Kaplansky density theorem

In the theory of von Neumann algebras, the Kaplansky density theorem states that if A is a *-subalgebra of the algebra B(H) of bounded operators on a Hilbert space H, then the strong closure of the unit ball of A in B(H) is the unit ball of the strong closure of A in B(H).

Use

This gives a strengthening of the von Neumann bicommutant theorem, showing that an element a of the double commutant of A, denoted by A′′, can be strongly approximated by elements of A whose norm is no larger than that of a.

Proof

The standard proof uses the fact that, when f is bounded, the continuous functional calculus a ↦ f(a) satisfies, for a net {a_α} of self adjoint operators

in the strong operator topology. This shows that self-adjoint part of the unit ball in A′′ can be approximated strongly by self-adjoint elements in the C*-algebra generated by A. A matrix computation then removes the self-adjointness restriction and proves the theorem.

See also

• Jacobson density theorem

References

• V.F.R.Jones von Neumann algebras; incomplete notes from a course.
• M. Takesaki Theory of Operator Algebras I ISBN 3-540-42248-X