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Let L be a truncated Archimedean vector lattice whose truncation is denoted by *. In a recent paper, we proved that there exists a locally compact Hausdorff space X such that L is a lattice isomorphic with a truncated vector lattice of functions in C∞(X) whose truncation is provided by meet with some characteristic function on X. This representation, no matter how interesting it is, has a major drawback, namely, C∞(X) need not be a vector lattice, unless X is extremally disconnected. The main purpose of this paper is to remedy this shortcoming by proving that, indeed, an extremally disconnected locally compact space X can be found such that L can be seen as an order dense vector sublattice of C∞(X) whose truncation is provided, again, by meet with some characteristic function on X. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a concept of norm-attainment in the projective symmetric tensor product of a Banach space X, which turns out to be naturally related to the classical norm-attainment of N -homogeneous polynomials on X. Due to this relation, we can prove that there exist symmetric tensors that do not attain their norms, which allows us to study the problem of when the set of norm-attaining elements in is dense. We show that the set of all normattaining symmetric tensors is dense in for a large set of Banach spaces such as Lp-spaces, isometric L1-predual spaces or Banach spaces with monotone Schauder basis, among others. Next, we prove that if X* satisfies the Radon-Nikodým and approximation properties, then the set of all norm-attaining symmetric tensors in is dense. From these techniques, we can present new examples of Banach spaces X and Y such that the set of all norm-attaining tensors in the projective tensor product is dense, answering positively an open question from the paper [10]. [ABSTRACT FROM AUTHOR]

Let P be a topological property. A space X is said to be star P if whenever is an open cover of X, there exists a subspace A ⊆ X with property P such that , where. In this paper, we investigate the relationships among star Lindelöf spaces, star almost Lindelöf spaces and star weakly Lindelöf spaces, and also study topological properties of star weakly Lindelöf spaces. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a class of ideals between z-ideals and z°-ideals (=d-ideals) namely zr-ideals. A zr-ideal is a z-ideal which is at the same time an r-ideal (an ideal I in a ring R is called an r-ideal if for each non-zerodivisor r ∈ R and each a ∈ R, ra ∈ I implies a ∈ I). In contrast to the sum of z-ideals in C(X) which is a z-ideal, the sum of zr-ideals need not be a zr-ideal. We prove that the sum of every two zr-ideals of C(X) is a zr-ideal if and only if X is a quasi F -space. In C(X) every z°-ideal is a zr-ideal and we characterize the spaces X for which the converse is also true. We observe that X is a cozero complemented space if and only if every (prime) r-ideal in C(X) is a z-ideal and whenever every (prime) z-ideal of C(X) is an r-ideal it is equivalent to X being an almost P-space. Using these facts it turns out that the set of all r-ideals and the set of all z-ideals of C(X) coincide if and only if X is a P-space. [ABSTRACT FROM AUTHOR]

In this paper, we show the following statements: (1) There exists a pseudocompact star Lindelöf Tychonoff space which is not star σ-compact. (2) There exists a Tychonoff pseudocompact star countable (hence, star Lindelöf) space having a pseudocompact, Gδ regular closed subspace which is not star Lindelöf. (3) Assuming , there exists a normal star countable (hence, star Lindelöf) space having a Gδ regular closed subspace which is not star Lindelöf. (4) Let X be a space, then A(X) is star Lindelöf if and only if e(X) ≤ ω. The statement (1) gives a negative answer to Song [13, Remark 2.3]. [ABSTRACT FROM AUTHOR]

In this paper, we investigate conditions on a pair of Banach lattices E and F that tells us when every positive almost L-weakly compact (resp. almost M- weakly compact) operator T : E → F is weakly compact. Also, we present some necessary conditions that tells us when every weakly compact operator T : E → F is almost M-weakly compact (resp. almost L-weakly compact). In particular, we will prove that if every weakly compact operator from a Banach lattice E into a Banach space X is almost L-weakly compact, then E is a KB-space or X has the Dunford-Pettis property and the norm of E is order continuous. [ABSTRACT FROM AUTHOR]