Necessary and sufficient conditions are given for a (complete) commutative algebra that is regular in the sense of von Neumann to have a non-zero derivation. In particular, it is shown that there exist non-zero derivations on the algebra L(M) of all measurable operators affiliated with a commutative von Neumann algebra M, whose Boolean algebra of projections is not atomic. Such derivations are not continuous with respect to measure convergence. In the classical setting of the algebra S[0,1] of all Lebesgue measurable functions on [0,1], our results imply that the first (Hochschild) cohomology group H1(S[0,1], S[0,1]) is non-trivial.
@article{urn:eudml:doc:41854,
title = {Non-trivial derivations on commutative regular algebras.},
journal = {Extracta Mathematicae},
volume = {21},
year = {2006},
pages = {107-147},
zbl = {1129.46056},
mrnumber = {MR2292743},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41854}
}
Ber, A. F.; Chilin, Vladimir I.; Sukochev, Fyodor A. Non-trivial derivations on commutative regular algebras.. Extracta Mathematicae, Tome 21 (2006) pp. 107-147. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41854/