Derivations with a Hereditary Domain

This paper is concerned with partially defined derivations on Banach algebras and particularly on C*-algebras and H*-algebras, a subject whose study was motivated by the question of time evolution and spatial translation in quantum physics (see [2, 7] for a full account of the theory). It is well known [4] that everywhere defined derivations on semisimple Banach algebras are automatically continuous. In contrast, there are examples of non-closable partially defined derivations on a C*-algebra (see [2 ; Example 1.4.4]). One of the fundamental problems in the theory of partially defined derivations on Banach algebras is to investigate algebraic restrictions on the domain of such derivations which make these maps automatically closable. We shall study the closability of derivations whose domains fulfil the algebraic requirements given in the Abstract. It should be pointed out that the condition dim [BkfRad(A)] !¢ is automatically satisfied in each of the following situations: 1. B is any subalgebra of a semisimple complex Banach algebra; 2. B is any subalgebra of an H*-algebra with possibly nonzero annihilator ; 3. B is any subalgebra contained in the socle of a semiprime complex Banach algebra.