Non-commutative conditional expectation

In mathematics, non-commutative conditional expectation is a generalization of the notion of conditional expectation in classical probability.

Formal definition

A positive, linear mapping $\Phi$ of a von Neumann algebra ${\mathcal {S}}$ onto a von Neumann algebra ${\mathcal {R}}$ (${\mathcal {S}}$ and ${\mathcal {R}}$ may be general C*-algebras as well) is said to be a conditional expectation (of ${\mathcal {S}}$ onto ${\mathcal {R}}$) when $\Phi (I)=I$ and $\Phi (R_{1}SR_{2})=R_{1}\Phi (S)R_{2}$ if $R_{1},R_{2}\in {\mathcal {R}}$ and $S\in {\mathcal {S}}$.