Let $(G,\tau )$ be a commutative Hausdorff locally solid lattice group. In this paper we prove the following: (1) If $(G,\tau )$ has the $A$(iii)-property, then its completion $(\widehat{G},\widehat{\tau })$ is an order-complete locally solid lattice group. (2) If $G$ is order-complete and $\tau$ has the Fatou property, then the order intervals of $G$ are $\tau$-complete. (3) If $(G,\tau )$ has the Fatou property, then $G$ is order-dense in $\widehat{G}$ and $(\widehat{G},\widehat{\tau })$ has the Fatou property. (4) The order-bound topology on any commutative lattice group is the finest locally solid topology on...

The main topic of the first section of this paper is the following theorem: let $A$ be an Archimedean $f$-algebra with unit element $e$, and $TA\to A$ a Riesz homomorphism such that $T^2\left(f\right)=T\left(fT\left(e\right)\right)$ for all $f\in A$. Then every Riesz homomorphism extension $\tilde{T}$ of $T$ from the Dedekind completion $A^{\delta }$ of $A$ into itself satisfies ${\tilde{T}}^2\left(f\right)=\tilde{T}\left(fT\left(e\right)\right)$ for all $f\in A^{\delta }$. In the second section this result is applied in several directions. As a first application it is applied to show a result about extensions of positive projections to the Dedekind completion. A second application...

In this paper we investigate conditions for a system of sequences of elements of a vector lattice; analogous conditions for systems of sequences of reals were studied by D. E. Peek.

Two problems posed by Choquet and Foias are solved:(i) Let $T$ be a positive linear operator on the space $C\left(X\right)$ of continuous real-valued functions on a compact Hausdorff space $X$. It is shown that if $n^{-1}{\sum }_{r=0}^{n-1}{T}^{r}1$ converges pointwise to a continuous limit, then the convergence is uniform on $X$.(ii) An example is given of a Choquet simplex $K$ and a positive linear operator $T$ on the space $A\left(K\right)$ of continuous affine real-valued functions on $K$, such that$$\inf \left\{\right(T^{n}1\left)\right(x):n\ge \}\<1$$for each $x$ in ${\partial }_{\ell }K$, but $\parallel T^{n}1\parallel$ does not converge to 0.

In this paper, we introduce and study the class of almost weak${}^{☆}$ Dunford-Pettis operators. As consequences, we derive the following interesting results: the domination property of this class of operators and characterizations of the wDP${}^{☆}$ property. Next, we characterize pairs of Banach lattices for which each positive almost weak${}^{☆}$ Dunford-Pettis operator is almost Dunford-Pettis.

Let $G$ be an Archimedean $\ell$-group. We denote by $G^d$ and $R_D\left(G\right)$ the divisible hull of $G$ and the distributive radical of $G$, respectively. In the present note we prove the relation ${\left(R_D\left(G\right)\right)}^d=R_D\left(G^d\right)$. As an application, we show that if $G$ is Archimedean, then it is completely distributive if and only if it can be regularly embedded into a completely distributive vector lattice.

We establish some sufficient conditions under which the subspaces of Dunford-Pettis operators, of M-weakly compact operators, of L-weakly compact operators, of weakly compact operators, of semi-compact operators and of compact operators coincide and we give some consequences.