Lipschitz Algebras and Derivations of von Neumann Algebras

Let M be an abelian von Neumann algebra andEan abelian operator M-bimodule. Then the domain of any ultraweakly closed derivation from M intoEis a Lipschitz algebra. Conversely, every Lipschitz algebra can be realized as such a domain. We discuss the relevance of this result to the theory of noncommutative metric spaces.