Abstract. Let M be a von Neumann algebra and 5 and T be commuting "-automorphisms on M satisfying the equation: 5 + S"1 = T + T~l. It is proved that M can be decomposed by a central projection p in M such that S = T on Mp and S= T-1 on M(\ -p).

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Abstract. In this note we investigate some properties of a-derivations on prime and semiprime rings. We establish some identities for a commuting a-derivation d on a semiprime ring R and show that d maps R into its center and obtain some well-known results as a consequence. We also generalize Posner's theorem on the composition of derivations for a-derivations and as an application resolve a functional equation of automorphisms on certain prime rings.

Stochastic generalizations of some fixed point theorems on a class of nonconvex sets in a locally bounded topological vector space are established. As applications, Brosowski-Meinardus type theorems about random invariant approximation are obtained. This work extends or provides stochastic versions of several well known results.

Abstract. Let X be a completely regular Hausdorff space, E a Hausdorff topological vector space, V a Nachbin family of weights on X, and CVi, (X,E) the weighted space of continuous /^-valued functions on X. Let B(E) be the vector space of all continuous linear mappings from E into itself, endowed with the topology of uniform convergence on bounded sets. If φ : X -* B(E) is a continuous mapping and / € CV^(X, E), let(χ ζ X). In this paper we give a necessary and sufficient condition for Μφ to be the multiplication operator (i.e. continuous selfmapping) on CV¡, (X, E), where E is a general space or a locally bounded space. These results extend recent work of Singh and Manhas to a non-locally convex setting and that of the authors where φ has been considered to be a complex or ^-valued map. IntroductionThe fundamental work on weighted space of continuous scalar-valued functions has been done mainly by Nachbin (X, E), assuming E to be a locally convex space or a normed space. This paper is a continuation of our earlier paper [3] in which we have studied, in the non-locally convex framework, multiplication operators on weighted function spaces which are induced by scalar-and vector-valued mappings.The purpose of this paper is to characterize those multiplication operators which are induced by operator-valued mappings. These results extend,

ABSTRACT. Let X be a completely regular Hausdorff space, E a topological vector space, V a Nachbin family of weights on X, and CVo(X, E) the weighted space of continuous E-valued functions on X. Let 0 X C be a mapping, f E CVo(X, E) and define Me(f) Of (pointwise). In case E is a topological algebra, p X E is a mapping then define M,(f) pf (pointwise). The main purpose of this paper is to give necessary and sufficient conditions for Me and M, to be the multiplication operators on CV0 (X, E) where E is a general topological space (or a suitable topological algebra) which is not necessarily locally convex. These results generalize recent work of Singh and Manhas based on the assumption that E is locally convex.

We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: let R be a semiprime ring with center Z(R), and let f , g be derivations of R such that f (x)x + xg(x) ∈ Z(R) for all x ∈ R, then f and g are central.As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivation f of a semiprime ring R is skew-commuting.2000 Mathematics Subject Classification: 47A50, 47B50. Introduction and preliminaries. Throughout, R denotes a ring with center Z(R).We write [x, y] for xy − yx. We will frequently use the identities Every central mapping is obviously commuting but not conversely, in general. A lot of work has been done on centralizing mappings (see, e.g., [3,4,5] and the references therein). A mapping f : R → R is called skew-centralizing if f (x)x + xf (x) ∈ Z(R)for all x ∈ R; in particular, if f (x)x + xf (x) = 0 for all x ∈ R, then it is called skew-commuting. We denote the radical of a Banach algebra A by rad(A).We now recall some facts concerning semiprime rings and their extended centroids. For any semiprime ring R, one can construct the ring of quotients Q of R [1]. As R can be embedded isomorphically in Q, we consider R as a subring of Q. If the element q ∈ Q commutes with every element in R, then q belongs to C, the center of Q. C contains the centroid of R and is called the extended centroid of R. In general, C is a von Neumann regular ring, and it is a field if and only if R is a prime [1, Theorem 5]. For more information on extended centroid of R, we refer to [2].Brešar [6, Theorem 2] has proved that if R is a prime ring of characteristic not 2 and f : R → R is an additive skew-commuting mapping (i.e., f satisfies f (x)x + xf (x) = 0 for all x ∈ R), then f = 0.Moreover, Brešar [5, Theorem 4.1] has considered a pair of derivations on a prime ring and has proved the following. Let R be a prime ring and U a nonzero left ideal of R. Suppose that the derivations d and g of R are such that d(u)u − ug(u) ∈ Z(R) for all u ∈ U . If d ≠ 0, then R is commutative.