Inner amenable locally compact groups

Lau, Anthony To Ming; Paterson, Alan L. T.

doi:10.1090/S0002-9947-1991-1010885-5

Inner amenable locally compact groups
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by Anthony To Ming Lau and Alan L. T. Paterson PDF

Trans. Amer. Math. Soc. 325 (1991), 155-169 Request permission

Abstract:

In this paper we study the relationship between amenability, inner amenability and property $P$ of a von Neumann algebra. We give necessary conditions on a locally compact group $G$ to have an inner invariant mean $m$ such that $m(V) = 0$ for some compact neighborhood $V$ of $G$ invariant under the inner automorphisms. We also give a sufficient condition on $G$ (satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdan’s property $T$, e.g., ${\text {SL}}(n,\mathbb {Z})$, $n \geq 3$) such that each linear form on ${L^2}(G)$ which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach $G$-modules is also obtained.

References

Marie Choda, The factors of inner amenable groups, Math. Japon. 24 (1979/80), no. 1, 145–152. MR 539400
Hisashi Choda and Marie Choda, Fullness, simplicity and inner amenability, Math. Japon. 24 (1979/80), no. 2, 235–246. MR 550002
Ching Chou, Anthony To Ming Lau, and Joseph Rosenblatt, Approximation of compact operators by sums of translations, Illinois J. Math. 29 (1985), no. 2, 340–350. MR 784527

Von Neumann algebras

A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
Mahlon M. Day, Fixed-point theorems for compact convex sets, Illinois J. Math. 5 (1961), 585–590. MR 138100
Edward G. Effros, Property $\Gamma$ and inner amenability, Proc. Amer. Math. Soc. 47 (1975), 483–486. MR 355626, DOI 10.1090/S0002-9939-1975-0355626-6
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
Frederick P. Greenleaf, Martin Moskowitz, and Linda Preiss Rothschild, Unbounded conjugacy classes in Lie groups and location of central measures, Acta Math. 132 (1974), 225–243. MR 425035, DOI 10.1007/BF02392116
F. P. Greenleaf, Amenable actions of locally compact groups, J. Functional Analysis 4 (1969), 295–315. MR 0246999, DOI 10.1016/0022-1236(69)90016-0
Siegfried Grosser and Martin Moskowitz, Compactness conditions in topological groups, J. Reine Angew. Math. 246 (1971), 1–40. MR 284541, DOI 10.1515/crll.1971.246.1
P. de la Harpe, Moyennabilité du groupe unitaire et propriété $P$ de Schwartz des algèbres de von Neumann, Algèbres d’opérateurs (Sém., Les Plans-sur-Bex, 1978) Lecture Notes in Math., vol. 725, Springer, Berlin, 1979, pp. 220–227 (French). MR 548116

Abstract harmonic analysis

Anthony To Ming Lau and Alan L. T. Paterson, Operator theoretic characterizations of $[IN]$-groups and inner amenability, Proc. Amer. Math. Soc. 102 (1988), no. 4, 893–897. MR 934862, DOI 10.1090/S0002-9939-1988-0934862-0
John R. Liukkonen, Dual spaces of groups with precompact conjugacy classes, Trans. Amer. Math. Soc. 180 (1973), 85–108. MR 318390, DOI 10.1090/S0002-9947-1973-0318390-5
V. Losert and H. Rindler, Asymptotically central functions and invariant extensions of Dirac measure, Probability measures on groups, VII (Oberwolfach, 1983) Lecture Notes in Math., vol. 1064, Springer, Berlin, 1984, pp. 368–378. MR 772419, DOI 10.1007/BFb0073651
V. Losert and H. Rindler, Conjugation-invariant means, Colloq. Math. 51 (1987), 221–225. MR 891290, DOI 10.4064/cm-51-1-221-225
Gary Hosler Meisters, Some problems and results on translation-invariant linear forms, Radical Banach algebras and automatic continuity (Long Beach, Calif., 1981) Lecture Notes in Math., vol. 975, Springer, Berlin, 1983, pp. 423–444. MR 697605, DOI 10.1007/BFb0064574

Følner’s condition for amenable semigroups

15

William L. Paschke, Inner amenability and conjugation operators, Proc. Amer. Math. Soc. 71 (1978), no. 1, 117–118. MR 473849, DOI 10.1090/S0002-9939-1978-0473849-5
Alan L. T. Paterson, Amenability, Mathematical Surveys and Monographs, vol. 29, American Mathematical Society, Providence, RI, 1988. MR 961261, DOI 10.1090/surv/029

Quasi-invariance intérieure sur les groupes localement compacts

Jean-Paul Pier, Amenable locally compact groups, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 767264

Amenable Banach algebras

Hans Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968. MR 0306811
Neil W. Rickert, Amenable groups and groups with the fixed point property, Trans. Amer. Math. Soc. 127 (1967), 221–232. MR 222208, DOI 10.1090/S0002-9947-1967-0222208-6
Joseph Rosenblatt, Translation-invariant linear forms on $L_p(G)$, Proc. Amer. Math. Soc. 94 (1985), no. 2, 226–228. MR 784168, DOI 10.1090/S0002-9939-1985-0784168-5
Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
J. T. Schwartz, $W^{\ast }$-algebras, Gordon and Breach Science Publishers, New York-London-Paris, 1967. MR 0232221
F. J. Yeadon, Fixed points and amenability: a counterexample, J. Math. Anal. Appl. 45 (1974), 718–720. MR 350444, DOI 10.1016/0022-247X(74)90062-6
Chuan Kuan Yuan, The existence of inner invariant means on $L^\infty (G)$, J. Math. Anal. Appl. 130 (1988), no. 2, 514–524. MR 929957, DOI 10.1016/0022-247X(88)90327-7
Robert J. Zimmer, On the von Neumann algebra of an ergodic group action, Proc. Amer. Math. Soc. 66 (1977), no. 2, 289–293. MR 460599, DOI 10.1090/S0002-9939-1977-0460599-3