491:≥ 0 looks like a jagged curve above a circle, which can be made by tearing off the end of a paper tube. The linear functional would then average the curve by snipping off some paper from one place and gluing it to another place, creating a flat top again. This is the invariant mean, i.e. the average value
580:
Left-invariance would mean that rotating the tube does not change the height of the flat top at the end. That is, only the shape of the tube matters. Combined with linearity, positivity, and norm-1, this is sufficient to prove that the invariant mean we have constructed is unique.
1475:
555:
709:
2011:
If every conjugacy class in a locally compact group has compact closure, then the group is amenable. Examples of groups with this property include compact groups, locally compact abelian groups, and
622:
2216:
Day's first published use of the word is in his abstract for an AMS summer meeting in 1949. Many textbooks on amenability, such as Volker Runde's, suggest that Day chose the word as a pun.
1863:
is amenable (a sequence of intervals of length tending to infinity is a Følner sequence). The existence of a shift-invariant, finitely additive probability measure on the group
1348:
157:
1765:
2076:
cannot be obtained by bootstrap constructions as used to construct elementary amenable groups. Since there exist such simple groups that are amenable, due to
Juschenko and
1731:
575:
183:
2115:, they cannot contain the free group on two generators. These groups are finitely generated, but not finitely presented. However, in 2002 Sapir and Olshanskii found
1480:
The equivalent conditions for amenability also become simpler in the case of a countable discrete group Γ. For such a group the following conditions are equivalent:
2026:
237:
209:
818:(1) ≥ 0. Valette improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function
1821:
Direct limits of amenable groups are amenable. In particular, if a group can be written as a directed union of amenable subgroups, then it is amenable.
1787:
of certain operators. For instance, the fundamental group of a closed
Riemannian manifold is amenable if and only if the bottom of the spectrum of the
2150:
a field either has a normal solvable subgroup of finite index (and therefore is amenable) or contains the free group on two generators. Although
1386:
2159:
494:
2577:
2611:
Guivarc'h, Yves (1990), "Produits de matrices aléatoires et applications aux propriétés géometriques des sous-groupes du groupes linéaire",
239:
of the whole group. Even though both the group and the subgroup has infinitely many elements, there is a well-defined sense of proportion.
2972:
268:. Then it is well known that it possesses a unique, up-to-scale left- (or right-) translation invariant nontrivial ring measure, the
630:
2057:. The first class of examples below can be used to exhibit non-elementary amenable examples thanks to the existence of groups of
773:
769:
750:
There is a left-invariant state on any separable left-invariant unital C*-subalgebra of the bounded continuous functions on
2768:
2895:
2877:
2859:
2814:
455:) with the space of finitely-additive Borel measures which are absolutely continuous with respect to the Haar measure on
1777:
1769:
80:
on two generators. Amenable groups cannot contain such groups, and do not allow this kind of paradoxical construction.
1357:. (Note that some of the properties of the Lebesgue integral fail here, since our measure is only finitely additive.)
2551:
591:
2187:
1825:
1360:
If a group has a left-invariant measure, it automatically has a bi-invariant one. Given a left-invariant measure
2987:
303:
measurable functions within this measure space (which is clearly independent of the scale of the Haar measure).
2163:
2727:
Olshanskii, Alexander Yu; Sapir, Mark V. (2002), "Non-amenable finitely presented torsion-by-cyclic groups",
2633:
Juschenko, Kate; Monod, Nicolas (2013), "Cantor systems, piecewise translations and simple amenable groups",
1776:
Note that A. Connes also proved that the von
Neumann group algebra of any connected locally compact group is
1264:, the measure can be thought of as answering the question: what is the probability that a random element of
1832:
2992:
121:, a simpler definition is used. In this setting, a group is amenable if one can say what proportion of
2066:
1316:
103:
2962:
1980:
could not be bounded). This implies that there is a well-defined norm-one linear form on the subspace
128:
2192:
2093:
subgroup on two generators, then it is not amenable. The converse to this statement is the so-called
2054:
1788:
772:
has a fixed point. For locally compact abelian groups, this property is satisfied as a result of the
62:
2501:
2120:
2116:
2019:
1815:
2705:
1868:
1128:
72:
The critical step in the Banach–Tarski paradox construction is to find inside the rotation group
42:
2826:(1989), "Unitary representations of fundamental groups and the spectrum of twisted Laplacians",
2927:
2759:
2496:
2197:
2175:
2166:. Analogues of the Tits alternative have been proved for many other classes of groups, such as
2094:
722:
contains a comprehensive account of the conditions on a second countable locally compact group
281:
99:
65:. In 1949 Mahlon M. Day introduced the English translation "amenable", apparently as a pun on "
1256:
is amenable if it has a finitely-additive left-invariant probability measure. Given a subset
560:
45:
under translation by group elements. The original definition, in terms of a finitely additive
2635:
1167:
830:
has non-negative integral with respect to Haar measure, where Δ denotes the modular function.
