Kaplansky density theorem

In the theory of von Neumann algebras, the Kaplansky density theorem, due to Irving Kaplansky, is a fundamental approximation theorem. The importance and ubiquity of this technical tool led Gert Pedersen to comment in one of his books that,
 * The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays.

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 &rarr; 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.