Induced actions of [formula omitted]-Volterra operators on regular bounded martingale spaces

Abstract

A positive operator T:E→E on a Banach lattice E with an order continuous norm is said to be B-Volterra with respect to a Boolean algebra B of order projections of E if the bands canonically corresponding to elements of B are left fixed by T. A linearly ordered sequence ξ in B connecting 0 to 1 is called a forward filtration. A forward filtration can be used to lift the action of the B-Volterra operator T from the underlying Banach lattice E to an action of a new norm continuous operator Tˆξ:Mr(ξ)→Mr(ξ) on the Banach lattice Mr(ξ) of regular bounded martingales on E corresponding to ξ. 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 0. This function canonically induces a norm continuous shift operator s between two Banach lattices of regular bounded martingales. Moreover, the operators Tˆξ and s commute. Utilizing this fact with inductive limits, we construct a categorical limit space MT,ξ which is called the associated space of the pair (T,ξ). We present new connections between theories of Boolean algebras, abstract martingales and Banach lattices.