Kaplansky density theorem

In the theory of von Neumann algebras, the Kaplansky density theorem states thatif "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.

ee also

*Jacobson density theorem

References

*V.F.R.Jones [http://math.berkeley.edu/~vfr/VonNeumann.pdf von Neumann algebras] ; incomplete notes from a course.
*M. Takesaki "Theory of Operator Algebras I" ISBN 3-540-42248-X