= Kaplansky density theorem
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= Kaplansky's density theorem
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For a <unital> <star-subalgebra> $A\subseteq B(H)$, its <unit ball> is dense in the <unit ball> of $A''$ in the <strong operator topology>. The same holds for <self-adjoint> unit balls and for positive unit balls, and the full unit balls may be approximated strongly together with their adjoints. Passing first to the <norm> <closure> allows <continuous functional calculus>; <resolvent> identities prove strong <continuity> of continuous functions vanishing at infinity, even for unbounded approximating nets of <self-adjoint operators>. A cutoff function supplies the <norm> bound, and a two-by-two <self-adjoint> <matrix> removes the <self-adjoint> restriction.
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