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

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 $\text{0}$ to $\text{1}$ 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 ${\hat{T}}_{\xi }:{\mathcal{M}}_r(\xi )\to {\mathcal{M}}_r(\xi )$ 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 $\text{0}$. This function canonically induces a norm continuous shift operator $\text{s}$ between two Banach lattices of regular bounded martingales. Moreover, the operators ${\hat{T}}_{\xi }$ and $\text{s}$ 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.

23 pages, 0 figures