Suppose $X$ is a locally solid vector lattice. It is known that there are several non-equivalent notions for compact operators on $X$. Furthermore, notion of the $AM$-property in $X$ as an extension for the $AM$-spaces in Banach lattices has been considered, recently. In this paper, we establish a variant of the known Krengel's theorem for different types of compact operators between locally solid vector lattices.
The statistical convergence is handled for sequences with the natural density, in general. In a recent paper, the statistical convergence for nets in Riesz spaces has been studied and investigated by developing topology-free techniques in Riesz spaces. In this paper, we introduce the statistically topological convergence for nets on locally solid Riesz spaces with solid topologies. Moreover, we introduce the statistical continuity on locally solid Riesz spaces.
The aim of this note is to consider different notions for the Banach-Saks property in locally solid vector lattices as an extension for the known concepts of the Banach-Saks property in Banach lattices. We investigate relations between them; in particular, we shall characterize spaces in which, these notions agree.
Let $(E,τ)$ be a locally solid vector lattice. A filter $\mathcal{F}$ on the set $E$ is said to be converge to a vector $e\in E$ if, each zero neighborhood set $U$ containing $e$, $U$ belongs to $\mathcal{F}$. We study on the concept of this convergence and give some basic properties of it.
Unbounded convergences have been applied successfully to locally solid topologies on vector lattices. In the present paper, we first expose several properties of various classes of Riesz pseudonorms on vector lattices. We accomplish this by abstracting some generalities of the norm of an AM-space with strong norm unit to locally solid topologies induced by a pseudonorm. By using these classes of pseudonorms, we study compactness properties of operators (not necessarily linear) between locally solid (not necessarily Hausdorff) topologies. We study new classes of operators such as pseudonorm com…
The main aim of the present note is to consider bounded orthomorphisms between locally solid vector lattices. We establish a version of the remarkable Zannen theorem regarding equivalence between orthomomorphisms and the underlying vector lattice to the case of all bounded orthomomorphisms. Furthermore, we investigate topological and ordered structures for these classes of orthomorphisms, as well.
In a topological Riesz space there are two types of bounded subsets: order bounded subsets and topologically bounded subsets. It is natural to ask (1) whether an order bounded subset is topologically bounded and (2) whether a topologically bounded subset is order bounded. A classical result gives a partial answer to (1) by saying that an order bounded subset of a locally solid Riesz space is topologically bounded. This paper attempts to further investigate these two questions. In particular, we show that (i) there exists a non-locally solid topological Riesz space in which every order bounded …
In this paper, we introduce the statistically multiplicative convergent sequences in locally solid Riesz algebras with respect to the algebra multiplication and the solid topology. We study on this concept and we give the notion of $\mathbb{st_m}$-bounded sequence, and also, we prove some relations between this convergence and the other convergences such as the order convergence and the statistical convergence in topological spaces. Also, we give some results related to semiprime $f$-algebras.
Let $X$ be a vector lattice and $(E,τ)$ be a locally solid vector lattice. An operator $T:X\to E$ is said to be $ob$-bounded if, for each order bounded set $B$ in $X$, $T(B)$ is topologically bounded in $E$. In this paper, we study on algebraic properties of $ob$-bounded operators with respect to the topology of uniform convergence and equicontinuous convergence.
Suppose $X$ is a locally solid vector lattice. It is known that there are several non-equivalent spaces of bounded operators on $X$. In this paper, we consider some situations under which these classes of bounded operators form locally solid vector lattices. In addition, we generalize some notions of $uaw$-Dunford-Pettis operators and $uaw$-compact operators defined on a Banach lattice to general theme of locally solid vector lattices. With the aid of appropriate topologies, we investigate some relations between topological and lattice structures of these operators. In particular, we character…
In this note, we consider the space of all continuous operators with respect to the unbounded topology on locally solid vector lattices. We investigate whether this space forms a band. In addition, we look into some situations under which, the modulus of a continuous operator has the same property. Some examples are given to make the context more understandable.
