L-functional analysis

TL;DR

This work constructs a comprehensive framework for functional analysis with $\mathbb{L}$-valued scalars, where $\mathbb{L}$ is a Dedekind complete unital $f$-algebra. By developing $\mathbb{L}$-normed spaces, $\mathbb{L}$-Banach spaces, $\mathbb{L}$-operator theory, and $\mathbb{L}$-Hilbert spaces, the authors establish $\mathbb{L}$-valued analogues of $\ell^p$ spaces, Hahn–Banach, duality, and Riesz representation, culminating in a representation theorem that decomposes $\mathbb{L}$-Hilbert spaces as $\ell^2$-sums of homogeneous components. The approach leverages representation theory to connect KH-modules with vector lattices, enabling order-convergence-centered analyses and spectral-type decompositions that parallel classical Hilbert space theory. The framework provides tools for extending classical results to a lattice-valued scalar setting, with potential applications in ergodic theory and random functional analysis where scalar fields are replaced by $f$-algebras. Overall, the paper offers a systematic foundation for $\mathbb{L}$-valued functional analysis, bridging KH-module ideas and vector-lattice probability.

Abstract

Inspired by the theories of Kaplansky-Hilbert modules and probability theory in vector lattices, we generalise functional analysis by replacing the scalars $\mathbb{R}$ or $\mathbb{C}$ by a real or complex Dedekind complete unital $f$-algebra $\mathbb{L}$; such an algebra can be represented as a suitable space of continuous functions. We set up the basic theory of $\mathbb{L}$-normed and $\mathbb{L}$-Banach spaces and bounded operators between them, we discuss the $\mathbb{L}$-valued analogues of the classical $\ell^p$-spaces, and we prove the analogue of the Hahn-Banach theorem. We also discuss the basics of the theory of $\mathbb{L}$-Hilbert spaces, including projections onto convex subsets, the Riesz Representation theorem, and representing $\mathbb{L}$-Hilbert spaces as a direct sum of $\ell^2$-spaces.

Theorems & Definitions (167)

• Remark 2.1.1
• Remark 2.1.2
• Remark 2.1.3
• Remark 2.1.4
• Lemma 2.2.1
• Lemma 2.2.2
• proof
• Lemma 2.2.3
• proof
• Lemma 2.3.1