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arxiv: 2508.12568 · v2 · submitted 2025-08-18 · 🧮 math.FA

Riesz representation theorems for vector lattices and Banach lattices of regular operators

Marcel de Jeu , Xingni Jiang This is my paper

Pith reviewed 2026-05-18 23:21 UTC · model grok-4.3

classification 🧮 math.FA
keywords Riesz representation theoremvector latticesregular operatorsE-valued measuresBorel measureslocally compact spacesnorm-to-order bounded operatorsBanach lattices
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The pith

The lattice of norm-to-order bounded operators from C_c(X) or C_0(X) into a Dedekind complete normal vector lattice E is isomorphic to the lattice of E-valued regular Borel measures.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes an isomorphism between the vector lattice of norm-to-order bounded operators mapping continuous functions with compact support or vanishing at infinity on a locally compact Hausdorff space X into E, where E is a Dedekind complete normal vector lattice, and the vector lattice of E-valued regular Borel measures on X. This generalizes the classical Riesz representation theorems for the duals of these function spaces, which are recovered as the special case when E is the real line. A reader would care because it provides a way to represent such operators via measures in a lattice setting, potentially aiding in the study of positivity and boundedness in functional analysis. When E is an order continuous Banach lattice, the isomorphism is isometric, preserving the norm structure as well.

Core claim

For a non-empty locally compact Hausdorff space X and a Dedekind complete normal vector lattice E, the vector lattice of norm to order bounded operators from C_c(X) or C_0(X) into E is isomorphic to the vector lattice of E-valued regular Borel measures on X. When E is an order continuous Banach lattice, the isomorphism is an isometric isomorphism between Banach lattices. When X is compact, every regular operator from C(X) into E is norm to order bounded. For some spaces E, such as KB-spaces or the regular operators on a KB-space, every regular operator from C_0(X) into E is norm to order bounded.

What carries the argument

The lattice isomorphism that associates each norm-to-order bounded operator to its corresponding E-valued regular Borel measure on X.

If this is right

  • When E is the real numbers the result reduces to the classical Riesz representation theorems for the order and norm duals of C_c(X) and C_0(X).
  • When X is compact every regular operator from C(X) into E is automatically norm-to-order bounded.
  • For E a KB-space or the lattice of regular operators on a KB-space every regular operator from C_0(X) into E is norm-to-order bounded.
  • Additional structural results hold for the full space of regular operators from C_c(X) into an order continuous Banach lattice E.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same correspondence may allow measure-theoretic arguments to be transferred directly to questions about positivity and order in lattices of operators.
  • It raises the possibility of similar representation theorems for operators on other function spaces such as continuous functions vanishing at infinity on non-locally-compact spaces.
  • Vector-valued regular measures arising this way could serve as a concrete model for integration against operator-valued kernels.

Load-bearing premise

E must be a Dedekind complete normal vector lattice, and order continuous for the isometric isomorphism to hold.

What would settle it

A specific Dedekind complete normal vector lattice E together with a locally compact Hausdorff space X and an explicit norm-to-order bounded operator from C_c(X) to E that cannot be represented by any E-valued regular Borel measure.

read the original abstract

For a non-empty locally compact Hausdorff space $X$ and a Dedekind complete normal vector lattice $E$, we show that the vector lattice of norm to order bounded operators from ${\text C}_{\text c}(X)$ or ${\text C}_0(X)$ into $E$ is isomorphic to the vector lattice of $E$-valued regular Borel measures on $X$. When $E$ is an order continuous Banach lattice, the isomorphism is an isometric isomorphism between Banach lattices. When $X$ is compact, every regular operator from $\mathrm{C}(X)$ into $E$ is norm to order bounded. For some spaces $E$, such as KB-spaces or the regular operators on a KB-space, every regular operator from ${\mathrm C}_0(X)$ into $E$ is norm to order bounded. Additional results are obtained for the whole space of regular operators from ${\text C}_{\text c}(X)$ into an order continuous Banach lattice. As a preparation, vector lattices and Banach lattices, resp. cones, of measures with values in a Dedekind complete vector lattice $E$, resp. in the extended positive cone of $E$, are investigated, as well as vector and Banach lattices of norm to order bounded operators. When $E$ is the real numbers, our results specialise to the well-known Riesz representation theorems for the order and norm duals of ${\text C}_{\text c}(X)$ and ${\text C}_0(X)$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper claims that for a non-empty locally compact Hausdorff space X and a Dedekind complete normal vector lattice E, the vector lattice of norm-to-order bounded operators from C_c(X) or C_0(X) into E is lattice-isomorphic to the vector lattice of E-valued regular Borel measures on X. When E is an order-continuous Banach lattice the isomorphism is isometric. Additional results include that every regular operator from C(X) into E is norm-to-order bounded when X is compact, that the same holds for all regular operators from C_0(X) into KB-spaces or regular operators on KB-spaces, and further results on the full space of regular operators from C_c(X) into order-continuous Banach lattices. The constructions reduce exactly to the classical Riesz theorems when E = ℝ.

Significance. If the mutually inverse order-preserving maps between the operator and measure lattices are correctly constructed, the work supplies a coherent lattice-theoretic extension of the classical Riesz representation theorems to E-valued measures and regular operators. The explicit reduction to the scalar case and the identification of natural classes (order-continuous Banach lattices, KB-spaces) where regularity and norm-to-order boundedness coincide are concrete strengths. The preparatory development of cones and lattices of E-valued measures supplies reusable infrastructure for further work in vector-lattice functional analysis.

minor comments (2)
  1. [Abstract] The abstract states that the isomorphism is isometric when E is order continuous, but the precise norm on the space of measures is not recalled in the abstract; a single sentence recalling the definition would improve readability.
  2. [Preparatory sections on measures] In the preparatory sections on E-valued measures, the notation for the extended positive cone of E is introduced without an explicit cross-reference to the definition of normality used later; adding the reference would clarify the dependence.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript. We are pleased that the referee recognizes the coherent lattice-theoretic extension of the classical Riesz representation theorems, the explicit reduction to the scalar case, and the identification of natural classes such as order-continuous Banach lattices and KB-spaces where regularity and norm-to-order boundedness coincide. The referee recommends minor revision, but the report does not list any specific major comments requiring detailed point-by-point rebuttal.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper constructs spaces of E-valued measures and norm-to-order bounded operators as preparatory material, then exhibits mutually inverse order-preserving maps between them to establish the lattice isomorphism. When E=ℝ the result reduces exactly to the classical Riesz representation theorems for C_c(X) and C_0(X), which are treated as external consistency checks rather than derived within the paper. No load-bearing self-citation, no parameter fitting renamed as prediction, and no definitional reduction where the claimed isomorphism is assumed in the inputs. The argument relies on standard vector lattice and measure theory constructions that are independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on these foundational assumptions from topology and the theory of vector lattices without introducing new free parameters or entities.

axioms (2)
  • domain assumption X is a non-empty locally compact Hausdorff space
    This topological condition is necessary for defining C_c(X), C_0(X), and regular Borel measures.
  • domain assumption E is a Dedekind complete normal vector lattice
    This property of E ensures the existence of suprema and the lattice operations needed for the isomorphism.

pith-pipeline@v0.9.0 · 5798 in / 1360 out tokens · 78134 ms · 2026-05-18T23:21:56.615608+00:00 · methodology

discussion (0)

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Reference graph

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