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AbstractThe purpose of this note is to give a correct proof of the theorem that concerns the domination property of the class of U-Dunford-Pettis operators obtained by O. Aboutafail, K. El Fahri, J. H’Michane and L. Zraoula [On the class of U-Dunford-Pettis operators, Quaestiones Mathematicae 45(4) (2022), 655–666]. Also, we investigate necessary and sufficient conditions under which U-Dunford–Pettis operators between Banach lattices must be Dunford–Pettis. [ABSTRACT FROM AUTHOR]

AbstractIn this work we consider natural generalizations of local complementation in Banach spaces, which include Lipschitz-local complementation. We show that all these notions are indeed equivalent to the classical notion of local complementation of Banach spaces. As an application, we show that local complementation is naturally preserved under certain free constructions in Functional Analysis, including Lipschitz-free spaces and free Banach lattices. [ABSTRACT FROM AUTHOR]

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension n contains a uniformly closed ideal of codimension at most $ 2n $ 2 n. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of Abramovich Y.A., Lipecki Z. [On ideals and sublattices in linear lattices and F-lattices. Math Proc Cambridge Philos Soc. 1990;108(1):79–87.; On lattices and algebras of simple functions. Comment Math Univ Carolin. 1990;31(4):627–635.], as well as Kakutani's characterization of closed sublattices of $ C(K) $ C (K) spaces. [ABSTRACT FROM AUTHOR]

In this paper, we characterize all N-dimensional hypercomplex numbers having unital Archimedean f-algebra structure. We use matrix representation of hypercomplex numbers to define an order structure on the matrix spectra. We prove that the unique (up to isomorphism) unital Archimedean f-algebra of hypercomplex numbers of dimension N ≥ 1 is that with real and simple spectrum. We also show that these number systems can be made into unital Banach lattice algebras and we establish some of their properties. Furthermore, we prove that every 2N-dimensional unital Archimedean f-algebra is the hyperbolization of that of dimension N. Finally, we consider hypercomplex number systems of dimension N = 2 , 3 , 4 , 6 and give their explicit matrix representation and eigenvalue operators. This work is a multidimensional generalization of the results obtained in Gargoubi and Kossentini (Adv Appl Clifford Algebras 26(4):1211–1233, 2016) and Bilgin and Ersoy S (Adv Appl Clifford Algebras 30:13, 2020) for, respectively, the two and four-dimensional systems. [ABSTRACT FROM AUTHOR]

The concept of weak orthogonality of order p ( 1 ≤ p ≤ ∞ ) in Banach lattices is introduced in order to obtain spaces with the weak fixed point property of order p. Moreover, various connections between a number of Banach space properties to imply the weak fixed point property, such as Opial condition, weak normal structure and property (M) are investigated. In particular, it is established that for each Banach space X and a suitable Banach lattice F, a Banach lattice M ⊂ K (X , F) has the weak fixed point property of order p, if each evaluation operator ψ y ∗ on M is a p-convergent operator for y ∗ ∈ F ∗ . [ABSTRACT FROM AUTHOR]

We introduce the space of holomorphic growth spaces with values in a Banach lattice. We provide norm and essential norm estimates of the embedding operator, and we completely characterize the bounded and compact embeddings of the growth spaces using vector-valued Carleson measures. As an application, we prove a characterization of weighted composition operators. [ABSTRACT FROM AUTHOR]

It was showed by Donner in [ Extension of positive operators and Korovkin theorems , Lecture Notes in Mathematics, vol. 904, Springer-Verlag, Berlin-New York, 1982] that every order complete vector lattice X may be embedded into a cone X^s, called the sup-completion of X. We show that if one represents the universal completion of X as C^\infty (K), then X^s is the set of all continuous functions from K to [-\infty,\infty ] that dominate some element of X. This provides a functional representation of X^s, as well as an easy alternative proof of its existence. [ABSTRACT FROM AUTHOR]