782:
All irreducible representations are weakly contained in the left regular representation λ on
761:
584:
As an example for locally compact groups, consider the group of integers. A bounded function
487:
As an example for compact groups, consider the circle group. The graph of a typical function
273:
162:
110:
95:
87:
46:
2674:
1856:
groups are amenable. The Haar measure is an invariant mean (unique taking total measure 1).
1354:
2798:
8:
2572:
2098:
2039:
2038:
It follows from the extension property above that a group is amenable if it has a finite
1876:
1761:
1102:
300:
265:
214:
188:
2678:
2828:
2736:
2644:
2524:
2477:
2171:
2155:
2004:), which is by construction a shift-invariant finitely additive probability measure on
1747:
1656:
is a symmetric probability measure on Γ with support generating Γ, then convolution by
91:
2697:
1000:
243:
2918:
2891:
2873:
2855:
2842:
2810:
2307:
2167:
330:
118:
31:
2942:
2913:
2837:
2823:
2794:
2777:
2746:
2714:
2693:
2682:
2654:
2620:
2559:
2546:
2534:
2506:
2310:
2131:
2058:
1849:
1814:
of an amenable group by an amenable group is again amenable. In particular, finite
1380:) is a right-invariant measure. Combining these two gives a bi-invariant measure:
1275:
It is a fact that this definition is equivalent to the definition in terms of
54:
38:
2590:
1811:
1784:
1515:
741:
262:
259:
58:
28:
2658:
2000:. By the Hahn–Banach theorem the latter admits a norm-one linear extension on ℓ(
2763:
2112:
2108:
2103:
2046:
1149:
625:
2947:
2751:
2625:
2487:
Ballmann, Werner; Brin, Michael (1995), "Orbihedra of nonpositive curvature",
125:
any given subset takes up. For example, any subgroup of the group of integers
2981:
2077:
2069:
are amenable. A suitable subsequence of balls will provide a Følner sequence.
2030:
1853:
77:
2766:(1987). "Entropy and isomorphism theorems for actions of amenable groups".
2719:
2151:
2127:
2073:
1845:
1780:, so the last condition no longer applies in the case of connected groups.
884:
there is an integrable non-negative function φ with integral 1 such that λ(
289:
269:
2519:
Bowen, Lewis (2012). "Every countably infinite group is almost
Ornstein".
1470:{\displaystyle \nu (A)=\int _{g\in G}\mu \left(Ag^{-1}\right)\,d\mu ^{-}.}
1090:
550:{\displaystyle \Lambda (f)=\int _{\mathbb {R} /\mathbb {Z} }f\ d\lambda }
20:
2665:
Leptin, H. (1968), "Zur harmonischen
Analyse klassenkompakter Gruppen",
2489:
Publications Mathématiques de l'Institut des Hautes Études
Scientifiques
1818:
of amenable groups are amenable, although infinite products need not be.
2781:
2686:
2563:
2538:
2511:
2473:
2090:
1754:
1727:
1529:
1522:
765:
16:
Locally compact topological group with an invariant averaging operation
2966:
2741:
2315:
2042:
amenable subgroup. That is, virtually amenable groups are amenable.
2549:(1981). "The fundamental group and the spectrum of the Laplacian".
2018:
By the direct limit property above, a group is amenable if all its
2012:
1792:
460:
2649:
2529:
1757:
is quasidiagonal (J. Rosenberg, A. Tikuisis, S. White, W. Winter).
1197:, the measure of the union of the sets is the sum of the measures.
2597:, North-Holland Mathematical Library, vol. 15, North-Holland
2415:
2403:
2123:
that have a periodic normal subgroup with quotient the integers.
1860:
2439:
395:) with respect to the left (respectively right) shift action of
211:
then the subgroup takes up 0 proportion. Otherwise, it takes up
2451:
2022:
subgroups are. That is, locally amenable groups are amenable.
1170:(also called a mean)—a function that assigns to each subset of
475:) induces a left-invariant, finitely additive Borel measure on
704:{\displaystyle \lim _{n}{\frac {1}{2n+1}}\sum _{k=-n}^{n}f(k)}
2336:
1294:
allows us to define integration of bounded functions on
73:
2604:
Invariant Means on
Topological Groups and Their Applications
61:
name "messbar" ("measurable" in
English) in response to the
2595:
C*-algebras (translated from the French by
Francis Jellett)
2523:. Contemporary Mathematics. Vol. 567. pp. 67–78.
94:. An intuitive way to understand this version is that the
2809:, Lecture Notes in Mathematics, vol. 1774, Springer,
2472:
This article incorporates material from
Amenable group on
1148:
The definition of amenability is simpler in the case of a
836:
There is a sequence of integrable non-negative functions φ
1804:
Every (closed) subgroup of an amenable group is amenable.
2324:
2027:
fundamental theorem of finitely generated abelian groups
976:), the distance between 0 and the closed convex hull in
714:
249:
2427:
2305:
2229:
2089:
If a countable discrete group contains a (non-abelian)
2080:, this provides again non-elementary amenable examples.
