Commutation theorem | Wikipedia audio article
Channel: wikipedia tts
Date: 31/12/2018

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Commutation_theorem


00:01:32 1 Commutation theorem for finite traces
00:02:27 1.1 Examples
00:04:52 2 Commutation theorem for semifinite traces
00:07:36 3 Hilbert algebras
00:11:21 3.1 Definition
00:13:49 3.2 Examples
00:14:10 3.3 Properties
00:15:33 4 See also
00:16:32 5 Notes
00:18:07 6 References
00:22:17 See also



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SUMMARY
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In mathematics, a  commutation theorem explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace. The first such result was proved by F.J. Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a discrete group or by the dynamical system associated with a
measurable transformation preserving a probability measure. Another important application is in the theory of unitary representations of unimodular locally compact groups, where the theory has been applied to the regular representation and other closely related representations. In particular this framework led to an abstract version of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and Forrest Stinespring and an abstract Plancherel theorem for spherical functions associated with a Gelfand pair due to Roger Godement. Their work was put in final form in the 1950s  by Jacques Dixmier as part of the theory of Hilbert algebras. It was not until the late 1960s, prompted partly by results in algebraic quantum field theory and quantum statistical mechanics due to the school of Rudolf Haag, that the more general non-tracial Tomita–Takesaki theory was developed, heralding a new era in the theory of von Neumann algebras.