On intermediate subalgebras of inclusions of von Neumann algebras having common Cartan subalgebras and their basic extensions
Volume 246 / 2019
Studia Mathematica 246 (2019), 295-320
MSC: Primary 46L10, 47L30; Secondary 46L99.
DOI: 10.4064/sm170915-13-2
Published online: 19 October 2018
Abstract
It is proved that given a separable von Neumann algebra $A$ which contains a Cartan subalgebra $D$, there always exists, for any intermediate von Neumann subalgebra $B$ with $D\subseteq B$, a faithful normal conditional expectation from $A$ onto $B$. Our proof is new and operator-algebraic in the sense that it is given without realizing $A$ as a von Neumann algebra associated with a discrete measured equivalence relation. We also show, using an operator-algebraic method, that the basic extension $A_{1}$ of the inclusion $B\subseteq A$ as above admits a Cartan subalgebra.
