Extreme points and linear isometries of the domain of a closed *-derivation in C(K)

摘要

Unbounded derivations in non-commutative C~*-algebras have been studied in detail by many authors, which are closely related to mathematical physics and especially are one of the natural frameworks for quantum dynamics in operator algebra context. In commutative C~*-algebras, unbounded derivations, which were studied systematically by Sakai in [21], are also very important and interesting object to study, because it plays a role of certain differential structure of underlying space. Indeed, known examples are given by (partial) differentiation on spaces with some differential structure. Since the differentiation d/dt on the space C~((1))([0, 1]) of continuously dif-ferentiable functions on [0, 1] is a typical example of closed derivations, for any closed derivation δ in a commutative unital C~*-algebra C(K) (K: a compact Hausdorff space) we may regard the domain D(δ) of δ as a generalization of the Banach space C~((1))([0, 1]). Moreover, if D(δ)=C(K), δ is bounded and hence δ≡0. Thus, we wish to look for unified approach to deal with C(K), C~((1))([0, 1]) and several other spaces of differentiable functions together.