The weak-type BMO-regularity property for couples of quasi-Banach lattices of measurable functions was recently introduced as a suitable substitute for the usual BMO-regularity in connection with characterization of the K-closedness of Hardy-type spaces on the unit circle and stability for the real interpolation. It was characterized in terms of the BMO-regularity of couples ((X, Y)α,p, (X, Y)β,q), 0 < α < β < 1, of the real interpolation spaces. In the present note, a natural characterization of this property similar to that of BMO-regularity for couples of Banach lattices (X, Y) in terms of the BMO-regularity of X′Y is extended to couples of lattices of measurable functions on homogeneous type spaces. We also derive equivalent conditions corresponding to the limit case where α = 0. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we show that if H is a Dedekind complete normed Riesz space with the local Egoroff property, then H is lattice isomorphic to an ideal of an AM-space if and only if for any order bounded norm null sequence in H. Based on this, several characterizations of discrete Banach lattices with order continuous norms are provided. As an application, we find the special regulator for every relative uniform convergent sequence in discrete Banach lattices with order continuous norms. [ABSTRACT FROM AUTHOR]

We work out the exact relationship between algebraic modular forms for a two-by-two general unitary group over a definite quaternion algebra, and those arising from genera of positive-definite quinary lattices, relating stabilisers of local lattices with specific open compact subgroups, paramodular at split places, and with Atkin-Lehner operators. Combining this with the recent work of Rösner and Weissauer, proving conjectures of Ibukiyama on Jacquet-Langlands type correspondences (mildly generalised here), provides an effective tool for computing Hecke eigenvalues for Siegel modular forms of degree two and paramodular level. It also enables us to prove examples of congruences of Hecke eigenvalues connecting Siegel modular forms of degrees two and one. These include some of a type conjectured by Harder at level one, supported by computations of Fretwell at higher levels, and a subtly different congruence discovered experimentally by Buzzard and Golyshev. [ABSTRACT FROM AUTHOR]

We introduce inner band projections in the space of regular operators on a Dedekind complete Banach lattice and study some structural properties of this class. In particular, we provide a new characterization of atomic order continuous Banach lattices as those for which all band projections in the corresponding space of regular operators are inner. We also characterize the multiplication operators L_AR_B which are band projections precisely as those with A,B being band projections up to a scalar multiple. [ABSTRACT FROM AUTHOR]

In this paper we introduce and study a new class of operators related to norm bounded sets on Banach Lattices and which brings together several classes of operators (as o-weakly compact, b-weakly compact, M-weakly compact, L-weakly compact and almost Dunford–Pettis operators). As applications, we find some new lattice approximation properties of these classes of operators. [ABSTRACT FROM AUTHOR]

Orthogonally biadditive operators preserving disjointness are studied. It is proved that, that for a Dedekind complete vector lattice and order ideals and in , the set of all orthogonally biadditive operators commuting with projections is a band in the Dedekind complete vector lattice of all regular orthogonally biadditive operators from the Cartesian product of and to . A general form of the order projection onto this band is obtained, and an operator version of the Radon–Nikodym theorem for disjointness-preserving positive orthogonally biadditive operators is proved. [ABSTRACT FROM AUTHOR]

A periodic lattice in Euclidean space is the infinite set of all integer linear combinations of basis vectors. Any lattice can be generated by infinitely many different bases. This ambiguity was partially resolved, but standard reductions remain discontinuous under perturbations modelling atomic displacements. This paper completes a continuous classification of 2-dimensional lattices up to Euclidean isometry (or congruence), rigid motion (without reflections), and similarity (with uniform scaling). The new homogeneous invariants allow easily computable metrics on lattices considered up to the equivalences above. The metrics up to rigid motion are especially non-trivial and settle all remaining questions on (dis)continuity of lattice bases. These metrics lead to real-valued chiral distances that continuously measure lattice deviations from higher-symmetry neighbours. The geometric methods extend the past work of Delone, Conway, and Sloane. [ABSTRACT FROM AUTHOR]

The purpose of this article is to study two classes of operators, which we call positively limited p-convergent operators, and weak * positively p-convergent operators. We discuss the relationship between these two classes of operators, and other known classes of operators such as p-convergent operators, limited p-convergent operators, disjoint p-convergent operators, etc. Moreover, the positive DP * property of order p is studied, and the behavior of these two classes of operators on Banach lattices with this property (with focus on Banach lattices with the positively limited p-Schur property) is investigated. In addition, the domination properties of positively limited p-convergent operators, and weak * positively p-convergent operators on Banach lattices are considered. [ABSTRACT FROM AUTHOR]

