The Mackey-Gleason problem

Let <italic>A</italic> be a von Neumann algebra with no direct summand of Type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>I</mml:mtext> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{{\text {I}}_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {P}(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be its lattice of projections. Let <italic>X</italic> be a Banach space. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m colon script upper P left-parenthesis upper A right-parenthesis right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>:</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m:\mathcal {P}(A) \to X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a bounded function such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m left-parenthesis p plus q right-parenthesis equals m left-parenthesis p right-parenthesis plus m left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>+</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">m(p + q) = m(p) + m(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whenever <italic>p</italic> and <italic>q</italic> are orthogonal projections. The main theorem states that <italic>m</italic> has a unique extension to a bounded linear operator from <italic>A</italic> to <italic>X</italic> . In particular, each bounded complex-valued finitely additive quantum measure on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">P</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {P}(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a unique extension to a bounded linear functional on <italic>A</italic> .

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