On derivable mappings

Abstract

A linear mapping δ from an algebra A into an A -bimodule M is called derivable at c ∈ A if δ ( a ) b + a δ ( b ) = δ ( c ) for all a , b ∈ A with a b = c . For a norm-closed unital subalgebra A of operators on a Banach space X, we show that if C ∈ A has a right inverse in B ( X ) and the linear span of the range of rank-one operators in A is dense in X then the only derivable mappings at C from A into B ( X ) are derivations; in particular the result holds for all completely distributive subspace lattice algebras, J -subspace lattice algebras, and norm-closed unital standard algebras of B ( X ) . As an application, every Jordan derivation from such an algebra into B ( X ) is a derivation. For a large class of reflexive algebras A on a Banach space X, we show that inner derivations from A into B ( X ) can be characterized by boundedness and derivability at any fixed C ∈ A , provided C has a right inverse in B ( X ) . We also show that if A is a canonical subalgebra of an AF C ∗ -algebra B and M is a unital Banach A -bimodule, then every bounded local derivation from A into M is a derivation; moreover, every bounded linear mapping from A into B that is derivable at the unit I is a derivation.