Top Qs
Timeline
Chat
Perspective

Non-commutative conditional expectation

From Wikipedia, the free encyclopedia

Remove ads
Remove ads

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

Remove ads

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 .

Remove ads

Applications

Summarize
Perspective

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.

Remove ads

Notes

References

Loading content...
Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads