Vol. 52, No. 2, 1974

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Perturbations of type I von Neumann algebras

John Phillips

Vol. 52 (1974), No. 2, 505–511

In a recent paper, R. V. Kadison and D. Kastler studied a certain metric on the family of von Neumann algebras defined on a fixed Hilbert space. The distance between two von Neumann algebras was defined to be the Hausdorff distance between their unit balls. They showed that if two von Neumann algebras were sufficiently close, then their central portions of type K(K = I,In,Π,II1,II,III) were also close. In the introduction to their paper, they conjectured that neighbouring von Neumann algebras must actually be unitarily equivalent. It is the purpose of this paper to prove this conjecture in the case that one of the algebras is of type I. The question of “inner” equivalence is left open. (Can the unitary equivalence be implemented by a unitary operator in the von Neumann algebra generated by the two neighbouring algebras?)

Mathematical Subject Classification 2000
Primary: 46L10
Received: 28 June 1973
Revised: 13 March 1974
Published: 1 June 1974
John Phillips