Additivity of vector Gleason measures (Q1185856)

The author studies the degree of additivity of orthogonal Hilbert-space- valued measures on the lattice \(L(H)\) of all projections acting on a Hilbert space \(H\). He gives criteria for such measures to be completely additive and establishes the connection between the additivity of orthogonal measures and the size of almost disjoint families on \(\dim H\). As corollaries the following results are proved: --- finitely additive measures distinguish dimension of Hilbert space; --- for cardinals \(\kappa\), \(\nu\) with \(\kappa>\nu\), \(\kappa\geq 3\), there is no Jordan homomorphism from type \(I_ \kappa\) factor into type \(I_ \nu\) factor; --- every lattice \(L(H)\) with \((\dim H)^ \omega=\dim H\) admits a non zero free orthogonal measure with values in \(H\).