Non-commutative conditional expectation

<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, non-commutative conditional expectation is a generalization of the notion of <a href="/facts/Conditional_expectation/3GxG1Q3x">conditional expectation</a> in classical <a href="/facts/Probability/fgYvMhwb">probability</a>. The space of essentially bounded measurable functions on a 
  
    
      
        σ
      
    
    {\displaystyle \sigma }
  
-finite measure space 
  
    
      
        (
        X
        ,
        μ
        )
      
    
    {\displaystyle (X,\mu )}
  
 is the canonical example of a <a href="/facts/Abelian_von_Neumann_algebra/Mljk27ba">commutative von Neumann algebra</a>. For this reason, the theory of von Neumann algebras is sometimes referred to as noncommutative measure theory. The intimate connections of <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> 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.
</p><p>For von Neumann algebras with a faithful normal tracial state, for example finite von Neumann algebras, the notion of conditional expectation is especially useful.
</p>

Non-commutative conditional expectation open-in-new

Non-commutative conditional expectation