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In the theory of [[von Neumann algebra]]s, 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<ref>Pg. 25; [[Pedersen, G. K.]], ''C*-algebras and their automorphism groups'', London Mathematical Society Monographs, {{isbn|978-0125494502}}.</ref> that, 
:''The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays.''

==Formal statement==
Let ''K''<sup>−</sup> denote the [[Strong operator topology|strong-operator closure]] of a set ''K'' in ''B(H)'', the set of bounded operators on the Hilbert space ''H'', and let (''K'')<sub>1</sub> denote the intersection of ''K'' with the unit ball of ''B(H)''.
:'''Kaplansky density theorem'''.<ref>Theorem 5.3.5; [[Richard Kadison]], ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. {{isbn|978-0821808191}}.</ref> If <math>A</math> is a self-adjoint algebra of operators in <math>B(H)</math>, then each element <math>a</math> in the unit ball of the strong-operator closure of <math>A</math> is in the strong-operator closure of the unit ball of <math>A</math>. In other words, <math>(A)_1^{-} = (A^{-})_1</math>. If <math>h</math> is a self-adjoint operator in <math>(A^{-})_1</math>, then <math>h</math> is in the strong-operator closure of the set of self-adjoint operators in <math>(A)_1</math>.

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''<sup>−</sup>)<sub>1</sub>, then ''h'' is in the strong-operator closure of the set of self-adjoint operators in (''A''<sup>+</sup>)<sub>1</sub>, where ''A''<sup>+</sup> 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<sup>−</sup>, 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.

==Proof==
The standard proof uses the fact that, a bounded continuous real-valued function ''f'' is strong-operator continuous. In other words, for a net {''a<sub>α</sub>''} of [[self adjoint operator]]s in ''A'', the [[continuous functional calculus]] ''a'' &rarr; ''f''(''a'') satisfies,

:<math>\lim f(a_{\alpha}) = f (\lim a_{\alpha})</math>

in the [[strong operator topology]]. This shows that self-adjoint part of the unit ball in ''A''<sup>−</sup> can be approximated strongly by self-adjoint elements in ''A''. A matrix computation in ''M''<sub>2</sub>(''A'') considering the self-adjoint operator with entries ''0'' on the diagonal and ''a'' and ''a''<sup>*</sup> at the other positions, then removes the self-adjointness restriction and proves the theorem.

==See also==
*[[Jacobson density theorem]]

==Notes==
{{Reflist}}

==References==
*[[Kadison, Richard]], ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. {{isbn|978-0821808191}}.
*V.F.R.Jones [http://math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf von Neumann algebras]; incomplete notes from a course.
*M. Takesaki ''Theory of Operator Algebras I'' {{isbn|3-540-42248-X}}

[[Category:Von Neumann algebras]]
[[Category:Theorems in functional analysis]]