A sequence $(x_n)$ in a locally solid Riesz space $(E,τ)$ is said to be statistically unbounded $τ$-convergent to $x\in E$ if, for every zero neighborhood $U$, $\frac{1}{n}\big\lvert\{k\leq n:\lvert x_k-x\rvert\wedge u\notin U\}\big\rvert\to 0$ as $n\to\infty$. In this paper, we introduce this concept and give the notions $st$-$u_τ$-closed subset, $st$-$u_τ$-Cauchy sequence, $st$-$u_τ$-continuous and $st$-$u_τ$-complete locally solid vector lattice. Also, we give some relations between the order convergence and the $st$-$u_τ$-convergence.
A linear operator $T$ between two lattice-normed locally solid Riesz spaces is said to be $p_τ$-continuous if, for any $p_τ$-null net $(x_α)$, the net $(Tx_α)$ is $p_τ$-null, and $T$ is also said to be $p_τ$-bounded operator if it sends $p_τ$-bounded subsets to $p_τ$-bounded subsets. They are generalize several known classes of operators such as continuous, order continuous, $p$-continuous, order bounded, $p$-bounded operators, etc. We also study $up_τ$-continuous operators between lattice-normed locally solids Riesz spaces.
Let $x_α$ be a net in a locally solid vector lattice $(X,τ)$; we say that $x_α$ is unbounded $τ$-convergent to a vector $x\in X$ if $\lvert x_α-x \rvert\wedge w \xrightarrowτ 0$ for all $w\in X_+$. In this paper, we study general properties of unbounded $τ$-convergence (shortly, $uτ$-convergence). $uτ$-Convergence generalizes unbounded norm convergence and unbounded absolute weak convergence in normed lattices that have been investigated recently. Besides, we introduce $uτ$-topology and study briefly metrizabililty and completeness of this topology.
Suppose $X$ is a vector lattice and there is a notion of convergence $x_α \rightarrow x$ in $X$. Then we can speak of an "unbounded" version of this convergence by saying that $(x_α)$ unbounded converges to $x\in X$ if $\lvert x_α-x\rvert \wedge u\rightarrow 0$ for every $u \in X_+$. In the literature, the unbounded versions of the norm, order and absolute weak convergence have been studied. Here we create a general theory of unbounded convergence but with a focus on uo-convergence and those convergences deriving from locally solid topologies.
Suppose $X$ is a locally solid vector lattice. In this paper, we introduce the notion "$AM$-property" in $X$ as an extension for $AM$-spaces in the category of all Banach lattices. With the aid of this concept, we characterize spaces in which bounded sets and order bounded sets agree. This, in turn, characterizes conditions under which each class of bounded operators on $X$ is order bounded and vice versa. Also, we show that under some natural assumptions, different types of bounded order bounded operators on $X$ have the Lebesgue or Levi property if and only if so is $X$.
Aliprantis's questions on locally solid topologies
Universidad Autónoma de Madrid（马德里自治大学）；ETH Zürich（苏黎世联邦理工学院）；Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（西班牙高等科学研究理事会数学科学研究所）
Let $(x_α)$ be a net in a vector lattice normed by locally solid lattice $(X,p,E_τ)$. We say that $(x_α)$ is unbounded $p_τ$-convergent to $x\in X$ if $p(\lvert x_α-x\rvert\wedge u)\xrightarrowτ 0$ for every $u\in X_+$. This convergence has been studied recently for lattice-normed vector lattices as the $up$-convergence in \cite{AGG,AEEM,AEEM2}, the $uo$-convergence in \cite{GTX}, and, as the $un$-convergence in \cite{DOT,GX,GTX,KMT,Tr2}. In this paper, we study the general properties of the unbounded $p_τ$-convergence.
