In this paper we study the classes of almost f-algebras and d-algebras. Apart from a survey of known properties we present some ‘new’ properties of almost f-algebras and d-algebras and we consider their connection with f-algebras. To be more precise, we show, among other things, in an elementary intrinsic manner, that every Archimedean almost f-algebra is commutative. This result is a considerable improvement upon the fact that every Archimedean f-algebra is commutative. Furthermore, we give a description of the set of nilpotent elements in both an Archimedean d-algebra and an Archimedean almost f-algebra.
Abstract. The paper deals mainly with the theory of algebra ideals and order ideals in /-algebras. Necessary and sufficient conditions are established for an algebra ideal to be prime, semiprime or idempotent. In a uniformly complete /-algebra with unit element every algebra ideal is an order ideal iff the /-algebra is normal. This result is based on the fact that the range of every orthomorphism in a uniformly complete normal Riesz space is an order ideal.1. Introduction. In the present paper we investigate how far several well-known results about ideal theory in the space C(X) of all real continuous functions on a completely regular Hausdorff space X can be extended to the most natural generalization of C(X), a uniformly complete Archimedean /-algebra with unit element. It should be observed that the class of these /-algebras contains the class of all C(X) as a proper subset. As an example we mention the space 91L([0, 1]) of all real Lebesgue measurable functions on [0, 1] with the usual identification of almost equal functions. This is a uniformly complete Archimedean /-algebra with unit element (with respect to the familiar operations and the natural partial ordering). As is well known (see e.g. Abstract /-ring theory and /-algebra theory have been studied by many authors (see e.g. [1,5,6,7,13,14,17, 21]). Some of these authors (see e.g. §8 in the paper [6] by G. Birkhoff and R. S. Pierce) define a /-ring as a lattice ordered ring with the property that u /\v = 0, w > 0 implies (uw) Ac = (wm) /\ v = 0. Others (see e.g. Definition 9.1.1 in the book [5] by A. Bigard, K. Keimel and S. Wolfenstein, and Chapter IX, §2 in the book [7] by L. Fuchs) define an /-ring as a lattice ordered ring which is isomorphic to a subdirect union of totally ordered rings. It is often
A complete description of positive projections in ideals of measurable functions is given in terms of conditional expectation-type operators. Introduction.As is well-known, conditional expectation operators on various function spaces exhibit a number of remarkable properties related either to the underlying order structure of the given function space, or to the metric structure when the function space is equipped with a norm. Such operators are necessarily positive projections which are averaging in a precise sense to be described below and in certain normed function spaces are contractive for the given norm. is based on the approach of Douglas for the case of L ι -spaces and makes essential use of the underlying metric structure via an appeal to a well-known interpolation theorem for rearrangement invariant KB-spaces. The approach to the present paper is on the other hand purely algebraic and uses the underlying order structure via a suitable adaptation of the ideas of Moy on averaging operators. The basic link which in essence reduces the more general problem of describing positive projections to the study of averaging operators is provided by a result of Kelley [K] which implies that each positive projection on an L°° -space with range a vector sublattice containing the constants is necessarily averaging.After gathering some preliminary notions in §2, we consider in §3 the relation between conditional expectations and averaging operators
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