Derivations of Murray–von Neumann algebras

Abstract In this paper, we answer in the affirmative the long-standing conjecture that the first cohomology group of the Murray–von Neumann algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>S</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi mathvariant="script">ℳ</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {S(\mathcal{M})} of all operators affiliated with a type <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mi>II</m:mi><m:mn>1</m:mn></m:msub></m:math> {\mathrm{II}_{1}} von Neumann algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi mathvariant="script">ℳ</m:mi></m:math> {\mathcal{M}} is 0. That is, we show that all derivations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow><m:mi>S</m:mi><m:mo>⁢</m:mo><m:mrow><m:mo stretchy="false">(</m:mo><m:mi mathvariant="script">ℳ</m:mi><m:mo stretchy="false">)</m:mo></m:mrow></m:mrow></m:math> {S(\mathcal{M})} are inner.

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