Celebratio Mathematica — Gleason

Andy Gleason and quantum mechanics

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A. M. Gleason: “Measures on the closed subspaces of a Hilbert space,” J. Math. Mech. 6 : 4 (1957), pp. 885–893. MR 0096113 Zbl 0078.28803 article

In his investigations of the mathematical foundations of quantum mechanics, Mackey [1957] has proposed the following problem: Determine all measures on the closed subspaces of a Hilbert space. A measure on the closed subspaces means a function \( \mu \) which assigns to every closed subspace a non-negative real number such that if \( \{A_i\} \) is a countable collection of mutually orthogonal subspaces having closed linear span \( B \), then \[ \mu(B) = \sum\mu(A_i). \] It is easy to see that such a measure can be obtained by selecting a vector \( v \) and, for each closed subspace \( A \), taking \( \mu(A) \) as the square of the norm of the projection of \( v \) on \( A \). Positive linear combinations of such measures lead to more examples and, passing to the limit, one finds that, for every positive semidefinite self-adjoint operator \( T \) of the trace class, \[ \mu(A) = \operatorname{trace}(TP_A), \] where \( P_A \) denotes the orthogonal projection on \( A \), defines a measure on the closed subspaces. It is the purpose of this paper to show that, in a separable Hilbert space of dimension at least three, whether real or complex, every measure on the closed subspaces is derived in this fashion.

Andrew M. Gleason: Glimpses of a life in mathematics. Edited by E. Bolker, P. Chernoff, C. Costes, and D. Lieberman. Privately printed, 1992. book

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