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I'm currently trying to find my way into the geometric description of Quantum Mechanics. I therefor started reading:

Geometry of state spaces. In: Entanglement and Decoherence (A. Buchleitner et al., eds.). Lecture Notes in Physics 768, Springer Verlag, Berlin, New York, 2009, 1-60.

A document that can also be found as a manuscript via: http://www.physik.uni-leipzig.de/~uhlmann/PDF/UC07.pdf

Even though I thought that I have a solid background in abstract algebra I somewhat got lost in Chapter 2 when he's trying to classify all the *-algebras that represent actual physical systems (starting at page 24 in the document).

Do you have some recommendations for texts that introduce the *-algebra language in Quantum Mechanics in a more 'detailed' way. Because I kind of have the feeling that at a certain point Uhlmann just keeps skipping steps and I also lack some of the physical intuition concerning partial traces, canonical traces, purification and all that. From time to time I'd also be happy to see a concrete example.

I'm looking forward to your responses.

Best regards.

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2 Answers 2

The original reference to Wedderburn's theorem is
J.H. Wedderburn, On hypercomplex numbers, Proceedings of the London Mathematical Society 2 (1908), 77-118.
http://plms.oxfordjournals.org/content/s2-6/1/77.short

For more on *-algebras, try Section 2.2 in Volume 3 and Section 1.4 of Volume 4 of Thirring, A course in mathematical physics.

Also of interest might be the following Wikipedia articles:
http://en.wikipedia.org/wiki/Artin%E2%80%93Wedderburn_theorem
http://en.wikipedia.org/wiki/Finite_dimensional_von_Neumann_algebra
http://en.wikipedia.org/wiki/Commutation_theorems#Hilbert_algebras
and more links in
http://en.wikipedia.org/wiki/Category:C*-algebras

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Thirring's exposition looks nice and seems to be a nice start. But I think he's not talking about partial trace, purification, etc. –  MrLee Oct 1 '12 at 22:11
    
@MrLee: A partial trace is simply a trace over some factors of a tensor product Hilbert space. I don't know what purification is; it doesn't seem to be a standard concept. –  Arnold Neumaier Oct 2 '12 at 8:09

OP wrote(v2):

I also lack some of the physical intuition concerning partial traces, canonical traces, purification and all that.

Apart from the mathematical subject of *-algebras, I get the impression that OP really wants to study quantum information rather than quantum mechanics. If this hunch is correct, then I can recommend for starters from the physics side, the textbook

M. Nielsen and I.L Chuang, Quantum Computation and Quantum Information, Cambridge University Pres (2000).

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