Topological reflexivity of the spaces of (α, β)-derivations on operator algebras

Abstract

Abstract. We prove that the spaces of (α, β)-derivations on certain operator algebras are topologically reflexive in the weak operator topology. Characterizations of automorphisms and (α, β)-derivations on reflexive algebras are also given. Introduction and preliminaries.The study of reflexive linear subspaces of the algebra B(X) of all bounded linear operators on the Banach space X represents a very active research area in operator theory (see [8] for a beautiful general view). The originators of this research direction are Kadison and Larson. In [11], Kadison studied local derivations from a von Neumann algebra R into a dual R-bimodule M. A linear map from R into M is called a local derivation if it agrees with some derivation at each point in the algebra (the derivations possibly varying from point to point). This investigation was motivated by the study of the Hochschild cohomology of operator algebras. Independently, Larson and Sourour [13] studied local derivations of B(X); they proved that every local derivation of B(X) is a derivation. Since then, a considerable amount of work has been done concerning local derivations of various algebras. See, for example, [4,5,9,10,18]. In this paper, we will extend this research to a more general setting.Let us now define our concept of topological reflexivity. Let X be a Banach space. We denote by L(X) and B(X) the algebras of all linear and all bounded linear operators on X respectively. Let A be a subalgebra of B(X). Given two subsets E ⊆ L(A) and F ⊆ B(A), if τ is a vector topology on A, we define

Hyperreflexivity Constants of the Bounded -Cocycle Spaces of Group Algebras and C*-Algebras

“…He showed that if G is a locally compact group with an open subgroup of polynomial growth and X a Banach L 1 (G)bimodule, then Z n (L 1 (G), X), the space of bounded n-cocycles from A into X, is reflexive. More results related to these questions can be found in [6,8,12,13,14,15,16,18,19,20,29].…”

“…13), implies that|ϕ( f e 1 , h) − ϕ( f, e 1 h)| = |Φ( f e 1 ⊗ h − f ⊗ e 1 h)| |φ( f e n , h) − φ( f, e n h)| ≤ 8π 3all f, h ∈ A(T), where e n denotes the function in A(T) defined by e n (s) = e ins (s ∈ R, n ∈ Z).…”