Well-behaved derivations on C[0, 1]

Ralph Jay De Laubenfels

Vol. 115 (1984), No. 1, 73–80

DOI: 10.2140/pjm.1984.115.73

Abstract

We give the following necessary and sufficient conditions for a ∗-derivation, A, on C[0,1], to generate a continuous group of ∗-automorphisms: A must be equivalent to the closure of pD, where (pD)f(x) ≡ p(x)f′(x), with (1∕p) not locally integrable at the zeroes of p. We give similar necessary and sufficient conditions for a well-behaved ∗-derivation to generate a positive contraction semigroup. We show that any ∗-derivation on C[0,1] has an extension (possibly on a larger space) that generates a continuous group of ∗-automorphisms.

Mathematical Subject Classification 2000

Primary: 46J10

Secondary: 46L40

Milestones

Received: 1 February 1983

Published: 1 November 1984