800:
is weakly contained in the left regular representation.
463:), the terminology becomes more natural: a mean in Hom(
1663:
If Γ acts by isometries on a (separable) Banach space
1487:
If Γ acts by isometries on a (separable) Banach space
2130:, however, the von Neumann conjecture is true by the
1973:
would be positive and bounded away from zero, whence
1506:
There is a left invariant norm-continuous functional
1389:
1319:
633:
594:
563:
497:
217:
191:
165:
131:
2904:
Tits, J. (1972), "Free subgroups in linear groups",
1152:, i.e. a group equipped with the discrete topology.
2158:, Guivarc'h later found an analytic proof based on
2360:
2256:
2246:
2244:
1469:
1342:
703:
616:
569:
549:
231:
203:
177:
151:
740:). The original definition, which depends on the
86:has many equivalent definitions. In the field of
2979:
2478:Creative Commons Attribution/Share-Alike License
2348:
1954:) has a distance larger than or equal to 1 from
1807:Every quotient of an amenable group is amenable.
731:Existence of a left (or right) invariant mean on
635:
440:if it admits a left- (or right-)invariant mean.
2928:"On Godement's characterisation of amenability"
2726:
2421:
2241:
1660:defines an operator of norm 1 on ℓ(Γ) (Kesten).
2632:
2409:
1852:with the discrete definition. More generally,
617:{\displaystyle f:\mathbb {Z} \to \mathbb {R} }
284:; there are both left and right measures when
2758:
2610:
2578:Bulletin of the American Mathematical Society
2445:
2342:
2013:discrete groups with finite conjugacy classes
1795:of the universal cover of the manifold is 0.
2486:
2457:
1831:Countable discrete amenable groups obey the
1067:with finite positive Haar measure such that
1021:with finite positive Haar measure such that
436:A locally compact Hausdorff group is called
2692:
1193:: given finitely many disjoint subsets of
1143:
325:if Λ has norm 1 and is non-negative, i.e.
37:carrying a kind of averaging operation on
2946:
2917:
2841:
2750:
2740:
2718:
2648:
2624:
2601:
2528:
2510:
2500:
2381:
2277:
1766:von Neumann algebras associated to groups
1499:* invariant, then Γ has a fixed point in
1450:
1333:
610:
602:
529:
519:
136:
2885:
2867:
2849:
2396:
2391:
2292:
2287:
1491:, leaving a weakly closed convex subset
1252:This definition can be summarized thus:
806:Every bounded positive-definite measure
2925:
2793:, Pure and Applied Mathematics, Wiley,
2589:
2262:
1535:There is a set of probability measures
1528:on any left invariant separable unital
770:locally convex topological vector space
2980:
2822:
2664:
2545:
2366:
2330:
2107:. Adyan subsequently showed that free
834:Day's asymptotic invariance condition.
760:Any action of the group by continuous
479:which gives the whole group weight 1.
2804:
2518:
2354:
2306:
1109:gives an operator of operator norm 1.
1051:For every finite (or compact) subset
1005:For every finite (or compact) subset
910:For every finite (or compact) subset
876:For every finite (or compact) subset
715:Equivalent conditions for amenability
588:is simply a bounded function of type
250:Definition for locally compact groups
2903:
2788:
2729:Publ. Math. Inst. Hautes Études Sci.
2613:Ergodic Theory and Dynamical Systems
2433:
2386:
2282:
2250:
726:that are equivalent to amenability:
719:
2698:"Zur allgemeinen Theorie des Maßes"
2570:
2521:Dynamical Systems and Group Actions
2235:
862:tends to 0 in the weak topology on
774:Markov–Kakutani fixed-point theorem
748:Existence of left-invariant states.
246:then it is automatically amenable.
13:
2973:An introduction to amenable groups
1828:; the converse is an open problem.
1244:is translated on the left by
1113:Johnson's cohomological condition.
498:
14:
3004:
2956:
2573:"Means on semigroups and groups"
2111:are non-amenable: since they are
2045:Furthermore, it follows that all
1343:{\displaystyle \int _{G}f\,d\mu }
1182:: the measure of the whole group
1131:, i.e. any bounded derivation of
2188:Uniformly bounded representation
1675:*) is a bounded 1-cocycle, i.e.
1166:if there is a finitely additive
152:{\displaystyle (\mathbb {Z} ,+)}
90:, the definition is in terms of
2888:Theory of Operator Algebras III
2791:Amenable locally compact groups
2372:
1719:·φ − φ for some φ in
1174:a number from 0 to 1—such that
888:)φ − φ is arbitrarily small in
2870:Theory of Operator Algebras II
2769:Journal d'Analyse Mathématique
2476:, which is licensed under the
2299:
2268:
2210:
2164:multiplicative ergodic theorem
2119:counterexamples: non-amenable
2084:
1399:
1393:
796:The trivial representation of
698:
692:
606:
507:
501:
146:
132:
1:
2852:Theory of Operator Algebras I
2065:Finitely generated groups of
1875:be the shift operator on the
1867:also follows easily from the
1798:
1732:the reduced group C*-algebra
1059:there is a measurable subset
1013:there is a measurable subset
309:A linear functional Λ in Hom(
159:is generated by some integer
2919:10.1016/0021-8693(72)90058-0
2843:10.1016/0040-9383(89)90015-3
2223:
2072:Finitely generated infinite
1833:Ornstein isomorphism theorem
1298:. Given a bounded function
1240:. That is, each element of
1228:denotes the set of elements
1129:amenable as a Banach algebra
962:Glicksberg−Reiter condition.
