The unitary group of a $$\textrm{II}_1$$ factor is SOT-contractible

Abstract We show that the unitary group of any SOT-separable $$\textrm{II}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>II</mml:mtext> <mml:mn>1</mml:mn> </mml:msub> </mml:math> factor M , with the strong operator topology, is contractible. Combined with several old results, this implies that the same is true for any SOT-separable von Neumann algebra with no type $$\textrm{I}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>I</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> direct summands ( $$n &lt; \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> ). The proof for the $$\textrm{II}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>II</mml:mtext> <mml:mn>1</mml:mn> </mml:msub> </mml:math> -factor case uses regularization via free convolution and Popa’s theorem on the existence of approximately free Haar unitaries in $$\textrm{II}_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>II</mml:mtext> <mml:mn>1</mml:mn> </mml:msub> </mml:math> factors.