nLab
bicommutant theorem

Contents

Idea

The bicommutant theorem characterizes concrete von Neumann algebras as those concrete C *C^*-algebras (C *C^*-algebras of bounded operators on some Hilbert space) that are the commutants of their own commutants.

Statement

The bicommutant theorem (as known as the double commutant theorem, or von Neumann’s double commutant theorem) is the following result:

Theorem

Let AB(H)A \subseteq B(H) be a sub-**-algebra of the algebra of bounded linear operators on a Hilbert space HH. Then AA is a von Neumann algebra (and therefore also a C *C^*-algebra) in HH if and only if A=AA = A'', where AA' denotes the commutant of AA.

Notice that the condition of AA being a von Neumann algebra (being closed in the weak operator topology; “weak” here can be replaced by “strong”, “ultrastrong”, or “ultraweak” as described in operator topology), which is a topological condition, is by this result equivalent to an algebraic condition (being equal to its bicommutant).

References

Revised on July 27, 2011 20:40:30 by Toby Bartels (64.89.62.147)