ABSTRACT
Structurable algebras were introduced by Allison, where the foundations for a fine structure theory of such algebras, later completed by Schafer and Smirnov, were laid. In fact, the results by Allison-Schafer-Smirnov give a complete structure theory for semisimple finite-dimensional structurable algebras. A long time before the appearance of structurable algebras, Steen introduced in particular types of complex Banach algebras called associative H*-algebras and Ambrose, under the assumption of zero annihilator obtained for these algebras a complete structure theory. The concept of differential operator can be extended verbatim to the context of modules over associative algebras. This chapter shows the characterization of the continuity of a differential operator on a Hilbert module in a way very similar to “algebraic” characterization of the continuity of differential operators on a left vector space over a division algebra paired with a right vector space.
