有限维张量积空间上的强可分算子

设F是一个域,a∈F~nF~m.若存在h∈F~m,k∈F~m,使得a=hk,则称a是可分的.空间F~nF~m上的线性算子A称为是强可分的,是指x∈F~nF~m,x可分Ax可分.本文证明了F~nF~n上的线性算子A是强可分的当且仅当存在F~n上的线性双射A_1与A_2,使得A=A_1A_2或A=A_1~T A_2;证明了F~nF~m(n≠m)上线性算子A是强可分的当且仅当存在F~n与F~m上的线性双射A_1与A_2,使得A=A_1A_2.最后,给出了可分算子、强可分算子和秩1保持映射之间的关系.     

一致覆盖和度量空间的紧映象

本文利用一致覆盖的概念,讨论了度量空间的序列覆盖紧映象的结构.主要结果有: (1)空间X是局部可分度量空间的序列覆盖紧映象当且仅当X具有由cosmic子空间构成的一致sn网; (2)空间X是局部可分度量空间的序列覆盖,商紧映象当且仅当X是度量空间的序列覆盖,商紧映象且是局部cosmic空间.     

局部可分度量空间闭s映象的注记

该文讨论局部可分度量空间闭s映象的分解定理, 证明了正则的Fréchet空间是局部可分度量空间的闭s映象当且仅当满足如下条件: 具有点可数的cs*网, 第一可数的闭子空间是局部可分的, 且Lindelof的闭子空间是可分的.     

σ-Porosity is separably determined

We prove a separable reduction theorem for σ-porosity of Suslin sets. In particular, if A is a Suslin subset in a Banach space X, then each separable subspace of X can be enlarged to a separable subspace V such that A is σ-porous in X if and only if A∩V is σ-porous in V. Such a result is proved for several types of σ-porosity. The proof is done using the method of elementary submodels, hence the results can be combined with other separable reduction theorems. As an application we extend a theorem of L. Zají?ek on differentiability of Lipschitz functions on separable Asplund spaces to the nonseparable setting.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

Sigma-compact,nowhere locally compact metric spaces can be densely imbedded in Hilbert space

It is shown that if H is a connected, locally contractible, separable, topologically complete metric space with the property that mappings of separable metric spaces into H are approximable by imbeddings (in particular, if H is Hilbert space), then every sigma-compact, nowhere locally compact metric space can be densely imbedded in H.     

Quasi-diagonal C1-algebras

We consider perturbations of C¹-algebras by compact operators. We show that if A is a separable liminal algebra of operators on a separable Hilbert space, then it is a subalgebra of a compact perturbation of a block diagonal algebra.     

Metrization and Semimetrization Theorems with Applications to Manifolds

This paper is a study of conditions under which a space with S₂ is metrizable, o-semimetrizable or semimetrizable. It is shown that: a wMN, w-space is metrizable if and only if it has S₂, a quasi--space is metrizable if and only if it is a pseudo wN-space with S₂, a separable manifold is metrizable if and only if it has S₂ with property (*), a perfectly normal manifold with q quasi-G ^* diagonal is metrizable and a separable manifold is a hereditarily separable metrizable if and only if it has -S₂.     

Bases of rearrangement invariant spaces

We prove that if E is a rearrangement-invariant space, then a boundedly complete basis exists in E, if and only if one of the following conditions holds: 1) E is maximal and E ≠ L ₁[0, 1]; 2) a certain (any) orthonormal system of functions from L _∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L _∞, represents a boundedly complete basis in E. As a corollary, we state the following assertion: Any (certain) orthonormal system of functions from L _∞[0, 1], possessing the properties of the Schauder basis for the space of continuous on [0, 1] functions with the norm L _∞, represents a spanning basis in a separable rearrangement-invariant space E, if and only if the adjoint space E* is separable. We prove that in any separable rearrangement-invariant space E the Haar system either forms an unconditional basis, or a strongly conditional one. The Haar system represents a strongly conditional basis in a separable rearrangement-invariant space, if and only if at least one of the Boyd indices of this space is trivial.     

迹稳定秩一的C~*-代数

本文引入了一类迹稳定秩一的C*-代数,证明了迹稳定秩一的C*-代数与AF-代数的张量积是迹稳定秩一的,得到了一个可分的单的有单位元的迹稳定秩一的,并且具有SP性质的C*-代数是稳定秩一的.同时,还讨论了迹稳定秩一的C*-代数的K-群的某些性质.     

