IUMJ

Title: Derivations of CSL algebras

Authors: Robert L. Moore

Issue: Volume 54 (2005), Issue 6, 1739-1750

Abstract:

We introduce the notion of quiltings for derivations acting on a CSL algebra $\mathcal{A}$. A quilting is a family of pairs $(X_{\alpha} , E_{\alpha})$, where $E_{\alpha}$ is an invariant subspace, $A_{\alpha}$ implements the derivation on the compression of the algebra to $E_{\alpha}$, and the norms of the $X$'s are bounded. We show that many derivations allow quiltings. As a byproduct, we can show that some derivations are spatially implemented, even when they act on algebras whose lattices are not completely distributive. If the identity in the algebra can be approximated by invariant projections $E_{\alpha}$ so that $(E_{\alpha})_- \neq I$, then every derivation from $\mbox{\upshape Alg}\, \mathcal{N}$ is spatially implemented.