The well-known logic first degree entailment logic (FDE), introduced by Belnap and Dunn, is defined with |$\wedge $|⁠ , |$\vee $| and |$\sim $| as the sole primitive connectives. The aim of this paper is to establish the lattice formed by the class of all 4-valued C-extending implicative expansions of FDE verifying the axioms and rules of Routley and Meyer's basic logic B and its useful disjunctive extension B |$^{\textrm {d}}$|⁠. It is to be noted that Boolean negation (so, classical propositional logic) is definable in the strongest element in the said class. [ABSTRACT FROM AUTHOR]

We present a lattice-based characterization of extreme points whithin well-defined convex subsets of positive operators, operating between Banach lattices with quasi-interior points. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and investigate a new class of operators known as almost unbounded L-weakly compact (in shortly, au L -weakly compact) and almost unbounded M-weakly compact (in shortly, au M -weakly compact) operators. We explore the lattice properties related to this class and examine their relationships with other established operator classes, such as L-weakly compact operators and almost L-weakly compact operators. We demonstrate that every L-weakly compact operator is an au L -weakly compact operator, but the reverse implication does not necessarily hold in all cases. [ABSTRACT FROM AUTHOR]

In this paper, we establish the stability of uaw-convergence under passing from sublattices. The various implications of this fact are presented through the paper. In particular, we show that if (x α) is an increasing net in a Banach lattice E and x α ⟶ uaw 0 in E then x α ⟶ un 0 in E ′ ′ . Furthermore, we deduce some results concerning uaw-completeness. Additionally, we present a new characterizations of KB-spaces (resp. reflexive Banach lattices), using the concepts of uaw-convergence and un-convergence. [ABSTRACT FROM AUTHOR]

In this paper, we investigate Banach lattices on which each positive semi-compact operator T : E → F is null almost L-weakly compact (rep. Null almost M-weakly compact). Additionally, we present certain sufficient and necessary conditions for a positive Null almost L-weakly compact operator to be semi-compact. [ABSTRACT FROM AUTHOR]

Inspired by locale theory, pointfree convex geometry was first proposed and studied by Yoshihiro Maruyama. In this paper, we shall continue to his work and investigate the related topics on pointfree convex spaces. Concretely, the following results are obtained: (1) A Hofmann–Lawson-like duality for pointfree convex spaces is established. (2) The M -injective objects in the category of S 0 -convex spaces are proved precisely to be sober convex spaces, where M is the class of strict maps of convex spaces; (3) A convex space X is sober iff there never exists a nontrivial identical embedding i : X ↪ Y such that its dualization is an isomorphism, and a convex space X is S D iff there never exists a nontrivial identical embedding k : Y ↪ X such that its dualization is an isomorphism. (4) A dual adjunction between the category CLat D of continuous lattices with continuous D-homomorphisms and the category CS D of S D -convex spaces with CP-maps is constructed, which can further induce a dual equivalence between CS D and a subcategory of CLat D ; (5) The relationship between the quotients of a continuous lattice L and the convex subspaces of cpt (L) is investigated and the collection Alg (Q (L)) of all algebraic quotients of L is proved to be an algebraic join-sub-complete lattice of Q (L) of all quotients of L, where cpt (L) denote the set of non-bottom compact elements of L. Furthermore, it is shown that Alg (Q (L)) is isomorphic to the collection Sob (P (cpt (L))) of all sober convex subspaces of cpt (L) ; (6) Several necessary and sufficient conditions for all convex subspaces of cpt (L) to be sober are presented. [ABSTRACT FROM AUTHOR]

A monomial in finite collection of homogeneous polynomials is a point-wise product of these polynomials each raised to some power. It was established in [14] and [15] that, given a finite set of positive linear operators acting from an Archimedean vector lattice into an f-algebra with multiplicative unit, the corresponding monomial is orthogonally additive polynomial if and only if the collection of these operators is a disjointness preserving set. It is shown in this paper that a similar result is also true in the case when the target space is a uniformly complete vector lattice. Moreover, the result remains valid if we replace the linear operators with homogeneous orthogonally additive polynomials. The correct definition of monomials in homogeneous polynomials is guaranteed by the homogeneous functional calculus and the concavification procedure. [ABSTRACT FROM AUTHOR]

