Amenable representations and finite injective von Neumann algebras

Alain Valette

Type: Article

Publication Date: 1997-01-01

Citations: 9

DOI: https://doi.org/10.1090/s0002-9939-97-03754-4

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">U(M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the unitary group of a finite, injective von Neumann algebra <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We observe that any subrepresentation of a group representation into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">U(M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is amenable in the sense of Bekka; this yields short proofs of two known results—one by Robertson, one by Haagerup—concerning group representations into <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U left-parenthesis upper M right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>M</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">U(M)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

Proceedings of the American Mathematical Society - View - PDF

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Works Cited by This (8)

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+	Injectivity and decomposition of completely bounded maps	1985	Uffe Haagerup
+	Amenable unitary representations of locally compact groups	1990	Mohammed E. B. Bekka
+	Property (T) forII 1 factors and unitary representations of Kazhdan groups	1993	A Robertson
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+	Kazhdan's property (T) and amenable representations	1993	Mohammed E. B. Bekka Alain Valette
+	CLASSIFICATION OF INJECTIVE FACTORS	1981	Theworkof Alain Connes
+	On the spectrum of the sum of generators for a finitely generated group	1993	Pierre de la Harpe A Robertson Alain Valette
+	Classification of Injective Factors Cases II 1 , II ∞ , III λ , λ � 1	1976	Alain Connes