In this paper it is shown that if a Banach lattice $E$ contains a copy of $c_0$, then it contains an almost lattice isometric copy of $c_0$. The above result is a lattice version of the well-known result of James concerning the almost isometric copies of $c_0$ in Banach spaces.

We characterize Banach lattices on which every weak Banach-Saks operator is b-weakly compact.

This paper will consider the closure of the set of operators which may be expressed as a sum of lattice homomorphisms whose range is contained in a Dedekind complete Banch lattice.

A sufficient and necessary condition for weak convergence of sequences in a class of Banach sequence lattices is obtained. As a direct application, a complete criterion of a weak convergence of sequences in l infinity is formulated.

In Agbeko (2012) the Hyers-Ulam-Aoki stability problem was posed in Banach lattice environments with the addition in the Cauchy functional equation replaced by supremum. In the present note we restate the problem so that it relates not only to supremum but also to infimum and their various combinations. We then propose some sufficient conditions which guarantee its solution.

It is known that a Banach lattice with order continuous norm contains a copy of ${\ell }_1$ if and only if it contains a lattice copy of ${\ell }_1$. The purpose of this note is to present a more direct proof of this useful fact, which extends a similar theorem due to R.C. James for Banach spaces with unconditional bases, and complements the $c_0$- and ${\ell }_{\infty }$-cases considered by Lozanovskii, Mekler and Meyer-Nieberg.

Let $G$ be a locally compact group, and let $\parallel {\parallel }_{\infty }^B$ be a function norm on $L^1(G{)}_{\mathrm{loc}}$ such that the space $L^{\infty }(G,B)$ of all locally integrable functions with finite $\parallel {\parallel }_{\infty }^B$-norm is an invariant solid Banach function space. Consider the space $L_{\mathrm{RUC}}(G,B)$ of all functions in $L^{\infty }(G,B)$ of which the right translation is a continuous map from $G$ into $L^{\infty }(G,B)$. Characterizations of the case where $L_{\mathrm{RUC}}(G,B)$ is a Riesz ideal of $L^{\infty }(G,B)$ are given in terms of the order-continuity of $\parallel {\parallel }_{\infty }^B$ on certain subspaces of $L^{\infty }\left(G\right)$. Throughout the paper, the discussion is carried out in the context...

We establish necessary and sufficient conditions under which the linear span of positive AM-compact operators (in the sense of Fremlin) from a Banach lattice $E$ into a Banach lattice $F$ is an order $\sigma$-complete vector lattice.

Let $X$ be a Banach lattice, and denote by $X_{+}$ its positive cone. The weak topology on $X_{+}$ is metrizable if and only if it coincides with the strong topology if and only if $X$ is Banach-lattice isomorphic to $l^1\left(\Gamma \right)$ for a set $\Gamma$. The weak${}^{*}$ topology on $X_{+}^{*}$ is metrizable if and only if $X$ is Banach-lattice isomorphic to a $C\left(K\right)$-space, where $K$ is a metrizable compact space.