Non-commutative conditional expectation

In mathematics, non-commutative conditional expectation is a generalization of the notion of conditional expectation in classical probability. The space of measurable functions on a $\sigma$-finite measure space $(X,\mu )$ is the canonical example of a commutative von Neumann algebra. For this reason, the theory of von Neumann algebras is sometimes referred to as noncommutative measure theory. The intimate connections of probability theory 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.

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 $S\in {\mathcal {S}}$.

References

• Kadison, R. V., Non-commutative Conditional Expectations and their Applications, Contemporary Mathematics, Vol. 365 (2004), pp. 143–179.