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.
Categories: math.FA