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The Riesz-Kantorovich formulas for $\mathbb{L}$-vector lattices

Tomas Chamberlain, Marten Wortel

TL;DR

The paper develops an $\mathbb{L}$-valued version of the Riesz-Kantorovich formulas for order-bounded $\mathbb{L}$-module homomorphisms, extending classical vector-lattice results to the setting of $\mathbb{L}$-vector lattices. It introduces an $\mathbb{L}$-valued extension lemma, analyzes Archimedean properties, and proves that the space of order-bounded operators $\mathcal{L}_b(X,Y)$ forms an $\mathbb{L}$-vector lattice equal to the regular operators $\mathcal{L}_r(X,Y)$, with explicit lattice operations given by pointwise sup/inf decompositions: $(S\vee T)(x)=\sup\{S(y)+T(x-y): y\in[0,x]\}$ and $(S\wedge T)(x)=\inf\{S(y)+T(x-y): y\in[0,x]\}$, along with $S^+(x)$, $S^-(x)$, and $|S|(x)$ formulas. The results establish Dedekind completeness of $\mathcal{L}_b(X,Y)$ and yield that the dual $X^\sim$ is a Dedekind-complete $\mathbb{L}$-vector lattice. This broadens the functional-analytic toolkit for $\mathbb{L}$-modules and $\mathbb{L}$-vector lattices, with potential impact on probabilistic and ergodic-theoretic contexts where conditional expectations and $\mathbb{L}$-valued norms arise.

Abstract

Let $\mathbb{L}$ be a Dedekind complete unital $f$-algebra. We prove the Riesz-Kantorovich formulas for order bounded $\mathbb{L}$-module homomorphisms from a directed partially ordered $\mathbb{L}$-module with the Riesz Decomposition Property into a Dedekind complete $\mathbb{L}$-vector lattice satisfying an additional mild condition.

The Riesz-Kantorovich formulas for $\mathbb{L}$-vector lattices

TL;DR

The paper develops an -valued version of the Riesz-Kantorovich formulas for order-bounded -module homomorphisms, extending classical vector-lattice results to the setting of -vector lattices. It introduces an -valued extension lemma, analyzes Archimedean properties, and proves that the space of order-bounded operators forms an -vector lattice equal to the regular operators , with explicit lattice operations given by pointwise sup/inf decompositions: and , along with , , and formulas. The results establish Dedekind completeness of and yield that the dual is a Dedekind-complete -vector lattice. This broadens the functional-analytic toolkit for -modules and -vector lattices, with potential impact on probabilistic and ergodic-theoretic contexts where conditional expectations and -valued norms arise.

Abstract

Let be a Dedekind complete unital -algebra. We prove the Riesz-Kantorovich formulas for order bounded -module homomorphisms from a directed partially ordered -module with the Riesz Decomposition Property into a Dedekind complete -vector lattice satisfying an additional mild condition.
Paper Structure (5 sections, 11 theorems, 6 equations)

This paper contains 5 sections, 11 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Archimedean Properties
  4. Extension Lemma
  5. The Riesz-Kantorovich Theorem

Key Result

Lemma 2.4

Let $\lambda \in \mathbb{L}$. Then there exists a sequence $\pi_n \ \mathord{\uparrow} \ 1$ in $\mathbb{P}$ such that $-n \leq \pi_n \lambda \leq n$ for all $n\in\mathbb{N}$. Furthermore, if $\lambda\in\mathbb{L}^+$, then $\pi_n \lambda \ \mathord{\uparrow} \ \lambda$.

Theorems & Definitions (24)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Definition 3.1
  • ...and 14 more