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[1] that,
The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays.
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.[2] If
A
B(H)
a
A
A
| - | |
| (A) | |
| 1 |
=(A-)1
h
(A-)1
h
(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.
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(a) satisfies,
\limf(a\alpha)=f(\lima\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.