Banach lattices with locally compact representation spaces

Feldman, William; Porter, J. F.

doi:10.1007/bf01161412

Cited by 7 publications

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“…Given a Banach lattice E with a locally compact representation space X, we shall identify E with a space of continuous extended real-valued functions on X (each finite on a dense set) containing C k (X) as a dense order ideal [3]. We recall [3] that E has a locally compact representation space if and only if E contains a topological order partition (t.o.p.). A special case of a t.o.p.…”

Section: Preliminariesmentioning

confidence: 99%

“…for G. (For [3, condition (iii)], use J^(z) = <£(zo T).) Thus G has a locally compact representation space Z, a space of non-zero lattice homomorphisms z on the ideal generated by T(C k (X)) (see, for example, [3]).…”

Section: Let E and F Be Banach Lattices Having Locally Compact Represmentioning

confidence: 99%

“…This broad class of Banach lattices includes, for example, all Banach lattices with quasi-interior points, topological orthogonal systems or topological order partitions. Such lattices have been studied in [3,4,11,12].…”

mentioning

confidence: 99%

“…These Banach lattices can be represented as continuous extended real-valued functions on their locally compact representation spaces (for example see [3,4]). Theorems 1 and 2 below describe the lattice homomorphisms or disjointness preserving operators on these Banach lattices as weighted composition operators with respect to the representation spaces, generalizing results known for the function lattices of the type C(K) where K is compact (see [15]).…”

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Operators on Banach Lattices as Weighted Compositions

Feldman

Porter

1986

Journal of the London Mathematical Society

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In this paper we study lattice (Riesz) homomorphisms or, more generally, disjointness preserving (Lamperti, see [2]) operators between real Banach lattices having locally compact representation spaces in the sense of Schaefer [12, III 5.4]. We recall that a Banach lattice E is said to have a locally compact representation space if E contains a copy of C k (X), the continuous real-valued functions on a locally compact space X with compact support, as a dense (order) ideal. This broad class of Banach lattices includes, for example, all Banach lattices with quasi-interior points, topological orthogonal systems or topological order partitions. Such lattices have been studied in [3,4,11,12].These Banach lattices can be represented as continuous extended real-valued functions on their locally compact representation spaces (for example see [3,4]). Theorems 1 and 2 below describe the lattice homomorphisms or disjointness preserving operators on these Banach lattices as weighted composition operators with respect to the representation spaces, generalizing results known for the function lattices of the type C(K) where K is compact (see [15]). and for quasi-interior point spaces (see [10]). These results should also be compared with those of Abramovich [1]. We show that, if T is a lattice homomorphism between Banach lattices E and F, then, given / in E, Tf is of the form r(/o ), where r is a continuous real-valued function on the representation space for F and ^ is an essentially continuous map between the appropriate representation spaces. These results permit an analysis of the principal order ideal ZT generated by a lattice homomorphism T. Corollary 1 states that 3T is equal to the orthomorphisms composed with T. It also generalizes a result of Nagel [10, 3.3], and complements Haid's Theorem 2.7 in [6]. Theorem 5 states that each positive element in the closure of ST can be expressed as the composition with T of what we call a' partial orthomorphism'. This result and Corollary 3 are analogues of a Radon-Nikodym type theorem proved by Luxemburg and Schep (see [8,9]). Corollary 4 provides sufficient conditions for the principal ideal &~ generated by T to be closed. PreliminariesGiven a Banach lattice E with a locally compact representation space X, we shall identify E with a space of continuous extended real-valued functions on X (each finite on a dense set) containing C k (X) as a dense order ideal [3]. We recall [3] that E has a locally compact representation space if and only if E contains a topological order partition (t.o.p.). A special case of a t.o.p. is a quasi-interior point, a positive element generating a dense order ideal.For a continuous function r ^ 0 on X, we let P r = {xeX: r(x) > 0}, Z r = {xeX: r(x) = 0} and we denote the interior of Z r by Z°r. We shall use the fact that P r U Z°r is dense in X.

