In this paper, we introduce the concept of quasi-point-separable topological vector spaces, which has the following important properties: 1.In general, the conditions for a topological vector space to be quasi-point-separable is not very difficult to check; 2.The class of quasi-point-separable topological vector spaces is very large that includes locally convex topological vector spaces and pseudonorm adjoint topological vector spaces as special cases; 3.Every quasi-point-separable Housdorrf topological vector space has the fixed point property (that is, every continuous self-mapping on any given nonempty closed and convex subset has a fixed point), which is the result of the main theorem of this paper (Theorem 4.1). Furthermore, we provide some concrete examples of quasi-point-separable topological vector spaces, which are not locally convex. It follows that the main theorem of this paper is a proper extension of Tychonoffs fixed point theorem on locally convex topological vector spaces.
The Separable Quotient Problem and the Lotz Property in Banach Spaces
Universidade Estadual da Paraíba（帕拉伊巴州立大学）；Instituto de Matemática Interdisciplinar (IMI)（交叉数学研究所）；Universidad Complutense de Madrid（马德里康普顿斯大学）；Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Centro de Ensino Superior do Seridó Universidade Federal do Rio Grande do Norte（塞里多高等教育中心北里奥格兰德联邦大学）
In this paper, we introduce the concepts of m-quasiconvex, originally m-quasiconvex,and generalized m-quasiconvex functionals on topological vector spaces. Then we extend the concept of point separable topological vector spaces (by the topological dual spaces) to quasi-point separable topological vector spaces by families of generalized m-quasiconvex functionals. We show that every pseudonorm adjoint topological vector space, which includes locally convex topological vector spaces as special cases, is quasi-point separable. By the Fan-KKM theorem, we prove a fixed point theorem on quasi-point …
Some results about the Schroeder-Bernstein Property for separable Banach spaces
我们构造了一个相互非同构的可分巴拿赫空间的连续统，这些空间是互补的。因此，这些空间的Schroeder-Bernstein指数都是$2^{\aleph_0}$。我们的构造是基于W. T. Gowers和B. 1997年，莫雷。我们还使用经典的描述集合论方法，如在V. Ferenczi和C. Rosendal等人的一些结果。Casazza和N. J. Kalton关于具有无条件有限维Schauder分解的空间的Schroeder-Bernstein性质。
It is shown that if the dual of a separable Banach space has Property($K^*$) then the original space has the weak fixed point property. This is an improvement of previously results.
The Bishop--Phelps--Bollobás Property for Extremally Disconnected Ranges: Separable and Low-Density Domains
Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）；Millsaps College（米尔斯普斯学院）；School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）；Hampton University（汉普顿大学）
In this paper we investigate operator Hilbert systems and their separable morphisms. We prove that the operator Hilbert space of Pisier is an operator system, which possesses the self-duality property. It is established a link between unital positive maps and Pietch factorizations, which allows us to describe all separable morphisms from an abelian C*-algebra to an operator Hilbert system. Finally, we prove a key property of entanglement breaking maps that involves operator Hilbert systems.
First we prove that if a separable Banach space $X$ contains an isometric copy of an infinite-dimensional space $A(S)$ of affine continuous functions on a Choquet simplex $S$, then its dual $X^*$ lacks the weak$^*$ fixed point property for nonexpansive mappings. Then, we show that the dual of a separable Lindenstrauss space $X$ fails the weak$^*$ fixed point property for nonexpansive mappings if and only if $X$ has a quotient isometric to some space $A(S)$. Moreover, we provide an example showing that "quotient" cannot be replaced by "subspace". Finally, it is worth to be mentioned that in our…
We are interested in a sufficient condition given in an article by P. Lefèvre, É. Matheron and A. Primot to obtain the Blum-Hanson Property and we then partially answer two questions asked in this same article on other possible conditions to have this property for a separable Banach space.
This paper deals with a property which is equivalent to generalised-lushness for separable spaces. It thus may be seemed as a geometrical property of a Banach space which ensures the space to have the Mazur-Ulam property. We prove that if a Banach space $X$ enjoys this property if and only if $C(K,X)$ enjoys this property. We also show the same result holds for $L_\infty(μ,X)$ and $L_1(μ,X)$.
Let $X$ and $Y$ be Banach spaces, and $T:X^*\to Y$ be an operator. We prove that if $X$ is Asplund and $Y$ has the approximation property, then for each Radon probability $μ$ on $(B_{X^*},w^*)$ there is a sequence of $w^*$-to-norm continuous operators $T_n:X^*\to Y$ such that $\|T_n(x^*)-T(x^*)\| \to 0$ for $μ$-a.e. $x^*\in B_{X^*}$; if $Y$ has the $λ$-bounded approximation property for some $λ\geq 1$, then the sequence can be chosen in such a way that $\|T_n\|\leq λ\|T\|$ for all $n\in \mathbb{N}$. The same conclusions hold if $X$ contains no subspace isomorphic to $\ell_1$, $Y$ has the appro…
The list of known Banach spaces whose linear geometry determines the (nonlinear) democracy functions of their quasi-greedy bases to the extent that they end up being democratic, reduces to $c_0$, $\ell_2$, and all separable $\mathcal{L}_1$-spaces. Oddly enough, these are the only Banach spaces that, when they have an unconditional basis, it is unique. Our aim in this paper is to study the connection between quasi-greediness and democracy of bases in nonlocally convex spaces. We prove that all quasi-greedy bases in $\ell_p$ for $0<p<1$ (which also has a unique unconditional basis) are democrati…
