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 -finite measure space 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.
For von Neumann algebras with a faithful normal tracial state, for example finite von Neumann algebras, the notion of conditional expectation is especially useful.
Formal definition
Let be von Neumann algebras ( and may be general C*-algebras as well), a positive, linear mapping of onto is said to be a conditional expectation (of onto ) when and if and .
Applications
Sakai's theorem
Let be a C*-subalgebra of the C*-algebra an idempotent linear mapping of onto such that acting on the universal representation of . Then extends uniquely to an ultraweakly continuous idempotent linear mapping of , the weak-operator closure of , onto , the weak-operator closure of .
In the above setting, a result[1] first proved by Tomiyama may be formulated in the following manner.
Theorem. Let be as described above. Then is a conditional expectation from onto and is a conditional expectation from onto .
With the aid of Tomiyama's theorem an elegant proof of Sakai's result on the characterization of those C*-algebras that are *-isomorphic to von Neumann algebras may be given.
Notes
- Tomiyama J., On the projection of norm one in W*-algebras, Proc. Japan Acad. (33) (1957), Theorem 1, Pg. 608
References
- Kadison, R. V., Non-commutative Conditional Expectations and their Applications, Contemporary Mathematics, Vol. 365 (2004), pp. 143–179.