7:
2659:10.4007/annals.2013.178.2.7
2422:Olshanskii & Sapir 2002
2181:
1923:) be the constant sequence
1839:
1514:(1) = 1 (this requires the
1495:of the closed unit ball of
1037:) is arbitrarily small for
984:) of the left translates λ(
104:irreducible representations
10:
3009:
2466:
2410:Juschenko & Monod 2013
1783:Amenability is related to
1521:There is a left invariant
1135:into the dual of a Banach
482:
288:is compact.) Consider the
2963:Some notes on amenability
2948:10.1017/s0004972700031506
2935:Bull. Austral. Math. Soc.
2752:10.1007/s10240-002-0006-7
2626:10.1017/S0143385700005708
2343:Ornstein & Weiss 1987
2121:finitely presented groups
2097:, which was disproved by
1762:von Neumann group algebra
1613:There are finite subsets
2789:Pier, Jean-Paul (1984),
2602:Greenleaf, F.P. (1969),
2458:Ballmann & Brin 1995
2203:
1883:), which is defined by (
1755:reduced group C*-algebra
1728:reduced group C*-algebra
1707:is a 1-coboundary, i.e.
942:is arbitrarily small in
570:{\displaystyle \lambda }
49:(or mean) on subsets of
2926:Valette, Alain (1998),
2807:Lectures on Amenability
2606:, Van Nostrand Reinhold
2126:For finitely generated
2053:All examples above are
1930: = 1 for all
1577:There are unit vectors
1144:Case of discrete groups
1083:) is arbitrarily small.
794:Trivial representation.
178:{\displaystyle p\geq 0}
2988:Geometric group theory
2720:10.4064/fm-13-1-73-116
2198:Von Neumann conjecture
2193:Kazhdan's property (T)
2176:non-positive curvature
2095:von Neumann conjecture
1848:are amenable. Use the
1645:| tends to 0 for each
1471:
1344:
1220:equals the measure of
762:affine transformations
705:
688:
624:, and its mean is the
618:
571:
551:
233:
205:
179:
153:
102:is the whole space of
100:regular representation
2886:Takesaki, M. (2013),
2868:Takesaki, M. (2002),
2850:Takesaki, M. (2001),
2636:Annals of Mathematics
2238:, pp. 1054–1055.
2067:subexponential growth
1472:
1345:
918:there is unit vector
766:compact convex subset
758:Fixed-point property.
706:
665:
619:
577:is Lebesgue measure.
572:
552:
274:Borel regular measure
234:
206:
180:
154:
111:discrete group theory
63:Banach–Tarski paradox
2971:Garrido, Alejandra.
2552:Comment. Math. Helv.
2172:simplicial complexes
2134:: every subgroup of
1824:Amenable groups are
1606:tends to 0 for each
1584:in ℓ(Γ) such that ||
1570:tends to 0 for each
1387:
1355:Lebesgue integration
1317:
908:Dixmier's condition.
631:
592:
561:
495:
215:
189:
163:
129:
53:, was introduced by
2760:Ornstein, Donald S.
2679:1968InMat...5..249L
2571:Day, M. M. (1949).
2460:, pp. 169–209.
2448:, pp. 483–512.
2436:, pp. 250–270.
2412:, pp. 775–787.
2333:, pp. 581–598.
2059:intermediate growth
2055:elementary amenable
1869:Hahn–Banach theorem
1180:probability measure
1139:-bimodule is inner.
1115:The Banach algebra
1103:probability measure
1049:Leptin's condition.
874:Reiter's condition.
842:with integral 1 on
804:Godement condition.
443:By identifying Hom(
301:essentially bounded
232:{\displaystyle 1/p}
204:{\displaystyle p=0}
2993:Topological groups
2805:Runde, V. (2002),
2782:10.1007/BF02790325
2687:10.1007/bf01389775
2564:10.1007/bf02566228
2512:10.1007/BF02698640
2424:, pp. 43–169.
2308:Weisstein, Eric W.
2168:fundamental groups
2156:algebraic geometry
2117:finitely presented
2101:in 1980 using his
2029:, it follows that
2020:finitely generated
1610:in Γ (J. Dixmier).
1467:
1340:
1087:Kesten's condition
701:
643:
614:
567:
547:
321:) is said to be a
229:
201:
175:
149:
92:linear functionals
57:in 1929 under the
2824:Sunada, Toshikazu
2345:, pp. 1–141.
