Is *-Algebra the Key to Understanding Quantum Probability Theory?

[micromass]

As I realized recently, the probability theory as used in quantum mechanics does not follow Kolmogorov's axioms. I am interested in a book that treats probability theory as it is done in quantum mechanics. Is this treated in books on quantum logic?

Any other good book on the mathematical foundations of quantum mechanics would be welcome.

 

[verty]

It is uncontroversial (though remarkable) that the formal apparatus of quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over by the “quantum logic” of projection operators on a Hilbert space.
...
The quantum-probabilistic formalism, as developed by von Neumann [1932], assumes that...

It seems these von Neumann (W*) algebras are the main invention in this new type of probability. They seem to be an extension of C* algebras; the two volumes by Kadison & Ringrose about operator algebras cover them. There are also separate volumes by Jacques Diximier on C* algebras and W* algebras.

I also looked into the probability side of it. I see two books that seem related to what you want:

https://www.amazon.com/dp/3642204376/?tag=pfamazon01-20
https://www.amazon.com/dp/354013915X/?tag=pfamazon01-20

It seems very strange to me, though, that the first one is from 2011 while the second is from 1985.

 

[verty]

Also have a look here, here, and here.

Strangely, there is a wikipedia entry for *-algebra, it was created in 2009 and as yet has no references listed. But someone obviously wrote it, so this may be a direction that things will go (I guess).