonseparable).     

Weakly continuously Urysohn spaces

We study weakly continuously Urysohn spaces, which were introduced in [P.L. Zenor, Continuously extending partial functions, Proc. Amer. Math. Soc. 135 (1) (2007) 305-312]. We show that every weakly continuously Urysohn wΔ-space has a base of countable order, that separable weakly continuously Urysohn spaces are submetrizable, hence continuously Urysohn, that monotonically normal weakly continuously Urysohn spaces are hereditarily paracompact, and that no linear extension of any uncountable subspace of the Sorgenfrey line is weakly continuously Urysohn. These results generalize various results in the literature concerning continuously Urysohn spaces.     

Simple algebraic and semialgebraic groups over real closed fields

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In Definably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

     

The group PSL(2,q) is not the multiplication group of a loop

ABSTRACT

We introduce the notion of weak transitivity for torsion-free abelian groups. A torsion-free abelian group G is called weakly transitive if for any pair of elements x, y ∈ G and endomorphisms ?, ψ ∈ End(G) such that x? = y, yψ = x, there exists an automorphism of G mapping x onto y. It is shown that every suitable ring can be realized as the endomorphism ring of a weakly transitive torsion-free abelian group, and we characterize up to a number-theoretical property the separable weakly transitive torsion-free abelian groups.     

On some algebraic properties of locally compact and weakly linearly compact topological Abelian groups

Locally compact and weakly linearly compact topological groups are studied. The notion of a weakly linearly compact topological Abelian group is a generalization of the notion of a weakly separable topological Abelian group, introduced by N. Ya. Vilenkin. Some algebraic properties of these groups are studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 41, Topology and Its Applications, 2006.     

Weak bimonoids in duoidal categories

Weak bimonoids in duoidal categories are introduced. They provide a common generalization of bimonoids in duoidal categories and of weak bimonoids in braided monoidal categories. Under the assumption that idempotent morphisms in the base category split, they are shown to induce weak bimonads (in four symmetric ways). As a consequence, they have four separable Frobenius base (co)monoids, two in each of the underlying monoidal categories. Hopf modules over weak bimonoids are defined by weakly lifting the induced comonad to the Eilenberg–Moore category of the induced monad. Making appropriate assumptions on the duoidal category in question, the fundamental theorem of Hopf modules is proven which says that the category of modules over one of the base monoids is equivalent to the category of Hopf modules if and only if a Galois-type comonad morphism is an isomorphism.     

Weakly quasi-o-minimal models

We introduce the notion of a weakly quasi-o-minimal model and prove that such models lack the independence property. We show that every weakly quasi-o-minimal ordered group is Abelian, every divisible Archimedean weakly quasi-o-minimal ordered group is weakly o-minimal, and every weakly o-minimal quasi-o-minimal ordered group is o-minimal. We also prove that every weakly quasi-o-minimal Archimedean ordered ring with nonzero multiplication is a real closed field that is embeddable into the field of reals.     

Hereditarily separable groups and monochromatic uniformization

We give a combinatorial equivalent to the existence of a non-free hereditarily separable group of cardinality ?₁. This can be used, together with a known combinatorial equivalent of the existence of a non-free Whitehead group, to prove that it is consistent that every Whitehead group is free but not every hereditarily separable group is free. We also show that the fact that ? is a p.i.d. with infinitely many primes is essential for this result.     

Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces

In this paper, we first show that for every mapping $f$ from a metric space $Ω$ to itself which is continuous off a countable subset of $Ω,$ there exists a nonempty closed separable subspace $S ⊂ Ω$ so that $f|_S$ is again a self mapping on $S.$ Therefore, both the fixed point property and the weak fixed point property of a nonempty closed convex set in a Banach space are separably determined. We then prove that every separable subspace of $c_0(\Gamma)$ (for any set $\Gamma$) is again lying in $c_0.$ Making use of these results, we finally presents a simple proof of the famous result: Every non-expansive self-mapping defined on a nonempty weakly compact convex set of $c_0(\Gamma)$ has a fixed point.     

On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.