von Neumann double commutant theorem
The von Neumann double commutant theorem is a remarkable result in the theory of self-adjoint algebras of operators on Hilbert spaces, as it expresses purely topological aspects of these algebras in terms of purely algebraic properties.
Theorem - von Neumann - Let be a Hilbert space (http://planetmath.org/HilbertSpace) and its algebra of bounded operators. Let be a *-subalgebra of that contains the identity operator. The following statements are equivalent:
, i.e. equals its double commutant.
is closed in the weak operator topology.
is closed in the strong operator topology.
Thus, a purely topological property of a , as being closed for some operator topology, is equivalent to a purely algebraic property, such as being equal to its double commutant.
This result is also known as the bicommutant theorem or the von Neumann density theorem.
|Title||von Neumann double commutant theorem|
|Date of creation||2013-03-22 18:40:27|
|Last modified on||2013-03-22 18:40:27|
|Last modified by||asteroid (17536)|
|Synonym||double commutant theorem|
|Synonym||von Neumann bicommutant theorem|
|Synonym||von Neumann density theorem|