Induced actions of $\mathfrak{B}$-Volterra operators on regular bounded martingale spaces

TL;DR

This paper explores how $rak{B}$-Volterra operators act on regular bounded martingale spaces, establishing their properties, associated shift operators, and a new categorical limit space linking Boolean algebras, martingales, and Banach lattices.

Contribution

It introduces a novel framework for lifting $rak{B}$-Volterra operators to martingale spaces and constructs a limit space connecting Boolean algebras, martingales, and Banach lattices.

Findings

Construction of a shift operator $ extbf{s}$ that commutes with $rak{B}$-Volterra actions.

Definition of a categorical limit space $rak{M}_{T, ext{ extbf{xi}}}$ for pairs $(T, ext{ extbf{xi}})$.

New connections established between Boolean algebras, abstract martingales, and Banach lattices.

Abstract

A positive operator $T:E\to E$ on a Banach lattice $E$ with an order continuous norm is said to be $\mathfrak{B}$-Volterra with respect to a Boolean algebra $\mathfrak{B}$ of order projections of $E$ if the bands canonically corresponding to elements of $\mathfrak{B}$ are left fixed by $T$. A linearly ordered sequence $\xi$ in $\mathfrak{B}$ connecting $\textbf{0}$ to $\textbf{1}$ is called a forward filtration. A forward filtration can be to used to lift the action of the $\mathfrak{B}$-Volterra operator $T$ from the underlying Banach lattice $E$ to an action of a new norm continuous operator $\hat{T}_{\xi}\colon \mathcal{M}_{r}(\xi) \to \mathcal{M}_{r}(\xi)$ on the Banach lattice $\mathcal{M}_{r}(\xi)$ of regular bounded martingales on $E$ corresponding to $\xi$. In the present paper, we study properties of these actions. The set of forward filtrations are left fixed by a function which erases the first order projection of a forward filtration and which shifts the remaining order projections towards $\textbf{0}$. This function canonically induces a norm continuous shift operator $\textbf{s}$ between two Banach lattices of regular bounded martingales. Moreover, the operators $\hat{T}_{\xi}$ and $\textbf{s}$ commute. Utilizing this fact with inductive limits, we construct a categorical limit space $\mathcal{M}_{T,\xi}$ which is called the associated space of the pair $(T,\xi)$. We present new connections between theories of Boolean algebras, abstract martingales and Banach lattices.