$L\sp{1}(I,\,X)$ with order convolution

Abstract

Abstract. It is shown that the maximal ideal space of /.'(/, X) is (0, 1] X ?M(X), where %Jl(X) denotes the maximal ideal space of the Banach algebra X. The Gelfand topology on the Carrier space (0, 1] x 3Jl(X) coincides with the topology which is the product of the interval topology in (0, 1] and the Gelfand topology on SDi(A'). Moreover, the Gelfand transform has the form of an indefinite integral.Let 7 denote the interval [0, 1] of real numbers. 7 is a totally ordered set with the semigroup structure obtained by defining xy = max{x, v}. When 7 is provided with the usual interval topology, 7 is a compact topological semigroup. Let C(7) denote the linear space of all complex-valued continuous functions on 7. We give C(7) the usual norm