We utilize degenerate perturbation theory to investigate continuous-time quantum search on second-order truncated simplex lattices. In this work, we show that the construction of the Hamiltonian must consider the structure of the lattice. This idea enables effective application of degenerate perturbation theory to third- and higher-order lattices. We identify two constraints on the reduction of the dimension of the Hamiltonian. In addition, we elucidate the influence of the distinct configurations of marked vertices on the quantum search. [ABSTRACT FROM AUTHOR]

Three‐way concept analysis is an important tool for information processing, and rule acquisition is one of the research hotspots of three‐way concept analysis. However, compared with three‐way concept lattices, three‐way semi‐concept lattices have three‐way operators with weaker constraints, which can generate more concepts. In this article, the problem of rule acquisition for three‐way semi‐concept lattices is discussed in general. The authors construct the finer relation of three‐way semi‐concept lattices, and propose a method of rule acquisition for three‐way semi‐concept lattices. The authors also discuss the set of decision rules and the relationships of decision rules among object‐induced three‐way semi‐concept lattices, object‐induced three‐way concept lattices, classical concept lattices and semi‐concept lattices. Finally, examples are provided to illustrate the validity of our conclusions. [ABSTRACT FROM AUTHOR]

The design of three-dimensional printing based conformal cooling channels (CCCs) in injection molding holds great significance. Compared to CCCs, conformal cooling (CC) cavity solutions show promise in delivering enhanced cooling performance for plastic products, although they have been underexplored. In this research, CC cavity is designed within the mold geometry, reinforced by body-centered cubic (BCC) lattice structures to enhance mechanical strength. Three distinct BCC lattice variations have been integrated into the CC cavity: the BCC structure, BCC with cubes, and BCC with pillars. The thermal performances of the BCC lattice-added CC cavity are assessed numerically after experimental validation. To provide feasible solutions from viewpoints of thermal performances, various BCC lattice structure thicknesses are analyzed in the range of 0.8–1.2 mm. Thermal simulation outcomes reveal that thicker lattice structures enhance mechanical strength but simultaneously lead to an increase in cooling time. Upon examining all the proposed CC cavity solutions supported by BCC, the cooling times range from 2.2 to 4 s, resulting in a reduction of 38.6% to 66.1% when compared to conventional straight-drilled channels. In contrast to CCCs, CC cavities have the potential to decrease the maximum temperature non-uniformity from 8.5 to 6 K. Nevertheless, the presence of lattice structures in CC cavity solutions results in an elevated pressure drop, reaching 2.8 MPa, whereas the results for CCCs remain below 2.1 MPa. Graphic Abstract [ABSTRACT FROM AUTHOR]

This paper is devoted to two classes of operators related to positively limited sets on Banach lattices, which we call positively limited operators and positively limited completely continuous operators, respectively. We study the relationship between these two classes of operators and other known classes of operators such as (L-, M-)weakly compact operators, almost limited operators, etc. Some lattice-topological properties of Banach lattices, the positive GP and the dual positive Schur property, are characterized in terms of positively limited operators or positively limited completely continuous operators. [ABSTRACT FROM AUTHOR]

We study the duality and norm completeness in the new classes of limitedly L-weakly compact and Dunford-Pettis L-weakly compact operators from Banach spaces to Banach lattices. [ABSTRACT FROM AUTHOR]

Let F q be the finite field with 푞 elements and consider the 푛-dimensional F q -vector space V = F q n . In this paper, we define a closure operator on the subgroup lattice of the group G = PGL ⁢ (V) . Let 휇 denote the Möbius function of this lattice. The aim is to use this closure operator to characterize subgroups 퐻 of 퐺 for which μ ⁢ (H , G) ≠ 0 . Moreover, we establish a polynomial bound on the number c ⁢ (m) of closed subgroups 퐻 of index 푚 in 퐺 for which the lattice of 퐻-invariant subspaces of 푉 is isomorphic to a product of chains. This bound depends only on 푚 and not on the choice of 푛 and 푞. It is achieved by considering a similar closure operator for the subgroup lattice of GL ⁢ (V) and the same results proven for this group. [ABSTRACT FROM AUTHOR]

