Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Commutation theorem for traces
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a commutation theorem for traces explicitly identifies the <a href="/facts/Commutant/AKhrEUGR">commutant</a> of a specific <a href="/facts/Von_Neumann_algebra/8NyWUIRH">von Neumann algebra</a> acting on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> in the presence of a <a href="/facts/Von_Neumann_algebra/8NyWUIRH">trace</a>. </p><p>The first such result was proved by <a href="/facts/Francis_Joseph_Murray/HsnrRXR1">Francis Joseph Murray</a> and <a href="/facts/John_von_Neumann/aUk6urof">John von Neumann</a> in the 1930s and applies to the von Neumann algebra generated by a <a href="/facts/Discrete_group/HIjkpJhS">discrete group</a> or by the <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> associated with a <a href="/facts/Ergodic_theory/rkqCbIy0">measurable transformation</a> preserving a <a href="/facts/Probability_measure/8BjhgoPR">probability measure</a>. </p><p>Another important application is in the theory of <a href="/facts/Unitary_representation/W1XRatG8">unitary representations</a> of <a href="/facts/Haar_measure/2IcmTcuU">unimodular</a> <a href="/facts/Locally_compact_group/AGPC8vnY">locally compact groups</a>, where the theory has been applied to the <a href="/facts/Regular_representation/dBND2lRj">regular representation</a> and other closely related representations. In particular this framework led to an abstract version of the <a href="/facts/Plancherel_theorem/IlpoAySd">Plancherel theorem</a> for unimodular locally compact groups due to <a href="/facts/Irving_Segal/SasFUqI8">Irving Segal</a> and <a href="/facts/W._Forrest_Stinespring/p8c2a0Ia">Forrest Stinespring</a> and an abstract <a href="/facts/Plancherel_theorem_for_spherical_functions/2XrDsy8V">Plancherel theorem for spherical functions</a> associated with a <a href="/facts/Gelfand_pair/1nr6GewY">Gelfand pair</a> due to <a href="/facts/Roger_Godement/jdjb03n0">Roger Godement</a>. Their work was put in final form in the 1950s by <a href="/facts/Jacques_Dixmier/eIgzkHvX">Jacques Dixmier</a> as part of the theory of Hilbert algebras. </p><p>It was not until the late 1960s, prompted partly by results in <a href="/facts/Algebraic_quantum_field_theory/XqBgUQva">algebraic quantum field theory</a> and <a href="/facts/Quantum_statistical_mechanics/oeD8Gp2R">quantum statistical mechanics</a> due to the school of <a href="/facts/Rudolf_Haag/M63w7y05">Rudolf Haag</a>, that the more general non-tracial <a href="/facts/Tomita%25E2%2580%2593Takesaki_theory/FkYYEB29">Tomita–Takesaki theory</a> was developed, heralding a new era in the theory of von Neumann algebras. </p>