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Non-commutative conditional expectation
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<h2 id="formal-definition">Formal definition</h2> <p>Let R ⊆ S {\displaystyle {\mathcal {R}}\subseteq {\mathcal {S}}} be von Neumann algebras ( S {\displaystyle {\mathcal {S}}} and R {\displaystyle {\mathcal {R}}} may be general <a href="/facts/C*-algebras/B3tnGHIo">C*-algebras</a> as well), a positive, linear mapping Φ {\displaystyle \Phi } of S {\displaystyle {\mathcal {S}}} onto R {\displaystyle {\mathcal {R}}} is said to be a <i>conditional expectation</i> (of S {\displaystyle {\mathcal {S}}} onto R {\displaystyle {\mathcal {R}}} ) when Φ ( I ) = I {\displaystyle \Phi (I)=I} and Φ ( R 1 S R 2 ) = R 1 Φ ( S ) R 2 {\displaystyle \Phi (R_{1}SR_{2})=R_{1}\Phi (S)R_{2}} if R 1 , R 2 ∈ R {\displaystyle R_{1},R_{2}\in {\mathcal {R}}} and S ∈ S {\displaystyle S\in {\mathcal {S}}} . </p> <h2 id="applications">Applications</h2> <h3>Sakai's theorem</h3> <p>Let B {\displaystyle {\mathcal {B}}} be a C*-subalgebra of the C*-algebra A , φ 0 {\displaystyle {\mathfrak {A}},\varphi _{0}} an idempotent linear mapping of A {\displaystyle {\mathfrak {A}}} onto B {\displaystyle {\mathcal {B}}} such that ‖ φ 0 ‖ = 1 , A {\displaystyle \|\varphi _{0}\|=1,{\mathfrak {A}}} acting on H {\displaystyle {\mathcal {H}}} the universal representation of A {\displaystyle {\mathfrak {A}}} . Then φ 0 {\displaystyle \varphi _{0}} extends uniquely to an ultraweakly continuous idempotent linear mapping φ {\displaystyle \varphi } of A − {\displaystyle {\mathfrak {A}}^{-}} , the weak-operator closure of A {\displaystyle {\mathfrak {A}}} , onto B − {\displaystyle {\mathcal {B}}^{-}} , the weak-operator closure of B {\displaystyle {\mathcal {B}}} . </p><p>In the above setting, a result<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> first proved by Tomiyama may be formulated in the following manner. </p><p>Theorem. Let A , B , φ , φ 0 {\displaystyle {\mathfrak {A}},{\mathcal {B}},\varphi ,\varphi _{0}} be as described above. Then φ {\displaystyle \varphi } is a conditional expectation from A − {\displaystyle {\mathfrak {A}}^{-}} onto B − {\displaystyle {\mathcal {B}}^{-}} and φ 0 {\displaystyle \varphi _{0}} is a conditional expectation from A {\displaystyle {\mathfrak {A}}} onto B {\displaystyle {\mathcal {B}}} . </p><p>With the aid of Tomiyama's theorem an elegant proof of <a href="/facts/Von_Neumann_algebra/8NyWUIRH">Sakai's result</a> on the characterization of those C*-algebras that are *-isomorphic to von Neumann algebras may be given. </p> <h2 id="notes">Notes</h2> <ul><li><a href="/facts/Kadison%2c_R._V./V6jkUQxG">Kadison, R. V.</a>, <i>Non-commutative Conditional Expectations and their Applications</i>, Contemporary Mathematics, Vol. 365 (2004), pp. 143–179.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Tomiyama J., On the projection of norm one in W*-algebras, Proc. Japan Acad. (33) (1957), Theorem 1, Pg. 608 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>