Kaplansky density theorem

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In the theory of von Neumann algebras, the Kaplansky density theorem states that if A is a *-subalgebra of the algebra B(H) of bounded operators on a Hilbert space H, then the strong closure of the unit ball of A in B(H) is the unit ball of the strong closure of A in B(H). This gives a strengthening of the von Neumann bicommutant theorem, showing that an element a of the double commutant of A, denoted by A′′, can be strongly approximated by elements of A whose norm is no larger than that of a.

The standard proof uses the fact that, when f is bounded, the continuous functional calculus a $\mapsto$ f(a) satisfies, for a net {a_α} of self adjoint operators

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

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 the C*-algebra generated by A. A matrix computation then removes the self-adjointness restriction and proves the theorem.