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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,

The density theorem is Kaplansky's great gift to mankind. It can be used every day, and twice on Sundays.


Formal statement

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).

Kaplansky density theorem.[2] If A is a self-adjoint algebra of operators in B(H), then each element a {\displaystyle a} a in the unit ball of the strong-operator closure of A is in the strong-operator closure of the unit ball of A. In other words, \( {\displaystyle (A)_{1}^{-}=(A^{-})_{1}} \). If h is a self-adjoint operator in \( {\displaystyle (A^{-})_{1}} \), then h is in the strong-operator closure of the set of self-adjoint operators in \( {\displaystyle (A)_{1}} \).

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−)1, then h is in the strong-operator closure of the set of self-adjoint operators in (A+)1, where A+ 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−, 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 {aα} of self adjoint operators in A, the continuous functional calculus a → f(a) satisfies,

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

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 A. 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.
See also

Jacobson density theorem

Notes

Pg. 25; Pedersen, G. K., C*-algebras and their automorphism groups, London Mathematical Society Monographs, ISBN 978-0125494502.

Theorem 5.3.5; Richard Kadison, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191.

References

Kadison, Richard, Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory, American Mathematical Society. ISBN 978-0821808191.
V.F.R.Jones von Neumann algebras; incomplete notes from a course.
M. Takesaki Theory of Operator Algebras I ISBN 3-540-42248-X

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