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice E. We prove that the class of regular operators for which the diagonal coincides with the atomic diagonal is always a band in , which contains the band of abstract integral operators. If E is also a Banach lattice, then contains positive Riesz and positive AM-compact operators. [ABSTRACT FROM AUTHOR]

Let W be the category of Archimedean lattice-ordered groups with distinguished weak unit and unit-preserving ℓ-group homomorphisms. We denote by P, wP, Loc, HA, respectively, the collections of projectable, weakly projectable, local, and hyperarchimedean W-objects. For any C ⊆ W, we say that a W-object H is hereditarilyC, denoted H ∈ hC, if G ∈ C whenever G is a W-subobject of H. Our focus is hP. Taking off from the known items P = wP ∩ Loc, h HA = HA ⊆ P, and C (X) ∈ P iff X is basically disconnected (BD), we show For C (X), equivalent are: hP; hLoc and X BD; X finite. If the Yosida space Y G is BD, then (1) If G is hLoc, then G is bounded; (2) If G is hP, then G is HA. hP = hwP, and every G ∈ hP with G ∉ HA has Yosida space Y G = αN, the one-point compactification of N. There are various G ∈ hP with Y G = αN, including: G bounded not HA; G not bounded with bounded part HA; G not bounded with bounded part not HA. [ABSTRACT FROM AUTHOR]

Relatively uniformly continuous (ruc) semigroups were recently introduced and studied by Kandic, Kramar-Fijavž, and the second-named author, in order to make the theory of one-parameter operator semigroups available in the setting of vector lattices, where no norm is present in general. In this article, we return to the more standard Banach lattice setting – where both ruc semigroups and C0 -semigroups are well-defined concepts – and compare both notions. We show that the ruc semigroups are precisely those positive C0 -semigroups whose orbits are order bounded for small times. We then relate this result to three different topics: (i) equality of the spectral and the growth bound for positive C0 -semigroups; (ii) a uniform order boundedness principle which holds for all operator families between Banach lattices; and (iii) a description of unbounded order convergence in terms of almost everywhere convergence for nets which have an uncountable index set containing a co-final sequence. [ABSTRACT FROM AUTHOR]

We study when Aron–Berner extensions of (separately) almost Dunford–Pettis multilinear operators between Banach lattices are (separately) almost Dunford–Pettis. For instance, for a σ -Dedekind complete Banach lattice F containing a copy of ℓ ∞ , we characterize the Banach lattices E 1 , ... , E m for which every continuous m-linear operator from E 1 × ⋯ × E m to F admits an almost Dunford–Pettis Aron–Berner extension. Illustrative examples are provided. [ABSTRACT FROM AUTHOR]

We consider discrete periodic operator on Z d with respect to lattices Γ ⊂ Z d of full rank. We describe the class of lattices Γ for which the operator may have a spectral gap for arbitrarily small potentials. We also show that, for a large class of lattices, the dimensions of the level sets of spectral band functions at the band edges do not exceed d - 2 . [ABSTRACT FROM AUTHOR]

We show that a positive operator between L^p-spaces is given by integration against a kernel function if and only if the image of each positive function has a lower semi-continuous representative with respect to a suitable topology. This is a consequence of a new characterization of kernel operators on general Banach lattices as those operators whose range can be represented over a fixed countable set of positive vectors. Similar results are shown to hold for operators that merely dominate a non-trivial kernel operator. [ABSTRACT FROM AUTHOR]

Karn and Sinha [Glasg. Math. J. 56 (2014), pp. 427–437] introduced the p- limited (1 \leq p < \infty) sets (see the definition below). We show that p-limited sets are preserved by continuous polynomials as well as by the projective tensor product and that scalar-valued polynomials are p-summable on p-limited sets. Considering the notion of p-limited set from the \ell _p-valued operators point of view, we introduce in Section \ref{sec:3} two weaker types of p-limitedness in the setting of Banach lattices and study their basic properties. [ABSTRACT FROM AUTHOR]

