# Quantum Mechanics in terms of *-algebras

I'm currently trying to find my way into the geometric description of Quantum Mechanics. I therefore 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.

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
http://en.wikipedia.org/wiki/Category:C*-algebras

• Thirring's exposition looks nice and seems to be a nice start. But I think he's not talking about partial trace, purification, etc. Commented Oct 1, 2012 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. Commented Oct 2, 2012 at 8:09
• @ArnoldNeumaier purification is that every mixed state can be reprensented as the partial trace of a pure state living in a higher dinensional space. (Not sure if this works in thr c* algebra) Commented Apr 1, 2018 at 11:39
• @lalala: Thanks for the explanation. The C^* analogue is the GNS construction, which represents each state as a pure state in some Hilbert space. Commented Apr 1, 2018 at 14:36

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).

A bit late reply, but maybe it is going to be helpful to others who have the same interests in studying Quantum Mechanics in terms of * algebras.

There is a beautiful book by F. Strocchi "An Introduction to the Mathematical Structure of Quantum Mechanics - A short course for Mathematicians" which properly introduces quantum mechanics in terms of * algebra.

Another classic, although a bit more advanced and maybe it would be better to read it after the book by Strocchi, is "Operator Algebras and Quantum Statistical Mechanics" volumes I and II, by D. Bratteli and A. Robinson.