On One-Parameter Subgroups of *-Automorphisms on Operator Algebras and the Corresponding Unbounded Derivations

Abstract

1. It is known that if a derivation in C*-algebras is everywhere defined, then it is bounded and is weakly inner ([8]). On the other hand, in mathematical physics we have to confront unbounded derivations which are defined as infinitesimal generators of oneparameter subgroups of *-automorphisms on C*-algebras. Under some assumptions (for example, the positivity of infinitesimal operators), we may reduce the study of those unbounded derivations to the one of bounded derivations ([9]). However there are many important derivations in mathematical physics which do not satisfy the positivity. In this paper we wish to initiate a study of unbounded derivations in C*-algebras. A main goal of the study is to show that unbounded derivations in some C*-algebras, which are important in mathematical physics can be approximated by inner derivations on them. Obviously we can not expect such results for an arbitrary C*-algebra. For example, if 9f = C0(cc, xc) is the C*-algebra of all continuous functions on (ox, xo), vanishing at infinity, then the only bounded derivation is the identically zero mapping. On the other hand, let 8 be the differential operator d/ dt corresponding to the one-parameter subgroup of translation operators; then it is an unbounded derivation, so that it can not be approximated by the identically zero mapping in any reasonable sense. Now we shall explain briefly the result obtained in this paper. Let W be a uniformly hyperfinite C*-algebra, (p(t)I ox p(t)(a) is norm-continuous for each a E X, and let 8 be the infinitesimal generator corresponding to