2170:of 2-dimensional
1723:* (B.E. Johnson).
1544:on Γ such that ||
1353:is defined as in
1286:Having a measure
1232:for each element
1216:, the measure of
1204:: given a subset
1191:finitely additive
1178:The measure is a
1158:A discrete group
1101:) by a symmetric
780:Irreducible dual.
768:of a (separable)
663:
634:
540:
242:If a group has a
119:discrete topology
39:bounded functions
32:topological group
3000:
2951:
2950:
2932:
2922:
2921:
2900:
2882:
2864:
2846:
2845:
2819:
2801:
2785:
2755:
2754:
2744:
2723:
2722:
2702:
2689:
2661:
2652:
2629:
2628:
2607:
2598:
2591:Dixmier, Jacques
2586:
2585:(11): 1054–1055.
2567:
2542:
2539:10.1090/conm/567
2532:
2515:
2514:
2504:
2461:
2455:
2449:
2443:
2437:
2431:
2425:
2419:
2413:
2407:
2401:
2376:
2370:
2364:
2358:
2352:
2346:
2340:
2334:
2328:
2322:
2321:
2320:
2311:"Discrete Group"
2303:
2297:
2272:
2266:
2260:
2254:
2248:
2239:
2233:
2217:
2214:
2132:Tits alternative
1994:tu + y
1850:counting measure
1620:of Γ such that |
1574:in Γ (M.M. Day).
1476:
1474:
1473:
1468:
1463:
1462:
1449:
1445:
1444:
1443:
1420:
1419:
1349:
1347:
1346:
1341:
1329:
1328:
1001:Følner condition
826:, the function Δ
710:
708:
707:
702:
687:
682:
664:
662:
645:
642:
623:
621:
620:
615:
613:
605:
576:
574:
573:
568:
556:
554:
553:
548:
538:
534:
533:
532:
527:
522:
414:) (respectively
355:) is said to be
343:A mean Λ in Hom(
282:second-countable
238:
236:
235:
230:
225:
210:
208:
207:
202:
184:
182:
181:
176:
158:
156:
155:
150:
139:
55:John von Neumann
3008:
3007:
3003:
3002:
3001:
2999:
2998:
2997:
2978:
2977:
2959:
2954:
2930:
2898:
2897:978-366210453-8
2880:
2879:978-354042914-2
2862:
2861:978-354042248-8
2817:
2816:978-354042852-7
2764:Weiss, Benjamin
2700:
2469:
2464:
2456:
2452:
2444:
2440:
2432:
2428:
2420:
2416:
2408:
2404:
2377:
2373:
2365:
2361:
2353:
2349:
2341:
2337:
2329:
2325:
2304:
2300:
2273:
2269:
2261:
2257:
2249:
2242:
2234:
2230:
2226:
2221:
2220:
2215:
2211:
2206:
2184:
2109:Burnside groups
2104:Tarski monsters
2087:
2047:solvable groups
1978:
1971:
1967:
1963:
1928:
1902:
1892:
1842:
1812:group extension
1801:
1785:spectral theory
1737:
1643:
1636:
1629:
1618:
1605:
1600:
1593:
1582:
1569:
1565:
1556:
1543:
1516:axiom of choice
1458:
1454:
1436:
1432:
1428:
1424:
1409:
1405:
1388:
1385:
1384:
1364:, the function
1324:
1320:
1318:
1315:
1314:
1310:, the integral
1208:and an element
1200:The measure is
1189:The measure is
1146:
861:
855:
841:
742:axiom of choice
717:
683:
669:
649:
644:
638:
632:
629:
628:
626:running average
609:
601:
593:
590:
589:
562:
559:
558:
528:
523:
518:
517:
513:
496:
493:
492:
485:
361:right-invariant
260:locally compact
252:
244:Følner sequence
221:
216:
213:
212:
190:
187:
186:
164:
161:
160:
135:
130:
127:
126:
29:locally compact
17:
12:
11:
5:
3006:
2996:
2995:
2990:
2976:
2975:
2969:
2958:
2957:External links
2955:
2953:
2952:
2923:
2912:(2): 250–270,
2901:
2896:
2883:
2878:
2865:
2860:
2847:
2836:(2): 125–132,
2820:
2815:
2802:
2786:
2756:
2724:
2694:von Neumann, J
2690:
2673:(4): 249–254,
2662:
2643:(2): 775–787,
2630:
2619:(3): 483–512,
2608:
2599:
2587:
2568:
2547:Brooks, Robert
2543:
2516:
2502:10.1.1.30.8282
2483:
2468:
2465:
2463:
2462:
2450:
2446:Guivarc'h 1990
2438:
2426:
2414:
2402:
2400:
2399:
2394:
2389:
2384:
2382:Greenleaf 1969
2371:
2359:
2347:
2335:
2323:
2298:
2296:
2295:
2290:
2285:
2280:
2278:Greenleaf 1969
2267:
2255:
2240:
2227:
2225:
2222:
2219:
2218:
2208:
2207:
2205:
2202:
2201:
2200:
2195:
2190:
2183:
2180:
2086:
2083:
2082:
2081:
2070:
2051:
2050:
2043:
2036:
2035:
2034:
2031:abelian groups
2016:
2009:
1976:
1969:
1968: - x
1965:
1961:
1938:. Any element
1926:
1897:
1888:
1877:sequence space
1871:this way. Let
1857:
1841:
1838:
1837:
1836:
1829:
1822:
1819:
1816:direct product
1808:
1805:
1800:
1797:
1774:
1773:
1758:
1751:
1735:
1724:
1691:) +
1661:
1650:
1649:in Γ (Følner).
