QUANTUM EXPANDERS AND QUANTIFIER REDUCTION FOR TRACIAL VONNEUMANN ALGEBRAS

Abstract We provide a complete characterization of theories of tracial von Neumann algebras that admit quantifier elimination. We also show that the theory of a separable tracial von Neumann algebra script upper M $\mathcal {M}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi mathvariant="script">M</mml:mi> </mml:math> is never model complete if its direct integral decomposition contains upper I upper I Subscript 1 $\mathrm {II}_1$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mstyle mathvariant="normal"> <mml:msub> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>I</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mstyle> </mml:math> factors script upper N $\mathcal {N}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi mathvariant="script">N</mml:mi> </mml:math> such that upper M 2 left parenthesis script upper N right parenthesis $M_2(\mathcal {N})$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mrow> <mml:msub> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false" form="prefix" fence="true">(</mml:mo> <mml:mi mathvariant="script">N</mml:mi> <mml:mo stretchy="false" form="postfix" fence="true">)</mml:mo> </mml:mrow> </mml:math> embeds into an ultrapower of script upper N $\mathcal {N}$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi mathvariant="script">N</mml:mi> </mml:math> . The proof in the case of upper I upper I Subscript 1 $\mathrm {II}_1$ <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mstyle mathvariant="normal"> <mml:msub> <mml:mrow> <mml:mi>I</mml:mi> <mml:mi>I</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:mstyle> </mml:math> factors uses an explicit construction based on random matrices and quantum expanders.