Kaplansky density theorem

Last updated September 03, 2021

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,

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

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,

(A)_{1}^{-}=(A^{-})_{1}

. If

h

is a self-adjoint operator in

(A^{-})_{1}

, then

h

is in the strong-operator closure of the set of self-adjoint operators in

(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 {a_α} 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.

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.

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

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

[1]

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