For many years physics and mathematics have had a fruitful influence on
one another. Classical mechanics and celestial mechanics have produced
very deep problems whose solutions have enhanced mathematics. On the
other hand, mathematics itself has found interesting theories which then
(sometimes after many years) have been reflected in physics, confirming
the thesis that nothing is more practical than a good theory. The same
is true for the younger physical discipline -of quantum mechanics. In
the 1930s two events, not at all random, became: The mathematical back-
grounds of both quantum mechanics and probability theory. In 1936, G.
Birkhoff and J. von Neumann published their historical paper "The logic
of quantum mechanics", in which a quantum logic was suggested. The
mathematical foundations of quantum mechanics remains an outstanding
problem of mathematics, physics, logic and philosophy even today. The
theory of quantum logics is a major stream in this axiomatical knowledge
river, where L(H), the system of all closed subspaces of a Hilbert space
H, due to J. von Neumann, plays an important role. When A.M. Gleason
published his solution to G. Mackey's problem showing that any state (=
probability measure) corresponds to a density operator, he probably did
not anticipate that his solution would become a cornerstone of ax iomati
cal theory of quantum mechanics nor that it would provide many
interesting applications to mathematics.
