Nonseparable disintegration theory: references - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T18:00:31Z http://mathoverflow.net/feeds/question/81780 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/81780/nonseparable-disintegration-theory-references Nonseparable disintegration theory: references Yulia Kuznetsova 2011-11-24T06:49:55Z 2013-04-27T04:52:43Z <p>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 $\pi=\int_P \pi_p d\mu(p)$.</p> <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?</p> <p>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.</p> <p>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 $p$.</p> <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.</p> <p>So this is my question: do you use this theory, and if yes, what authors do you refer to?</p> http://mathoverflow.net/questions/81780/nonseparable-disintegration-theory-references/118068#118068 Answer by Sergio A. Yuhjtman for Nonseparable disintegration theory: references Sergio A. Yuhjtman 2013-01-04T17:00:16Z 2013-01-04T17:00:16Z <p>In case this recent article arxiv.org/abs/1212.6192 is correct, it is surely relevant.</p> http://mathoverflow.net/questions/81780/nonseparable-disintegration-theory-references/128889#128889 Answer by Pedro Lauridsen Ribeiro for Nonseparable disintegration theory: references Pedro Lauridsen Ribeiro 2013-04-27T04:52:43Z 2013-04-27T04:52:43Z <p>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...</p>