Vector valued absolutely continuous functions on idempotent semigroups

Abstract

In this paper the concept of vector valued, absolutely continuous functions on an idempotent semigroup is studied. For F a function of bounded variation on the semigroup S of semicharacters with values of F in the Banach space X, let A = AC(S, X, F) be all those functions of bounded variation which are absolutely continuous with respect to F. A representation theorem is obtained for linear transformations from the space A to a Banach space which are continuous in the BV-norm. A characterization is also obtained for the collection of functions of A which ate Lipschitz with respect to F. With regards to the new integral being utilized it is shown that all absolutely continuous functions are integtable. Introduction. Absolutely continuous functions have been extensively studied in the literature. For example in [5] the dual space of the space of absolutely continuous functions is characterized. In [7], T. Hildebrandt gives a representation theorem for the linear functionals on BV[0, l] which are continuous in the weak topology. In [6] a representation theorem for linear functionals continuous in the variation norm on BV[0, l] is given. This representation is in terms of a socalled v-integral. The techniques of that paper, however, make strong use of the order on [O, l]. In [8] absolutely continuous functions and functions of bounded variation on idempotent semigroups are defined and these functions are identified with a certain class of finitely additive set functions. In [l], the identification in [8] is used to obtain a representation theorem. A characterization of the so-called Lipschitz functions in the setting of [8] is also obtained by the authors. The techniques of [l] depend on a result of Darst [2] which states that if u and v are two finitely additive real valued set functions with v « u, then v is the limit in the variation norm of finitely additive set functions zz defined by u (A) = f .s du, where s is a simple function. This re77 J n J A n n r suit does not in general hold true when u and v are vector valued. In this paper we study the concept of vector valued, absolutely continuous Received by the editors January 26, 1972. AMS(MOS) subject classifications (1970). Primary 46G10, 46E40; Secondary 28A25, 28A45.