Disjoint sequence methods from the theory of Riesz spaces are used to study measures of weak non-compactness in L_{1}(\mu)-spaces. A principal new result of the present paper is the following: Let E be an abstract M-space. Then \begin{align*} \omega (B)&=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {disjoint} \}\\ &=\inf \{\varepsilon >0:\exists x^{*}\in E^{*}_{+} \operatorname {so}\operatorname {that} B\subseteq [-x^{*},x^{*}]+\varepsilon B_{E^{*}}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {weakly}\operatorname {null} \}\\ &=\sup \{\operatorname {ca}_{\rho _{B}}((x_{n})_{n}):(x_{n})_{n}\subseteq (B_{E})_{+} \operatorname {increasing} \}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\|x^{*}_{n}\|:(x^{*}_{n})_{n}\subseteq \operatorname {Sol}(B)\operatorname {disjoint}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\sup \limits _{x^{*}\in B}|\langle x^{*},x_{n}\rangle |:(x_{n})_{n}\subseteq B_{E}\operatorname {disjoint} \}\\ \end{align*} for every norm bounded subset B of E^{*}. [ABSTRACT FROM AUTHOR]

Given a positive integer d and a 1 , ⋯ , a r a family of vectors in R d , { k 1 a 1 + ⋯ + k r a r : k 1 , ⋯ , k r ∈ Z } ⊂ R d is the so-called lattice generated by the family. In high dimensional integration, prescribed lattices are used for constructing reliable quadrature schemes. The quadrature points are the lattice points lying on the integration domain, typically the unit hypercube [ 0 , 1) d or a rescaled shifted hypercube. It is crucial to have a cost-effective method for enumerating lattice points within such domains. Undeniably, the lack of such fast enumeration procedures hinders the applicability of lattice rules. Existing enumeration procedures exploit intrinsic properties of the lattice at hand, such as periodicity, orthogonality, recurrences, etc. In this paper, we unveil a general-purpose fast lattice enumeration algorithm based on linear programming (named FLE-LP). [ABSTRACT FROM AUTHOR]

Discrete versions of the Laplace and Dirac operators have been studied in the context of combinatorial models of statistical mechanics and quantum field theory. In this paper, we introduce several variations of the Laplace and Dirac operators on graphs, and we investigate graph-theoretic versions of the Schrödinger and Dirac equations. We provide a combinatorial interpretation for solutions of the equations and we prove gluing identities for the Dirac operator on lattice graphs, as well as for graph Clifford algebras. [ABSTRACT FROM AUTHOR]

We elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation T=Ta+Tc$T = T_a + T_c$, where Ta$T_a$ is the sum of an absolutely order summable family of disjointness preserving operators and Tc$T_c$ is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators. [ABSTRACT FROM AUTHOR]

Let X and Y be compact Hausdorff spaces with Y hyperstonean. In this paper, we prove that if C(X, R) and C(Y, R) are lattice isomorphic then these Banach spaces are linearly isometric, and, consequently, X and Y are homeomorphic, which in turn implies that X is also hyperstonean. Actually, we prove more than what is ann [ABSTRACT FROM AUTHOR]

We implement the Boolean valued analysis of Banach spaces. The realizations of Banach spaces in a Boolean valued universe are lattice normed spaces. We present the basic techniques of studying these objects as well as the Boolean valued approach to injective Banach lattices. [ABSTRACT FROM AUTHOR]

Let W be the category of Archimedean lattice-ordered groups with weak order unit, Comp the category of compact Hausdorff spaces, and W Y -! Comp the Yosida functor, which represents a W-object A as consisting of extended real-valued functions A = D(Y A) and uniquely for various features. This yields topological mirrors for various algebraic (W-theoretic) properties providing close analysis of the latter. We apply this to the subclasses of a-projectable, and laterally a-complete objects, denoted P(a) and L(a), where a is a regular infinite cardinal or 8. Each W-object A has unique minimum essential extensions A = p(a)A = l(a)A in the classes P(a) and L(a), respectively, and the spaces Y p(a)A and Y l(a)A are recognizable (for the most part); then we write down what p(a)A and l(a)A are as functions on these spaces. The operators p(a) and l(a) are compared: we show that both preserve closure under all implicit functorial operations which are finitary. The cases of A = C(X) receive special attention. In particular, if (p < a) l(a)C(X) = C(Y l(a)C(X)), then X is finite. But (p = a) for infinite X, p(a)C(X) sometimes is, and sometimes is not, C(Y p(a)C(X)). [ABSTRACT FROM AUTHOR]

