On the Relative Reflexivity of Finitely Generated Modules of Operators

Abstract

Abstract. Let 32 be a von Neumann algebra on a Hubert space X with commutant 32 and centre £?. For each subspace 5? of 32 let ref^ (S")

Commutators and Square-Zero Elements in Banach Algebras

“…Another way of stating Theorem 6.5 is that every non-central element c ∈ A gives rise to a nonzero square-zero element in A of the form xcy for some x, y ∈ A such that xy = yx = 0. We remark that this is a generalization of the result by Magajna [22,Corollary 2.8], stating that for every noncentral element c in a C * -algebra A there exist x, y ∈ A such that xcy = 0 and (xcy) 2 = 0.…”

Algebraic reflexivity of linear transformations

“…The main result in [6] states that if S is an n-dimensional subspace of B(H), where H is a separable complex Hilbert space, then S is algebraically √ 2n -reflexive. While the above result answered a question raised in [7], the proof given in [6] is quite technical and relies on results from operator theory on Hilbert spaces. The main purpose of this paper is to generalize the above result to linear transformations on vector spaces and provide a simpler and more self-contained proof.…”

Complete positivity of elementary operators

“…Recently Z. Pan and the author resolved Magajna's problem [4]. We prove that if S is an n-dimensional subspace of L(H), then S is [ √ 2n]-reflexive and k(n) = [ √ 2n].…”