Integrator induced homomorphisms on Banach algebra valued regulated functions

Abstract

A function from a closed interval [a, b] to a Banach space X is regulated if all one-sided limits exist at each point of the interval. A function $$\alpha$$ from [a, b] to the space of all bounded linear transformations from X to a Banach space Y is an integrator for the regulated functions if, for each regulated function f, the Riemann-Stieltjes sums of f, with sampling points from the interiors of subintervals, converge to a vector in Y. When X and Y are Banach ( $$C^{^{*}}$$ -)algebras, we give a complete description of the class of all integrators that induce Banach (resp. $$C^{^{*}}$$ -)algebra homomorphisms. Each multiplicative integrator is associated with a nested family of idempotents (resp. selfadjoint projections). The main result of Fernandes and Arbach (Ann Funct Anal 3(2):21–31, 2012) exhibits a very special subclass of such integrators (which have $$\left\{ 0 \preceq \mathbb {1} _{_{\mathcal {B}}}\right\}$$ as the associated family of projections).