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Amenable representations and finite injective von Neumann algebras

Author(s): Alain Valette
Journal: Proc. Amer. Math. Soc. 125 (1997), 1841-1843.
MSC (1991): Primary 22D25; Secondary 46L10
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Abstract: Let $U(M)$ be the unitary group of a finite, injective von Neumann algebra $M$. We observe that any subrepresentation of a group representation into $U(M)$ 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 $U(M)$.

References:

[Be]
M. E. B. Bekka, Amenable representations of locally compact groups, Invent. Math. 100 (1990), 383-401. MR 91g:22007

[BV]
M. E. B. Bekka & A. Valette, Kazhdan's property (T) and amenable representations, Math. Z. 212 (1993), 293-299. MR 94a:22006

[Co]
A. Connes, Classification of injective factors, Ann. of Math. 104 (1976), 585-609.

[Ha]
U. Haagerup, Injectivity and decomposition of completely bounded maps, in Operator algebras and their connections with topology and ergodic theory (Proceedings, Busteni, Romania 1983), LNM 1132, Springer 1985, 170-222. MR 87i:46133

[HRV]
P. de la Harpe, A. G. Robertson & A. Valette, On the spectrum of the sum of generators for a finitely generated group, Israel J. Math. 81 (1993), 65-96. MR 94j:22007

[Ki]
E. Kirchberg, Discrete groups with Kazhdan's property (T) and factorization property are residually finite, Math. Ann. 299 (1994), 551-563. MR 95h:46083

[Ro]
A. G. Robertson, Property (T) for $\mathrm {II}_1$-factors and unitary representations of Kazhdan groups, Math. Ann. 296 (1993), 547-555. MR 94f:22006

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Additional Information:

Alain Valette
Affiliation: Institut de Mathématiques, Université de Neuchâtel, Rue Emile Argand 11, CH-2007 Neuchâtel, Switzerland
Email: valette@maths.unine.ch

DOI: 10.1090/S0002-9939-97-03754-4
PII: S 0002-9939(97)03754-4
Keywords: Amenable representations, finite injective von Neumann algebra, Kazhdan's property (T)
Received by editor(s): October 6, 1995
Received by editor(s) in revised form: December 5, 1995
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1997, American Mathematical Society

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