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?)
