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 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.

Formal definition edit

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 edit

Sakai's theorem edit

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 edit

  1. ^ Tomiyama J., On the projection of norm one in W*-algebras, Proc. Japan Acad. (33) (1957), Theorem 1, Pg. 608

References edit

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