1641:
1634:
1627:
1616:
1611:
1603:
1598:
1591:
1580:
1575:
1567:
1561:
1552:
1539:
1533:
1519:
1504:
1485:
1484:Γ is amenable.
1478:
1477:
1466:
1461:
1457:
1453:
1448:
1442:
1439:
1435:
1431:
1427:
1423:
1418:
1415:
1412:
1408:
1404:
1401:
1398:
1395:
1392:
1351:
1350:
1339:
1336:
1332:
1327:
1323:
1250:
1249:
1202:left-invariant
1198:
1187:
1150:discrete group
1145:
1142:
1141:
1140:
1110:
1084:
1046:
997:
959:
930:) such that λ(
905:
871:
857:
851:
837:
831:
801:
791:
777:
755:
745:
716:
713:
700:
697:
694:
691:
686:
681:
678:
675:
672:
668:
661:
658:
655:
652:
648:
641:
637:
612:
608:
604:
600:
597:
566:
546:
543:
537:
531:
526:
521:
516:
512:
509:
506:
503:
500:
484:
481:
359:(respectively
357:left-invariant
251:
248:
228:
224:
220:
200:
197:
194:
174:
171:
168:
148:
145:
142:
138:
134:
25:amenable group
15:
9:
6:
4:
3:
2:
3005:
2994:
2991:
2989:
2986:
2985:
2983:
2974:
2970:
2968:
2964:
2961:
2960:
2949:
2944:
2940:
2936:
2929:
2924:
2920:
2915:
2911:
2907:
2902:
2899:
2893:
2889:
2884:
2881:
2875:
2871:
2866:
2863:
2857:
2853:
2848:
2844:
2839:
2835:
2831:
2830:
2825:
2821:
2818:
2812:
2808:
2803:
2800:
2796:
2792:
2787:
2783:
2779:
2775:
2771:
2770:
2765:
2761:
2757:
2753:
2748:
2743:
2738:
2734:
2730:
2725:
2721:
2716:
2713:(1): 73–111,
2712:
2708:
2707:
2699:
2695:
2691:
2688:
2684:
2680:
2676:
2672:
2668:
2667:Invent. Math.
2663:
2660:
2656:
2651:
2646:
2642:
2638:
2637:
2631:
2627:
2622:
2618:
2615:(in French),
2614:
2609:
2605:
2600:
2596:
2592:
2588:
2584:
2580:
2579:
2574:
2569:
2565:
2561:
2557:
2554:
2553:
2548:
2544:
2540:
2536:
2531:
2526:
2522:
2517:
2513:
2508:
2503:
2498:
2494:
2490:
2485:
2484:
2482:
2481:
2479:
2475:
2459:
2454:
2447:
2442:
2435:
2430:
2423:
2418:
2411:
2406:
2398:
2397:Takesaki 2002
2395:
2393:
2392:Takesaki 2001
2390:
2388:
2385:
2383:
2380:
2379:
2375:
2368:
2363:
2356:
2351:
2344:
2339:
2332:
2327:
2318:
2317:
2312:
2309:
2302:
2294:
2293:Takesaki 2002
2291:
2289:
2288:Takesaki 2001
2286:
2284:
2281:
2279:
2276:
2275:
2271:
2264:
2259:
2252:
2247:
2245:
2237:
2232:
2228:
2213:
2209:
2199:
2196:
2194:
2191:
2189:
2186:
2185:
2179:
2177:
2173:
2169:
2165:
2161:
2157:
2154:' proof used
2153:
2149:
2145:
2141:
2137:
2133:
2129:
2128:linear groups
2124:
2122:
2118:
2114:
2110:
2106:
2105:
2100:
2096:
2092:
2079:
2075:
2074:simple groups
2071:
2068:
2064:
2063:
2062:
2060:
2056:
2049:are amenable.
2048:
2044:
2041:
2037:
2033:are amenable.
