Applications of Spaces with Filters to Archimedean ℓ-Groups with Weak Unit

1989 ◽  
pp. 99-112 ◽  
Author(s):  
Richard N. Ball ◽  
Anthony W. Hager
Keyword(s):  
Weak Unit

scholarly journals Epicompletetion of archimedean l–groups and vector lattices with weak unit

1990 ◽  
Vol 48 (1) ◽  
pp. 25-56 ◽  
Author(s):  
Richard N. Ball ◽  
Anthony W. Hager
Keyword(s):  
Vector Lattice ◽  
Boolean Algebras ◽  
Weak Order ◽  
Minimal Extension ◽  
Order Unit ◽  
Weak Unit ◽  
Vector Lattices ◽  
Measurable Functions ◽  
Baire Functions

AbstractIn the category W of archimedean l–groups with distinguished weak order unit, with unitpreserving l–homorphism, let B be the class of W-objects of the form D(X), with X basically disconnected, or, what is the same thing (we show), the W-objects of the M/N, where M is a vector lattice of measurable functions and N is an abstract ideal of null functions. In earlier work, we have characterized the epimorphisms in W, and shown that an object G is epicomplete (that is, has no proper epic extension) if and only if G ∈ B. This describes the epicompletetions of a give G (that is, epicomplete objects epically containing G). First, we note that an epicompletion of G is just a “B-completion”, that is, a minimal extension of G by a B–object, that is, by a vector lattice of measurable functions modulo null functions. (C[0, 1] has 2c non-eqivalent such extensions.) Then (we show) the B–completions, or epicompletions, of G are exactly the quotients of the l–group B(Y(G)) of real-valued Baire functions on the Yosida space Y(G) of G, by σ-ideals I for which G embeds naturally in B(Y(G))/I. There is a smallest I, called N(G), and over the embedding G ≦ B(Y(G))/N(G) lifts any homorphism from G to a B–object. (The existence, though not the nature, of such a “reflective” epicompletion was first shown by Madden and Vermeer, using locales, then verified by us using properties of the class B.) There is a unique maximal (not maximum) such I, called M(Y(G)), and B(Y(G))/M(Y(G)) is the unique essentialBcompletion. There is an intermediate σ -ideal, called Z(Y(G)), and the embedding G ≦ B(y(G))/Z(Y(G)) is a σ-embedding, and functorial for σ -homomorphisms. The sistuation stands in strong analogy to the theory in Boolean algebras of free σ -algebras and σ -extensions, though there are crucial differences.

The Yosida space and representation of the projectable hull of an archimedean ℓ-group with weak unit

2017 ◽  
Vol 40 (1) ◽  
pp. 57-61
Author(s):  
Anthony W. Hager ◽  
Warren Wm. McGovern
Keyword(s):  
Weak Unit

scholarly journals The HSP-Classes of Archimedean l-groups with Weak Unit

2010 ◽  
Vol 19 (S1) ◽  
pp. 13-24
Author(s):  
Bernhard Banaschewski ◽  
Anthony Hager
Keyword(s):  
Weak Unit

On hull classes of ℓ-groups and covering classes of spaces

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
Ricardo Carrera ◽  
Anthony Hager
Keyword(s):  
Hausdorff Spaces ◽  
Central Idea ◽  
Weak Unit ◽  
Hull Class ◽  
Continuous Maps ◽  
The Relationship

AbstractW denotes the category of archimedean ℓ-groups with designated weak unit and ℓ-homomorphisms that preserve the weak unit. Comp denotes the category of compact Hausdorff spaces with continuous maps. The Yosida functor is used to investigate the relationship between hull classes in W and covering classes in Comp. The central idea is that of a hull class whose hull operator preserves boundedness. We demonstrate how the Yosida functor may be used to identify hull classes in W and covering classes in Comp. In addition, we exhibit an array of order preserving bijections between certain families of hull classes and all covering classes, one of which was recently produced by Martínez. Lastly, we apply our results to answer a question of Knox and McGovern about the class of all feebly projectable ℓ-groups.

