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,

Formal statement

Let K⁻ denote the strong-operator closure of a set K in B, the set of bounded operators on the Hilbert space H, and let ₁ denote the intersection of K with the unit ball of B.
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 ₁, then h is in the strong-operator closure of the set of self-adjoint operators in 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 of self-adjoint operators in A, the continuous functional calculus a → f satisfies,
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 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.