Annihilator-preserving maps, multipliers, and derivations

For a commutative subspace lattice L in a von Neumann algebra N and a bounded linear map f : N boolean AND alg L -> B(H), we show that if Af (B)C = 0 for all A, B, C is an element of N boolean AND alg L satisfying AB = BC = 0, then f is a generalized derivation. For a unital C*-algebra A, a unital Banach A-bimodule M, and a bounded linear map f : A -> M, we prove that if f (A) B = 0 for all A, B is an element of A with AB = 0. then f is a left multiplier; as a consequence, every bounded local derivation from a C*-algebra to a Banach A-bimodule is a derivation. We also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier. (C) 2009 Elsevier Inc. All rights reserved.