Commutation and Splitting Theorems in von Neumann Algebras

Abstract

Motivated by a recent result of Ge & Kadison (Invent.Math. 1996) for
factors and using some of our old results concerning Dixmier sets, we extend
appropriately the Splitting result of Ge & Kadison to general von Neumann
algebras. For this we need also a Commutation Theorem for “tensor products
over a commutative subalgebra”. Eventually, both our Splitting Theorem
and Commutation Theorem are direct particular cases of a General Commutation
Theorem for tensor products over subalgebras which may look as
( R^1 tensor R^2 over R)’ = (R^1)’ tensor (R^2)’ over R’.