Centralizers of J-subspace lattice algebras

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 . A left (right) centralizer of A is an additive map Φ:A→A satisfying Φ(AB)=Φ(A)B(Φ(AB)=AΦ(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=PΦ(P) for every idempotent P in F(L), then ϕ is a centralizer.