2032:
2028:
2024:
2023:
2021:
2017:
2014:
2010:
2007:
2003:
1999:
1995:
1991:
1987:
1983:
1979:
1972:
1957:
1953:
1950: −
1949:
1945:
1941:
1937:
1934: ∈
1933:
1929:
1922:
1918:
1914:
1910:
1906:
1900:
1896:
1893: =
1891:
1886:
1882:
1878:
1874:
1870:
1866:
1862:
1859:The group of
1858:
1855:
1851:
1847:
1846:Finite groups
1844:
1843:
1834:
1830:
1827:
1823:
1820:
1817:
1813:
1809:
1806:
1803:
1802:
1796:
1794:
1790:
1786:
1781:
1779:
1771:
1767:
1763:
1759:
1756:
1752:
1749:
1745:
1743:
1739:
1729:
1725:
1722:
1718:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1686:
1682:
1678:
1674:
1670:
1666:
1662:
1659:
1655:
1651:
1648:
1644:
1637:
1630:
1623:
1619:
1612:
1609:
1601:
1595: −
1594:
1587:
1583:
1576:
1573:
1564:
1560:
1557: −
1555:
1551:
1547:
1542:
1538:
1534:
1531:
1530:C*-subalgebra
1527:
1524:
1520:
1517:
1513:
1510:on ℓ(Γ) with
1509:
1505:
1502:
1498:
1494:
1490:
1486:
1483:
1482:
1481:
1464:
1459:
1455:
1451:
1446:
1440:
1437:
1433:
1429:
1425:
1421:
1416:
1413:
1410:
1406:
1402:
1396:
1390:
1383:
1382:
1381:
1379:
1375:
1371:
1367:
1363:
1358:
1356:
1337:
1334:
1330:
1325:
1321:
1313:
1312:
1311:
1309:
1305:
1301:
1297:
1293:
1289:
1284:
1282:
1278:
1273:
1271:
1267:
1263:
1259:
1255:
1247:
1243:
1239:
1235:
1231:
1227:
1223:
1219:
1215:
1211:
1207:
1203:
1199:
1196:
1192:
1188:
1185:
1181:
1177:
1176:
1175:
1173:
1169:
1165:
1161:
1157:
1153:
1151:
1138:
1134:
1130:
1126:
1122:
1118:
1114:
1111:
1108:
1104:
1100:
1096:
1092:
1088:
1085:
1082:
1078:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1047:
1044:
1040:
1036:
1032:
1028:
1024:
1020:
1016:
1012:
1008:
1004:
1002:
998:
995:
991:
987:
983:
979:
975:
971:
967:
963:
960:
957:
953:
949:
945:
941:
937:
933:
929:
925:
921:
917:
913:
909:
906:
903:
899:
895:
891:
887:
883:
879:
875:
872:
869:
865:
860:
854:
849:
845:
840:
835:
832:
829:
825:
821:
817:
813:
809:
805:
802:
799:
795:
792:
789:
785:
781:
778:
775:
771:
767:
763:
759:
756:
753:
749:
746:
743:
739:
735:
732:
729:
728:
727:
725:
721:
712:
695:
689:
684:
679:
676:
673:
670:
666:
659:
656:
653:
650:
646:
639:
627:
598:
595:
587:
582:
578:
564:
544:
541:
535:
524:
514:
510:
504:
490:
480:
478:
474:
470:
466:
462:
458:
454:
450:
446:
441:
439:
435:
434:Definition 3.
431:
429:
425:
421:
417:
413:
410:
406:
402:
398:
394:
390:
386:
382:
378:
374:
370:
366:
362:
358:
354:
350:
346:
342:
341:Definition 2.
338:
336:
332:
328:
324:
320:
316:
312:
308:
307:Definition 1.
304:
302:
298:
294:
291:
287:
283:
279:
275:
272:. (This is a
271:
267:
264:
261:
257:
247:
245:
240:
226:
222:
218:
198:
195:
192:
172:
169:
166:
143:
140:
124:
120:
116:
112:
107:
105:
101:
97:
93:
89:
85:
81:
79:
78:free subgroup
75:
70:
68:
64:
60:
56:
52:
48:
44:
40:
36:
33:
30:
26:
22:
2938:
2934:
2909:
2905:
2890:, Springer,
2887:
2872:, Springer,
2869:
2854:, Springer,
2851:
2833:
2827:
2806:
2790:
2773:
2767:
2742:math/0208237
2732:
2728:
2710:
2704:
2670:
2666:
2640:
2634:
2616:
2612:
2603:
2594:
2582:
2576:
2555:
2550:
2520:
2492:
2488:
2471:
2470:
2453:
2441:
2429:
2417:
2405:
2374:
2362:
2350:
2338:
2326:
2314:
2301:
2270:
2263:Valette 1998
2258:
2231:
2212:
2160:V. Oseledets
2147:
2143:
2139:
2135:
2125:
2102:
2088:
2052:
2005:
2001:
1997:
1993:
1989:
1985:
1981:
1974:
1959:
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1924:
1920:
1916:
1912:
1908:
1904:
1898:
1894:
1889:
1884:
1880:
1872:
1864:
1826:unitarizable
1782:
1775:
1772:(A. Connes).
