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[1] that, 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).The Kaplansky density theorem can be used to formulate some approximations with respect to the strong operator topology.In other words, for a net {aα} of self-adjoint operators in A, the continuous functional calculus a → f(a) satisfies, in the strong operator topology.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.