A continuous operator T between two Banach lattices E and F is called almost order-weakly compact, whenever for each almost order bounded subset A of E, T(A) is a relatively weakly compact subset of F. We show that the positive operator T from E into a Dedekind complete Banach lattice F is almost order-weakly compact iff T(xn)... 0 in F for each disjoint almost order bounded sequence {xn} in E. In this manuscript, we study some properties of this class of operators and its relationships with the others known classes of operators. [ABSTRACT FROM AUTHOR]

In this note it is established that a finite family of positive linear operators acting from an Archimedean vector lattice into an Archimedean -algebra with unit is disjointness preserving if and only if the polynomial presented in the form of the product of powers of these operators is orthogonally additive. A similar statement is established for the sum of polynomials represented as products of powers of positive operators. [ABSTRACT FROM AUTHOR]

An elliptic second-order differential operator on is considered, where , is a positive definite and bounded matrix-valued function periodic with respect to some lattice, and is a matrix first-order differential operator. Approximations for small of the operator-functions and in various operator norms are obtained. The results can be applied to study the behavior of the solution of the Cauchy problem for the hyperbolic equation . [ABSTRACT FROM AUTHOR]

In this paper, we study the long term dynamics of the stochastic Schrödinger delay lattice systems when the nonlinear drift and diffusion terms are both locally Lipschitz continuous. Based on the well-posedness of the system, we first prove the existence and uniqueness of weak pullback mean random attractors in a product Hilbert space. We then show the tightness of distribution laws of solutions and the existence of invariant measures. We further prove the set of all invariant measures of the delay system is tight and every limit point of invariant measures of the delay system must be an invariant measure of the limiting system as time delay approaches zero. The idea of uniform tail-estimates is employed to to establish the tightness of distributions of solutions as well as the set of invariant measures of the delay system. [ABSTRACT FROM AUTHOR]

We introduce a method based on Lipschitz pointwise transformations to define a distance on a Banach function space from its norm. We show how some specific lattice geometric properties (p-convexity, p-concavity, p-regularity) or, equivalently, some types of summability conditions (for example, when the terms of the terms in the sums in the range of the operator are restricted to the interval [ − 1 , 1 ] ) can be studied by adapting the classical analytical techniques of the summability of operators on Banach lattices, which recalls the work of Maurey. We show a technique to prove new integral dominations (equivalently, operator factorizations), which involve non-homogeneous expressions constructed by pointwise composition with Lipschitz maps. As an example, we prove a new family of integral bounds for certain operators on Lorentz spaces. [ABSTRACT FROM AUTHOR]

Gleason's theorem (Gleason 1957 J. Math. Mech. 6 885) is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice extend to positive linear functionals on the algebra of bounded operators . Over many years, and by the effort of various authors, the theorem has been broadened in its scope from type I to arbitrary von Neumann algebras (without type I 2 factors). Here, we prove a generalisation of Gleason's theorem to composite systems. To this end, we strengthen the original result in two ways: first, we extend its scope to dilations in the sense of Naimark (1943 Dokl. Akad. Sci. SSSR 41 359) and Stinespring (1955 Proc. Am. Math. Soc. 6 211) and second, we require consistency with respect to dynamical correspondences on the respective (local) algebras in the composition (Alfsen and Shultz 1998 Commun. Math. Phys. 194 87). We show that neither of these conditions changes the result in the single system case, yet both are necessary to obtain a generalisation to bipartite systems. [ABSTRACT FROM AUTHOR]

In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification EC$E_{\mathbb {C}}$ of a real vector lattice E and introduce the lateral order on EC$E_{\mathbb {C}}$. Our first main result asserts that the set of all fragments Fv$\mathfrak {F}_v$ of an element v∈EC$v\in E_{\mathbb {C}}$ of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification EC$E_{\mathbb {C}}$ of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator T:EC→X$\mathcal {T}: E_{\mathbb {C}} \rightarrow X$ from the complexification EC$E_{\mathbb {C}}$ of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator T$\mathcal {T}$ from EC$E_{\mathbb {C}}$ to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from EC$E_{\mathbb {C}}$ to a complex Banach lattice (ℓp(D)C$(\ell _p(\mathcal {D})_{\mathbb {C}}$ is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators. [ABSTRACT FROM AUTHOR]