| dbo:abstract |
In mathematics, a commutation theorem for traces 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 Francis Joseph 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. (en) |
| dbo:wikiPageExternalLink |
https://archive.org/details/calgebras0000dixm |
| dbo:wikiPageID |
18404411 (xsd:integer) |
| dbo:wikiPageLength |
20129 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID |
1077496337 (xsd:integer) |
| dbo:wikiPageWikiLink |
dbr:Probability_space dbr:Roger_Godement dbr:Rudolf_Haag dbr:Trace_class_operator dbr:John_von_Neumann dbr:Unitary_operator dbr:Dynamical_system dbr:Von_Neumann_algebra dbc:Theorems_in_functional_analysis dbr:Mathematics dbr:Gelfand_pair dbr:Quantum_statistical_mechanics dbr:Bounded_operator dbr:Crossed_product dbr:State_(functional_analysis) dbr:Matrix_coefficient dbr:Countable dbr:Haar_measure dbr:Locally_compact_group dbr:Ergodic_theory dbr:Francis_Joseph_Murray dbr:Discrete_group dbr:Hilbert–Schmidt_operator dbr:Positive_operator dbr:Regular_representation dbr:Hilbert_space dbc:Representation_theory_of_groups dbr:Irving_Segal dbr:Jacques_Dixmier dbr:Tensor_product dbr:Abelian_von_Neumann_algebra dbc:Von_Neumann_algebras dbc:Ergodic_theory dbr:Affiliated_operator dbr:Tomita–Takesaki_theory dbc:Theorems_in_representation_theory dbr:Inner_product_space dbr:Strong_operator_topology dbr:Unitary_representation dbr:Plancherel_theorem dbr:Finite_group dbr:Multiplication_operator dbr:Plancherel_theorem_for_spherical_functions dbr:Probability_measure dbr:W._Forrest_Stinespring dbr:Equivalence_of_measures dbr:Algebraic_quantum_field_theory dbr:Commutant |
| dbp:wikiPageUsesTemplate |
dbt:Citation dbt:Harvtxt dbt:Pb dbt:Reflist dbt:See_also dbt:Short_description dbt:Visible_anchor dbt:Functional_analysis |
| dct:subject |
dbc:Theorems_in_functional_analysis dbc:Representation_theory_of_groups dbc:Von_Neumann_algebras dbc:Ergodic_theory dbc:Theorems_in_representation_theory |
| rdf:type |
owl:Thing |
| rdfs:comment |
In mathematics, a commutation theorem for traces 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 Francis Joseph 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. (en) |
| rdfs:label |
Commutation theorem for traces (en) |
| rdfs:seeAlso |
dbr:Tomita–Takesaki_theory |
| owl:sameAs |
wikidata:Commutation theorem for traces https://global.dbpedia.org/id/4i6cx |
| prov:wasDerivedFrom |
wikipedia-en:Commutation_theorem_for_traces?oldid=1077496337&ns=0 |
| foaf:isPrimaryTopicOf |
wikipedia-en:Commutation_theorem_for_traces |
| is dbo:wikiPageRedirects of |
dbr:Commutation_theorem dbr:Semilinearity_(operator_theory) dbr:Commutation_theorems dbr:Commutation_theory |
| is dbo:wikiPageWikiLink of |
dbr:Roger_Godement dbr:John_von_Neumann dbr:Commutation_theorem dbr:Hilbert_algebra dbr:Irving_Segal dbr:Tomita–Takesaki_theory dbr:Semilinearity_(operator_theory) dbr:Commutation_theorems dbr:Commutation_theory |
| is rdfs:seeAlso of |
dbr:Tomita–Takesaki_theory |
| is foaf:primaryTopic of |
wikipedia-en:Commutation_theorem_for_traces |