LINEAR MAPPINGS THAT PRESERVE THE DERIVATIONAL STRUCTURE OF C*-ALGEBRAS

Abstract

Given a C*-algebra A and a suitable set of derivations on A, we consider the algebras A _n of n-differentiable elements of A as described in [B], before passing to an analysis of important classes of bounded linear maps between two such spaces. We show that even in this general framework, all the main features of the theory for the case C^(m)(U) → C ^(p) (V) where U and V are open balls in suitable Banach spaces, are preserved (see for example [A-G-L], [Gu-L], [Ja] and [L]). As part of the theory developed we obtain a non-trivial extension of the Kleinecke-Shirokov theorem in the category of C*-algebras to unbounded partially defined *-derivations. This indicates the existence of a single mathematical principle governing both the non-increasibility of differentiability by continuous homomorphisms and the untenability of the Heisenberg Uncertainty Principle for bounded observables.