Normal expectations in von Neumann algebras

André (Piotrowsky) De Korvin

Vol. 27 (1968), No. 2, 333–338

DOI: 10.2140/pjm.1968.27.333

Abstract

Let h and k be two Hilbert spaces, h ⊗ k will denote the tensor product of h and k. Let 𝒜 be a von Neumann algebra acting on h. Let ψ be an ampliation of 𝒜 in h ⊗ k, i.e., ψ is a map of 𝒜 into bounded linear operators of h ⊗ k and ψ(𝒜) = 𝒜⊗ I_k (I_k is the identity map on k). Let 𝒜 be the image of 𝒜 by ψ.

The purpose of this paper is to prove the following result: If ℬ is a subalgebra of 𝒜 and if ℬ is the range of a normal expectation φ defined on 𝒜, then there exists an ampliation of 𝒜 in h ⊗ k, independent of ℬ and of φ, such that φ ⊗ I_k is a spatial isomorphism of 𝒜.

Mathematical Subject Classification

Primary: 46.65

Milestones

Received: 13 November 1967

Published: 1 November 1968