Kaplansky Density Theorem

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 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.Theorem 5.3.5; Richard Kadison, ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. . 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 of self-adjoint operators in ''A'', the continuous functional calculus ''a'' → ''f''(''a'') satisfies,

:$\lim f(a_) = f (\lim a_)$

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 ''M''₂(''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

References

* Kadison, Richard, ''Fundamentals of the Theory of Operator Algebras, Vol. I : Elementary Theory'', American Mathematical Society. .
* V.F.R.Jone
von Neumann algebras
incomplete notes from a course.
* M. Takesaki ''Theory of Operator Algebras I''

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Von Neumann algebras
Theorems in functional analysis