## Nonseparable disintegration theory: references

I mean a theorem of the following kind. Let $A$ be a C*-algebra, and let $\pi: A\to B(H)$ be its representation. Then there exist a set $P$ with a positive measure $\mu$, a field of Hilbert spaces such that $H\simeq \int_P H_p d\mu(p)$, and irreducible representations $\pi_p: A\to B(H_p)$ such that $\pi=\int_P \pi_p d\mu(p)$.

In classical references (Dixmier/Takesaki/Kadison...) both $A$ and $H$ are assumed to be separable. Is there a canonical reference for the nonseparable case?

I have found two articles, not counting particular cases: S. Teleman On reduction theory. {\it Rev. Roumaine Math. Pures Appl.} {\bf 21}, no.~4 (1976), 465--486. and R. Henrichs Decomposition of invariant states and nonseparable C*-algebras. Publ. Res. Inst. Math. Sci. 18, 159-181 (1982). Both use definition of fields of Hilbert spaces given by W. Wils in Direct integrals of Hilbert spaces I. {\it Math. Scand.} {\bf 26} (1970), 73--88.

Both prove the theorem above (Henrichs for the unital case), with one main difference: in Teleman's version, $P$ is a subset of pure states of $A$, but $\mu$ may not be regular (not every set is approximated by compacts from inside). In Henrichs', $\mu$ is regular but one and the same irrep can repeat, even for every $p$.

In the history of this question there were lots or erroneous articles, so I treat these two also with caution. I've gone through Teleman's proof(because it is self-contained). It seems correct, but it turns out that $\pi_p$ may be zero, and this is not indicated in the paper. Through Henrichs I didn't go in detail. He relies on a rarely used theorem of Tomita, for which he however gives an independent proof.

So this is my question: do you use this theory, and if yes, what authors do you refer to?

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## 2 Answers

A related problem (if you look at it from the viewpoint of disintegration of states in C*-algebras) is the problem of disintegration of measures in a nonseparable setting. There is a recent paper which revisits this old problem by M. Kosiek and K. Rudol, "Fibers of the $L^\infty$ Algebra and Disintegration of Measures". Archiv der Mathematik 97 (2011) 559-567. Supposing it is also correct, it may help...

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In case this recent article arxiv.org/abs/1212.6192 is correct, it is surely relevant.

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Thank you... At the first glance, looks unconvincing and reminds the works of Mukul S. Patel (found in ArXiv), who has also claimed to solve lots of old problemds. But this is worth checking with more attention. – Yulia Kuznetsova Jan 9 at 14:51