Existence of Invariant Weak Units in Banach Lattices: Countably Generated Left Amenable Semigroup of Operators

1993 ◽  
Vol 45 (6) ◽  
pp. 1299-1312
Author(s):  
K. Prabaharan
Keyword(s):  
Linear Operators ◽  
Banach Lattices ◽  
Dual Cone ◽  
Sufficient Condition ◽  
Semigroup Of Operators ◽  
Necessary And Sufficient Condition ◽  
Amenable Semigroup ◽  
Weak Unit ◽  
Necessary And Sufficient ◽  
Uniformly Bounded

AbstractLet Σ be a countably generated left amenable semigroup and ﹛Tσ|σ ∈ Σ﹜ be a representation of Σ as a semigroup of positive linear operators on a weakly sequentially complete Banach lattice E with a weak unit e. It is assumed Tσ are uniformly bounded. It is shown that a necessary and sufficient condition for the existence of a weak unit invariant under ﹛Tσ | σ ∈ Σ﹜ is that inf σ∈Σ H(Tσe) > 0 for all nonzero H in the positive dual cone of E.

Functorial polar functions

2011 ◽  
Vol 61 (3) ◽  
Author(s):  
Ricardo Carrera
Keyword(s):  
Hausdorff Spaces ◽  
Weak Unit ◽  
Closed Sets ◽  
Polar Function ◽  
Uncountable Cardinal ◽  
Functorial Polar Functions ◽  
Skeletal Maps ◽  
Polar Functions ◽  
Regular Closed

AbstractW∞ denotes the category of archimedean ℓ-groups with designated weak unit and complete ℓ-homomorphisms that preserve the weak unit. CmpT2,∞ denotes the category of compact Hausdorff spaces with continuous skeletal maps. This work introduces the concept of a functorial polar function on W∞ and its dual a functorial covering function on CmpT2,∞.We demonstrate that functorial polar functions give rise to reflective hull classes in W ∞ and that functorial covering functions give rise to coreflective covering classes in CmpT 2,∞. We generate a variety of reflective and coreflecitve subcategories and prove that for any regular uncountable cardinal α, the class of α-projectable ℓ-groups is reflective in W ∞, and the class of α-disconnected compact Hausdorff spaces is coreflective in CmpT 2,∞. Lastly, the notion of a functorial polar function (resp. functorial covering function) is generalized to sublattices of polars (resp. sublattices of regular closed sets).

A Mean Ergodic Theorem for Multiparameter Superadditive Processes on Banach Lattices

1990 ◽  
Vol 42 (6) ◽  
pp. 1018-1040
Author(s):  
Felix Lee
Keyword(s):  
Ergodic Theorem ◽  
Banach Lattice ◽  
Measure Space ◽  
Invariant Function ◽  
Positive Cone ◽  
Banach Lattices ◽  
Weak Unit ◽  
Mean Ergodic Theorem ◽  
Maximal Invariant

LetEbe a Banach Lattice. We will considerEto be weakly sequentially complete and to have a weak unitu. Thus we may representEas a lattice of real valued functions defined on a measure space (χ,,μ). There is a setR⊂χsuch thatRsupports a maximal invariant functionΦfor a postive contractionTonE[5]. LetN=χ—Rbe the complement ofR. Akcoglu and Sucheston showed thatwhereE+is the positive cone ofE. If in addition a monotone condition (UMB) is satisfied, then the same authors showed [4] thatconverges in norm.

scholarly journals Operations that preserve integrability, and truncated Riesz spaces

2020 ◽  
Vol 32 (6) ◽  
pp. 1487-1513
Author(s):  
Marco Abbadini
Keyword(s):  
Real Number ◽  
Finite Measure ◽  
Variety Of Algebras ◽  
Riesz Spaces ◽  
Weak Unit ◽  
Measure Spaces ◽  
Concrete Model

AbstractFor any real number {p\in[1,+\infty)}, we characterise the operations {\mathbb{R}^{I}\to\mathbb{R}} that preserve p-integrability, i.e., the operations under which, for every measure μ, the set {\mathcal{L}^{p}(\mu)} is closed. We investigate the infinitary variety of algebras whose operations are exactly such functions. It turns out that this variety coincides with the category of Dedekind σ-complete truncated Riesz spaces, where truncation is meant in the sense of R. N. Ball. We also prove that {\mathbb{R}} generates this variety. From this, we exhibit a concrete model of the free Dedekind σ-complete truncated Riesz spaces. Analogous results are obtained for operations that preserve p-integrability over finite measure spaces: the corresponding variety is shown to coincide with the much studied category of Dedekind σ-complete Riesz spaces with weak unit, {\mathbb{R}} is proved to generate this variety, and a concrete model of the free Dedekind σ-complete Riesz spaces with weak unit is exhibited.

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