Induced actions of B-Volterra operators on regular bounded martingale spaces

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 T ˆ ξ : M r ( ξ ) → M r ( ξ ) on the Banach lattice M r ( 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 T ˆ ξ and s commute. Utilizing this fact with inductive limits, we construct a categorical limit space M T , 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.