Kaplansky density theorem

Formal statement

Let K− denote the strong-operator closure of a set K in B(H), the set of bounded operators on the Hilbert space H, and let (K)1 denote the intersection of K with the unit ball of B(H). :Kaplansky density theorem. If A is a self-adjoint algebra of operators in B(H), then each element a in the unit ball of the strong-operator closure of A is in the strong-operator closure of the unit ball of A. In other words, (A)_1^{-} = (A^{-})_1. If h is a self-adjoint operator in (A^{-})_1, then h is in the strong-operator closure of the set of self-adjoint operators in (A)_1.

The Kaplansky density theorem can be used to formulate some approximations with respect to the strong operator topology.

1. If h is a positive operator in (A−)1, then h is in the strong-operator closure of the set of self-adjoint operators in (A+)1, where A+ denotes the set of positive operators in A.

2. If A is a C*-algebra acting on the Hilbert space H and u is a unitary operator in A−, then u is in the strong-operator closure of the set of unitary operators in A.

In the density theorem and 1) above, the results also hold if one considers a ball of radius r 0, instead of the unit ball.

Proof

The standard proof uses the fact that a bounded continuous real-valued function f is strong-operator continuous. In other words, for a net {aα} of self-adjoint operators in A, the continuous functional calculus a → f(a) satisfies,

:\lim f(a_{\alpha}) = f (\lim a_{\alpha})

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 A. A matrix computation in M2(A) considering the self-adjoint operator with entries 0 on the diagonal and a and a* at the other positions, then removes the self-adjointness restriction and proves the theorem.

Notes

References

• Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. .
• V.F.R.Jones von Neumann algebras; incomplete notes from a course.
• M. Takesaki Theory of Operator Algebras I

References

1. Pg. 25; [[Pedersen, G. K.]], ''C*-algebras and their automorphism groups'', London Mathematical Society Monographs, {{isbn. 978-0125494502.
2. Theorem 5.3.5; [[Richard Kadison]], ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. {{isbn. 978-0821808191.