1741:
1733:
1720:
1716:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1676:
1672:
1668:
1664:
1657:
1653:
1646:
1639:
1632:
1625:
1621:
1614:
1607:
1596:
1589:
1585:
1578:
1571:
1562:
1558:
1553:
1549:
1545:
1540:
1536:
1525:
1511:
1507:
1500:
1496:
1492:
1488:
1479:
1377:
1373:
1369:
1365:
1361:
1359:
1352:
1307:
1303:
1299:
1295:
1291:
1287:
1285:
1280:
1276:
1274:
1269:
1265:
1261:
1257:
1253:
1251:
1245:
1241:
1237:
1233:
1229:
1225:
1221:
1217:
1213:
1209:
1205:
1201:
1194:
1190:
1183:
1179:
1171:
1163:
1159:
1155:
1154:
1147:
1136:
1132:
1124:
1120:
1116:
1112:
1106:
1098:
1094:
1086:
1080:
1076:
1072:
1068:
1064:
1060:
1056:
1052:
1048:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
999:
993:
989:
985:
981:
977:
973:
969:
965:
961:
955:
951:
947:
943:
939:
935:
931:
927:
923:
919:
915:
911:
907:
901:
897:
893:
889:
885:
881:
877:
873:
867:
863:
858:
852:
847:
846:such that λ(
843:
838:
833:
827:
823:
819:
815:
811:
807:
803:
797:
793:
787:
783:
779:
757:
751:
747:
737:
733:
730:
723:
718:
585:
583:
579:
488:
486:
476:
472:
468:
464:
456:
452:
448:
444:
442:
437:
433:
432:
427:
423:
419:
415:
411:
408:
404:
400:
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
356:
352:
348:
344:
340:
339:
334:
326:
322:
318:
314:
310:
306:
305:
296:
292:
290:Banach space
285:
277:
270:Haar measure
255:
253:
241:
122:
114:
108:
83:
82:
71:
66:
50:
34:
24:
18:
2941:: 153–158,
2706:Fund. Math.
2558:: 581–598.
2495:: 169–209,
2367:Leptin 1968
2331:Brooks 1981
2085:Nonexamples
1958:(otherwise
1911:), and let
1778:hyperfinite
1770:hyperfinite
1156:Definition.
1091:convolution
720:Pier (1984)
84:Amenability
21:mathematics
2982:Categories
2906:J. Algebra
2799:0621.43001
2735:: 43–169,
2474:PlanetMath
2355:Bowen 2012
2099:Olshanskii
1907: ∈ ℓ(
1799:Properties
1768:) of Γ is
814:satisfies
375:) for all
333:implies Λ(
2967:Terry Tao
2776:: 1–141.
2650:1204.2132
2530:1103.4424
2497:CiteSeerX
2434:Tits 1972
2387:Pier 1984
2316:MathWorld
2283:Pier 1984
2251:Pier 1984
2224:Citations
1964: = x
1789:Laplacian
1683:) =
1460:−
1456:μ
1438:−
1422:μ
1414:∈
1407:∫
1391:ν
1338:μ
1322:∫
992:equals |∫
677:−
667:∑
607:→
565:λ
545:λ
515:∫
499:Λ
263:Hausdorff
170:≥
43:invariant
2829:Topology
2696:(1929),
2593:(1977),
2236:Day 1949
2182:See also
2113:periodic
1946::=range(
1942: ∈
1915: ∈
1903:for all
1861:integers
1840:Examples
1793:L2-space
1703:), then
1671:in ℓ(Γ,
1532:of ℓ(Γ).
1164:amenable
964:For any
461:ba space
438:amenable
117:has the
113:, where
88:analysis
41:that is
2675:Bibcode
2467:Sources
2146:) with
2025:By the
1992:taking
1984:u
1854:compact
1791:on the
1748:nuclear
1168:measure
1089:. Left
483:Example
422:(x) =
363:) if Λ(
337:) ≥ 0.
98:of the
96:support
47:measure
2894:
2876:
2858:
2813:
2797:
2499:
1988:
1268:is in
950:) for
896:) for
557:where
539:
403:(x) =
383:, and
371:) = Λ(
59:German
2931:(PDF)
2737:arXiv
2701:(PDF)
2645:arXiv
2525:arXiv
2378:See:
2274:See:
2204:Notes
2078:Monod
2040:index
1764:(see
1746:) is
1730:(see
1638:| / |
1523:state
1186:is 1.
1127:) is
764:on a
299:) of
276:when
266:group
258:be a
185:. If
74:SO(3)
27:is a
23:, an
2892:ISBN
2874:ISBN
2856:ISBN
2811:ISBN
2152:Tits
2091:free
1760:The
1753:The
1726:The
1715:) =
1667:and
1372:) =
1224:. (
1079:)/m(
1033:)/m(
430:)).
331:a.e.
329:≥ 0
323:mean
254:Let
67:mean
2965:by
2943:doi
2914:doi
2838:doi
2795:Zbl
2778:doi
2747:doi
2715:doi
2683:doi
2655:doi
2641:178
2621:doi
2560:doi
2535:doi
2507:doi
2174:of
1996:to
1966:i+1
1652:If
1290:on
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