Integral functionals on $C^*$-algebra of vector-valued regulated functions

Abstract

‎In this paper we deal with the notion of regulated functions with‎ ‎values in a $C^*$-algebra $ \mathcal A $ and present examples using‎ ‎a special bi-dimensional $C^*$-algebra of triangular matrices‎. ‎We‎ ‎consider the Dushnik integral for these functions and shows that a‎ ‎convenient choice of the integrator function produces an integral‎ ‎homomorphism on the $C^*$-algebra of all regulated functions $‎ ‎G([a,b],\mathcal A) $‎. ‎Finally we construct a family of linear‎ ‎integral functionals on this $C^*$-algebra‎.