Vol. 100, No. 2, 1982

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Gleason’s theorem for type I von Neumann algebras

Jürgen Tischer

Vol. 100 (1982), No. 2, 473–488
Abstract

For a von Neumann algebra A a G-measure m on A is defined as a map from the projections of A to the positive reals which satisfies the equation

   ∑       ∑
m (   Pi) =   m (Pi)

for every family (Pi) of pairwise orthogonal projections. We prove the following generalization of Gleason’s theorem: If m is a G-measure on a type I von Neumann algebra A not containing a type I2 direct summand, then there exists an extension of m to a positive normal linear for on A.

Mathematical Subject Classification
Primary: 46L50, 46L50
Secondary: 81B10
Milestones
Received: 12 June 1980
Revised: 8 October 1980
Published: 1 June 1982
Authors
Jürgen Tischer