A linear operator $T$ between two vector lattices normed by locally solid Riesz spaces is said to be $p_τ$-continuous if, for any $p_τ$-null net $(x_α)$, the net $(Tx_α)$ is $p_τ$-null, and $T$ is said to be $p_τ$-bounded operator if it sends $p_τ$-bounded subsets to $p_τ$-bounded subsets. Also, $T$ is called $p_τ$-compact if, for any $p_τ$-bounded net $(x_α)$, the net $(Tx_α)$ has a $p_τ$-convergent subnet. They generalize several known classes of operators such as norm continuous, order continuous, $p$-continuous, order bounded, $p$-bounded, compact and AM-compact operators. We study the gen…
Let E be a locally solid vector lattice. In this paper, we consider two particular vector subspaces of the space of all order bounded operators on E. With the aid of two appropriate topologies, we show that under some conditions, they establish both, locally solid vector lattices and topologically complete topological algebras.
Motivated by the recent definition of $AM$-property in locally solid vector lattices [O. Zabeti, arXiv: 1912.00141v2 [math.FA]], in this note, we try to investigate those results in the category of all locally solid lattice rings. In fact, we characterize locally solid lattice rings in which order bounded sets and bounded sets agree. Furthermore, with the aid of $AM$-property, we find conditions under that, order bounded group homomorphisms and different types of bounded group homomorphisms coincide. Moreover, we show that each class of bounded order bounded group homomorphisms on a locally so…
A net $(x_γ)_{γ\inΓ}$ in a locally solid Riesz space $(X,τ)$ is said to be unbounded $τ$-convergent to $x$ if $|x_γ-x|\wedge u\mathop{\oversetτ{\longrightarrow}} 0$ for all $u\in X_+$. We recall that there is a locally solid linear topology $\mathfrak{u}τ$ on $X$ such that unbounded $τ$-convergence coincides with $\mathfrak{u}τ$-convergence, and moreover, $\mathfrak{u}τ$ is characterised as the weakest locally solid linear topology which coincides with $τ$ on all order bounded subsets. It is with this motivation that we introduce, for a uniform lattice $(L,u)$, the weakest lattice uniformity $…
Suppose $X$ is a locally solid lattice ring. It is known that there are three classes of bounded group homomorphisms on $X$ whose topological structures make them again topological rings. In this note, we consider lattice structure on them; more precisely, we show that, under some mild assumptions, they are locally solid lattice rings.
We investigate the $oτ$-continuous/bounded/compact and Lebesgue operators from vector lattices to topological vector spaces; the KB operators between locally solid lattices and topological vector spaces; and the Levi operators from locally solid lattices to vector lattices. The main idea of operator versions of notions related to vector lattices lies in redistributing topological and order properties of a topological vector lattice between the domain and range of an operator under investigation. Domination properties for these classes of operators are studied.
A mixed lattice vector space is a partially ordered vector space with two partial orderings, generalizing the notion of a Riesz space. Whereas the algebraic theory of mixed lattice structures dates back to the 1970s, the topological theory of mixed lattice spaces remains largely unexplored. The purpose of this paper is to develop the basic topological theory of mixed lattice spaces. A vector topology is said to be compatible with the mixed lattice structure if the mixed lattice operations are continuous. We obtain a characterization of compatible mixed lattice topologies, which is similar to t…
Suppose $G$ is a locally solid lattice group. It is known that there are non-equivalent classes of bounded homomorphisms on $G$ which have topological structures. In this paper, our attempt is to assign lattice structures on them. More precisely, we use of a version of the remarkable Riesz-Kantorovich formulae and Fatou property for bounded order bounded homomorphisms to allocate the desired structures. Moreover, we show that unbounded convergence on a locally solid lattice group is topological and we investigate some applications of it. Also, some necessary and sufficient conditions for compl…
Given a von Neumann algebra $M$ with a faithful normal semi-finite trace $τ,$ let $L(M, τ)$ be the algebra of all $τ$-measurable operators affiliated with $M.$ We prove that if $A$ is a locally convex reflexive complete metrizable solid $\ast$-subalgebra in $L(M, τ),$ which can be embedded into a locally bounded weak Fréchet $M$-bimodule, then any derivation on $A$ is inner.
