Conditional expectations in von Neumann algebras

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Let M be a von Neumann algebra and N its von Neumann subalgebra. Let ϑ be a faithful, semifinite, normal weight on M + such that the restriction ϑ ¦ N of ϑ onto N is semifinite. The first main result is that 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 M onto N such that ϑ ̇ (x) = ϑ ̇ ∘ ϵ(x) for every xϵ M ϑ. The second result is that a von Neumann algebra M is finite if and only if any maximal abelian self-adjoint subalgebra of 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.