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Section: Preliminariesmentioning

confidence: 99%

Section: Let E and F Be Banach Lattices Having Locally Compact Represmentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

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Operators on Banach Lattices as Weighted Compositions

Feldman

Porter

1986

Journal of the London Mathematical Society

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“…We recall (see [2] or [5]) that a real Banach lattice G is said to have a locally compact representation space X if the space Ck(X ,K), all real-valued continuous functions on X with compact support, can be identified with a dense (order) ideal in G. In fact, G has a locally compact representation space if and only if E contains a topological order partition (see [2]). We will say that a complex Banach lattice (in the sense of Schaefer [5]) has a locally compact representation space X if E = G + iG and G has a locally compact representation space X.…”

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Compact weighted composition operators on Banach lattices

Feldman¹

1990

Proc. Amer. Math. Soc.

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Abstract.A characterization of compact (and M-weakly compact) weighted composition operators on real and complex Banach lattices which can be appropriately realized as function spaces is provided.The compact weighted composition operators on C (X), all bounded continuous complex valued functions on X with the supremum norm, have been characterized in the case of compact X by Kamowitz [4], and for completely regular X by Singh and Summers [6]. We recall that an operator F on C (X) is called a weighted composition operator if there exists an r on C (X) and a continuous map from X to X so that for each / in C (X), we haveIn this note we will characterize a class of compact weighted composition operators (and compact composition operators) on more general function spaces, which we will call Banach F-lattices. These include the Lp -spaces and the Banach lattices, both real and complex, with quasi-interior points (more generally, topological order partitions). We also demonstrate that these operators are equivalent to the A/-weakly compact weighted composition operators.We will call a complex Banach space E a Banach F-lattice if there exists a real Banach lattice G, which can be identified with equivalence classes of real valued functions on a completely regular space X, and E = G + iG can be identified with equivalence classes of complex valued functions on X satisfying the following conditions:(i) If / is equivalent to g , then {x. G X: f(x) = g(x)} is dense.(ii) Each continuous complex valued bounded function on X (or function with compact support in the case of a locally compact X ) represents an element in E. (iii) The vector space operations on E and the lattice operations on G correspond to the pointwise defined operations.

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Carleman operators on Banach lattices

Feldman

1988

Math Z

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An operator, not necessarily linear, will be called a Carleman operator if the image of the positive elements in the unit ball are bounded in the universal completion of the range space. For certain Banach lattices, a class of (not necessarily linear) Carleman operators is characterized in terms of an integral representation and in a more general setting as operators satisfying a pointwise finiteness condition. These operators though not linear are orthogonally additive and monotone. The notion of a Carleman operator from a normed space into a Riesz space was formulated by Grobler and van Eldik [4]. Namely, a linear map T from E into F is a Carleman operator if the image of the unit ball under T is order bounded in the universal completion of F. For a general operator T , we define T to be a Carleman operator if the image of the positive elements in the unit ball are order bounded in the universal completion of F. In this note, a class of operators that are orthogonally additive, monotone, and satisfy a condition with respect to scalar multiplication is considered. We provide a characterization of a Carleman operator among the continuous operators in this class as a type of integral operator (Theo-rem 1) and in Theorem 2, we provide a characterization of a Carleman operator for operators satisfying a translation property. In the absence of a measure, Theorem 3 provides a characterization of Carleman operators in terms of the finiteness condition. For linear Carleman operators, characterizations have been provided by the author in [3]. Orthogonally additive operators have been considered in a variety of settings, including several papers by Mazón and De León analyzing order bounded orthogonally additive operators (e.g., see [2], and [5]). Preliminaries We will assume that the Banach lattice E is representable in the sense of Schaefer [6], that is, there exists a locally compact topological space X with the property that E is Riesz isomorphic to an ideal in C ∞ (X), all the continuous extended real-valued functions (range [−∞, ∞]), each finite on a dense subset of X. We will study those Banach lattices with quasi-interior points and thus representable on a compact space X. Specifically, e in E is a quasi-interior point if the order ideal generated by e is dense in E. The ideal generated by e is then identified with C(X), the continuous real-valued functions on a compact space X. Here, the space

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Banach lattices with locally compact representation spaces

Cited by 7 publications

References 3 publications

Operators on Banach Lattices as Weighted Compositions

Operators on Banach Lattices as Weighted Compositions

Compact weighted composition operators on Banach lattices

Carleman operators on Banach lattices

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