Centralizers of [formula omitted]-subspace lattice algebras

Abstract

Let L be a J -subspace lattice on a Banach space and let A be a subalgebra of Alg L which contains F ( L ) , where F ( L ) denotes the algebra of all finite rank operators in Alg L . A left (right) centralizer of A is an additive map Φ : A → A satisfying Φ ( AB ) = Φ ( A ) B ( Φ ( AB ) = A Φ ( B ) ) for all A , B ∈ A , and a centralizer of A is a both left and right centralizer. In this paper, we describe the general form of a centralizer of A , and show that every linear local left (right) centralizer of A is a left (right) centralizer. Also, it is proved that if a linear map Φ : F ( L ) → F ( L ) satisfies Φ ( P ) = Φ ( P ) P = P Φ ( P ) for every idempotent P in F ( L ) , then ϕ is a centralizer.