abstract: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.