Conditional expectations in von Neumann algebras

Abstract

Let $\text{M}$ be a von Neumann algebra and $\text{N}$ its von Neumann subalgebra. Let ϑ be a faithful, semifinite, normal weight on $\text{M}$₊ such that the restriction ϑ ¦ $\text{N}$ of ϑ onto $\text{N}$ is semifinite. The first main result is that $\text{N}$ is invariant under the modular automorphism group σ_t^ϑ associated with ϑ if and only if there exists a σ-weakly continuous faithful projection ϵ of norm one from $\text{M}$ onto $\text{N}$ such that $\text{ϑ}\text{̇}\text{(x) =}\text{ϑ}\text{̇}\text{∘ ϵ(x)}$ for every xϵ$\text{M}$_ϑ. The second result is that a von Neumann algebra $\text{M}$ is finite if and only if any maximal abelian self-adjoint subalgebra of $\text{M}$ is the range of a σ-weakly continuous projection of norm one. This result is an answer for the question which Kadison raised in the author's talk at the International Congress of Mathematicians in Nice, 1970.