Sequential + separable vs sequentially separable and another variation on selective separability

A space X is sequentially separable if there is a countable D ? X such that every point of X is the limit of a sequence of points from D. Neither “sequential + separable” nor “sequentially separable” implies the other. Some examples of this are presented and some conditions under which one of the two implies the other are discussed. A selective version of sequential separability is also considered.     

On connection between the switching separability of a graph and its subgraphs

A graph of order n ≥ 4 is called switching separable if its modulo-2 sum with some complete bipartite graph on the same set of vertices is divided into two mutually independent subgraphs, each having at least two vertices. We prove the following: If removing any one or two vertices of a graph always results in a switching separable subgraph then the graph itself is switching separable. On the other hand, for each odd order greater than 4, there is a graph that is not switching separable, but removing a vertex always results in a switching separable subgraph. We show some connection with similar facts on the separability of Boolean functions and the reducibility of n-ary quasigroups.     

A characterization of dual Banach lattices

In this paper we give a characterization of dual Banach lattices. In fact, we prove that a Banach function space E on a separable measure space which has the Fatou property is a dual Banach lattice if and only if all positive operators from L¹(0,1) into E are abstract kernel operators, hence extending the fact, proved by M. Talagrand, that separable Banach lattices with the Radon-Nikodym property are dual Banach lattices.     

Characterizations of intervals via continuous selections

We prove that: (i) a pathwise connected, Hausdorff space which has a continuous selection is homeomorphic to one of the following four spaces: singleton, [0,1), [0,1] or the long lineL, (ii) a locally connected (Hausdorff) space which has a continuous selection must be orderable, and (iii) an infinite connected, Hausdorff space has exactly two continuous selections if and only if it is compact and orderable. We use these results to give various characterizations of intervals via continuous selections. For instance, (iv) a topological spaceX is homeomorphic to [0,1] if (and only if)X is infinite, separable, connected, Hausdorff space and has exactly two continuous selections, and (v) a topological spaceX is homeomorphic to [0,1) if (and only if) one of the following equivalent conditions holds: (a)X is infinite, Hausdorff, separable, pathwise connected and has exactly one continuous selection; (b)X is infinite, separable, locally connected and has exactly one continuous selection; (c)X is infinite, metric, locally connected and has exactly one continuous selection. Three examples are exhibited which demonstrate the necessity of various assumptions in our results.     

Separable and low-rank continuous games

In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game. This research was funded in part by National Science Foundation grants DMI-0545910 and ECCS-0621922 and AFOSR MURI subaward 2003-07688-1.     

Are Biseparable Extensions Frobenius?

In Secion 1 we describe what is known of the extent to which a separable extension of unital associative rings is a Frobenius extension. A problem of this kind is suggested by asking if three algebraic axioms for finite Jones index subfactors are dependent. In Section 2 the problem in the title is formulated in terms of separable bimodules. In Section 3 we specialize the problem to ring extensions, noting that a biseparable extension is a two-sided finitely generated projective, split, separable extension. Some reductions of the problem are discussed and solutions in special cases are provided. In Section 4 various examples are provided of projective separable extensions that are neither finitely generated nor Frobenius and which give obstructions to weakening the hypotheses of the question in the title. In Section 5 we show that characterizations of the separable extensions among Frobenius extensions are special cases of a result for adjoint functors. Mathematics Subject Classifications (2000): 16L60, 16H05.     

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.     

Universal measurability of the identity mapping of a Banach space in certain topologies

If X is a Banach space and X is its conjugate, then a subset Y of X is called madmissible for X if a) the topology (X, Y) is Hausdorff, b) the identity embedding of (X, (X, Y)) into X is universally measurable (Ref. Zh. Mat., 1975, 8B 75 8K). If X is separable, then the existence of an m-admissible set is well known. In this note it is shown that there exist nonseparable X having separable m-admissible sets. The properties of spaces with separable m-admissible sets are considered. It is proved, in particular, that a separable normalizing subset Y of X is m-admissible for X if and only if every (X, Y)-compact set is separable in X.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 305–314, February, 1978.     

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Möbius function of an interval [σ,π] in the poset of permutations ordered by pattern containment in the case where π is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,…,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Möbius function in the case where σ and π are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142.We also show that the Möbius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Möbius function of an interval [σ,π] of separable permutations is bounded by the number of occurrences of σ as a pattern in π. Another consequence is that for any separable permutation π the Möbius function of (1,π) is either 0, 1 or −1.