Non-commutative conditional expectation explained

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

\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.

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

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

Applications

Sakai's theorem

Let

l{B} be a C*-subalgebra of the C*-algebra ak{A}, \varphi _{0} an idempotent linear mapping of ak{A} onto l{B} such that \|\varphi _{0\|} = 1 , ak{A} acting on l{H} the universal representation of ak{A} . Then \varphi _{0} extends uniquely to an ultraweakly continuous idempotent linear mapping \varphi of ak{A} ^{-} , the weak-operator closure of ak{A} , onto l{B} ^{-} , the weak-operator closure of l{B} .

In the above setting, a result^[1] first proved by Tomiyama may be formulated in the following manner.

Theorem. Let

ak{A}, l{B}, \varphi, \varphi _{0} be as described above. Then \varphi is a conditional expectation from ak{A} ^{-} onto l{B} ^{-} and \varphi _{0} is a conditional expectation from ak{A} onto l{B} .

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.

References

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

Notes and References

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