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While studying the $C^*$-algebraic formulation of the recently solved Kadison-Singer problem, I was wondering about maximal abelian subalgebras:

Let $\mathcal{A}$ be a unital C* algebra. There seems to be a theorem (see e.g. Kadison/Singer "Extension of Pure States", however, I couldn't find a version not behind the paywall) that there are only three types of maximal abelian subalgebras (+ direct sums)

  • the continuous maximal abelian subalgebra $\mathcal{A}_c$, which I believe is isomorphic to $L^{\infty}([0,1])$ acting on the Hilbert space $L^2([0,1])$.
  • the discrete maximal abelian subalgebra $\mathcal{A}_d$, which I believe is isomorphic to $\mathcal{D}(\ell_2)$, the diagonal matrices of square integrable functions.
  • finite dimensional versions of the latter algebra $\mathcal{A}_n$ for all $n\in\mathbb{N}$ (basically, diagonal matrices).

This theorem seems to be general knowledge, however I could not find a reference to a proof. Could someone either provide one or point me to the main ideas and/or intuitions to prove this? I can only see how this should be the case for finite dimensional systems (where only the third case survives).

Furthermore, this may be a long shot, but such a decomposition reminds me of spectral decompositions into continuous and pure point spectrum, so I would like to find out, whether there is maybe a relationship?

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It is not hard to see that it is as you say when $\mathcal A$ is a von Neumann algebra.

But there are C$^*$-algebras with few to no projections, and so none of those masas can live there. Consider for instance $C^*_r(\mathbb F_2)$, the reduced C$^*$-algebra of the free group on two generators $\mathbb F=\langle a,b\rangle$. This algebra is known to be projectionless. But you can consider $C^*(a)\subset C^*_r(\mathbb F_2)$. I think this is a masa; but even if it isn't, it is contained in some masa by an easy application of Zorn's Lemma, and this masa is a proper subalgebra as the full algebra is non-abelian. So we have a masa in a projectionless C$^*$-algebra, which cannot be any kind of direct sums like the examples in the question, as all these have many projections.

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  • $\begingroup$ mhh... so maybe the "theorem" is wrong? I am quite surprised, but thanks for the von Neumann idea. I guess you can see that the masa will be a vN-algebra, too, hence an abelien vN-algebra and if I read this correctly, there is a one-to-one correspondence between certain measure spaces and masa's of vN-algebras. Then I guess, something akin to Lebesgue's decomposition theorem and Radon-Nikodym tells us that we have such a decomposition. Nevertheless, would you have any reference at all that studies masa's of $C^*$-algebras? $\endgroup$ – Martin Dec 5 '13 at 23:00
  • $\begingroup$ As for references, not really. There are papers, for example by Renault and also by Wassermann, but they deal with more specific questions, I think. $\endgroup$ – Martin Argerami Dec 5 '13 at 23:06
  • $\begingroup$ The masas in Kadison-Singer's paper are masas in $B(H)$, so masas in a von Neumann algebra. I don't think they mention masas of $C^*$-algebras. Masas of von Neumann algebras are of course von Neumann algebras themselves, but that doesn't hold for masas of C$^*$-algebras. $\endgroup$ – Martin Argerami Dec 5 '13 at 23:07
  • $\begingroup$ right. True, I thought it shouldn't make a difference, but that's of course wrong. Thanks, that sorts out a few things in my head. $\endgroup$ – Martin Dec 5 '13 at 23:24

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