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with measure theory suggest that one may be able to extend the classical ideas in probability to a noncommutative setting by studying those ideas on general von
Neumann algebras.
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For von
Neumann algebras with a faithful normal tracial state, for example finite von Neumann algebras, the notion of conditional expectation is especially useful.
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190:. For this reason, the theory of von Neumann algebras is sometimes referred to as noncommutative measure theory. The intimate connections of
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on the characterization of those C*-algebras that are *-isomorphic to von
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In the above setting, a result first proved by
Tomiyama may be formulated in the following manner.
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extends uniquely to an ultraweakly continuous idempotent linear mapping
1041:{\displaystyle {\mathfrak {A}},{\mathcal {B}},\varphi ,\varphi _{0}}
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Non-commutative
Conditional Expectations and their Applications
1252:, Contemporary Mathematics, Vol. 365 (2004), pp. 143–179.
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134:. The space of essentially bounded measurable functions on a
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With the aid of
Tomiyama's theorem an elegant proof of
232:{\displaystyle {\mathcal {R}}\subseteq {\mathcal {S}}}
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523:{\displaystyle \Phi (R_{1}SR_{2})=R_{1}\Phi (S)R_{2}}
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1232:, Proc. Japan Acad. (33) (1957), Theorem 1, Pg. 608
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774:{\displaystyle \|\varphi _{0}\|=1,{\mathfrak {A}}}
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573:{\displaystyle R_{1},R_{2}\in {\mathcal {R}}}
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677:{\displaystyle {\mathfrak {A}},\varphi _{0}}
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106:Learn how and when to remove this message
55:"Non-commutative conditional expectation"
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124:non-commutative conditional expectation
647:be a C*-subalgebra of the C*-algebra
291:as well), a positive, linear mapping
126:is a generalization of the notion of
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44:adding citations to reliable sources
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1092:{\displaystyle {\mathfrak {A}}^{-}}
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603:{\displaystyle S\in {\mathcal {S}}}
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701:{\displaystyle {\mathfrak {A}}}
188:commutative von Neumann algebra
31:needs additional citations for
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640:{\displaystyle {\mathcal {B}}}
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404:{\displaystyle {\mathcal {R}}}
380:{\displaystyle {\mathcal {S}}}
352:{\displaystyle {\mathcal {R}}}
328:{\displaystyle {\mathcal {S}}}
280:{\displaystyle {\mathcal {R}}}
256:{\displaystyle {\mathcal {S}}}
186:is the canonical example of a
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1048:be as described above. Then
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439:{\displaystyle \Phi (I)=I}
239:be von Neumann algebras (
1061:{\displaystyle \varphi }
869:{\displaystyle \varphi }
179:{\displaystyle (X,\mu )}
1266:Conditional probability
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361:conditional expectation
147:{\displaystyle \sigma }
128:conditional expectation
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304:{\displaystyle \Phi }
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40:improve this article
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51:Find sources:
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29:This article
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614:Applications
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38:Please help
33:verification
30:
289:C*-algebras
132:probability
120:mathematics
1240:References
781:acting on
732:such that
66:newspapers
1139:φ
1116:−
1085:−
1056:φ
1030:φ
1023:φ
948:−
893:−
864:φ
838:φ
753:‖
744:φ
740:‖
666:φ
591:∈
561:∈
499:Φ
454:Φ
419:Φ
299:Φ
220:⊆
171:μ
142:σ
1260:Category
993:Theorem.
931:, onto
829:. Then
411:) when
80:scholar
82:
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1216:Notes
1181:onto
1099:onto
708:onto
387:onto
335:onto
87:JSTOR
73:books
1130:and
995:Let
623:Let
580:and
446:and
363:(of
263:and
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59:news
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530:if
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36:.
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