The Connes embedding property for quantum group von Neumann algebras
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Abstract
For a compact quantum group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Kac type, we study the existence of a Haar trace-preserving embedding of the von Neumann algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity Baseline left-parenthesis double-struck upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^\infty (\mathbb {G})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into an ultrapower of the hyperfinite II <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 1"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -factor (the Connes embedding property for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity Baseline left-parenthesis double-struck upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^\infty (\mathbb {G})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ). We establish a connection between the Connes embedding property for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity Baseline left-parenthesis double-struck upper G right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^\infty (\mathbb {G})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the structure of certain quantum subgroups of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and use this to prove that the II <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Subscript 1"> <mml:semantics> <mml:msub> <mml:mi/> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -factors <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity Baseline left-parenthesis upper O Subscript upper N Superscript plus Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>O</mml:mi> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^\infty (O_N^+)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript normal infinity Baseline left-parenthesis upper U Subscript upper N Superscript plus Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msubsup> <mml:mi>U</mml:mi> <mml:mi>N</mml:mi> <mml:mo>+</mml:mo> </mml:msubsup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L^\infty (U_N^+)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> associated to the free orthogonal and free unitary quantum groups have the Connes embedding property for all <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N greater-than-or-equal-to 4"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">N \ge 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . As an application, we deduce that the free entropy d
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| Category | Codex | Gemma |
|---|---|---|
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