The class $$C_{0}$$ and the closability property

Abstract A (not necessarily closed) algebra of operators $$\mathcal{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> on a Hilbert space is said to have the closability property if every densely defined linear transformation that commutes with $$\mathcal{A}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> is closable. We show that an algebra of the form $$\{u(T):u\in H^{\infty}\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi>u</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> , where T is an operator of class $$C_{0}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , has the closability property precisely when T has a certain finiteness property, usually know as property (P). An analogous result is proved for commutative von Neumann algebras.