Closed derivations in commutative [formula omitted] algebras

Abstract

This paper is a study of closed derivations in commutative C ∗ algebras. A partial characterization is given of the domain of a closed ∗ derivation in a C ∗ algebra C( X), as a Banach ∗ subalgebra of C( X). The structure of closed quasi-well behaved ∗ derivations in C([0, 1]) is discussed. In particular, a complete description is given of closed derivations in C([0, 1]) which extend the derivative, with domain C 1([0, 1)). Such a derivation D is necessarily a ∗ derivation, D ( D)=([0,1])+ker( D), and ker(D) = C ∗(φ) for some “generalized Cantor function” φ. These results generalize to algebras C([0, 1] × Ω), where Ω is compact Hausdorff, and to C( T ) and C o( R ). It is shown that a closed derivation D in C o( R ) such that D ( D) is a normal subalgebra of C o( R ) and D commutes with translations is a constant multiple of the derivative; this is an analogue of a theorem of Sakai for C( T ). Further generalizations are obtained concerning closed derivations commuting with a C ∗ dynamics.