Vol. 36, No. 3, 1971

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The quotient algebra of a finite von Neumann algebra

Masamichi Takesaki

Vol. 36 (1971), No. 3, 827–831
Abstract

We will prove the following: Let M be a finite von Neumann algebra with center Z and A a von Neumann subalgebra of Z. Let Ω be the spectrum space of A and identify A with C(Ω). Let 𝜖 be a σ-weakly continuous linear map of M onto A such that 𝜖(xx) = 𝜖(xx) 0 for every x M,𝜖(ax) = a𝜖(x) for every a A and x M,𝜖(1) = 1 and 𝜖(xx)0 for every nonzero x M. For each ω Ω, let mω denote the set of all x M with 𝜖(xx)(ω) = 0. Then mω is a closed ideal and the quotient C-algebla M∕mω is a finite von Neumann algebra. Furthermore, if πω denote the canonical homomorphism of M onto M∕mω, then πω(N) is a von Neumann subalgebra of M∕mω for every von Neumann subalgebra N containing A.

Mathematical Subject Classification
Primary: 46.65
Milestones
Received: 8 June 1970
Published: 1 March 1971
Authors
Masamichi Takesaki