Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The amenability constant of the Fourier algebra


Author: Volker Runde
Journal: Proc. Amer. Math. Soc. 134 (2006), 1473-1481
MSC (2000): Primary 46H20; Secondary 20B99, 22D05, 22D10, 43A40, 46J10, 46J40, 46L07, 47L25, 47L50
DOI: https://doi.org/10.1090/S0002-9939-05-08164-5
Published electronically: October 18, 2005
MathSciNet review: 2199195
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a locally compact group $ G$, let $ A(G)$ denote its Fourier algebra and $ \hat{G}$ its dual object, i.e., the collection of equivalence classes of unitary representations of $ G$. We show that the amenability constant of $ A(G)$ is less than or equal to $ \sup \{ \deg(\pi) : \pi \in \hat{G} \}$ and that it is equal to one if and only if $ G$ is abelian.


References [Enhancements On Off] (What's this?)

  • [Con et al.] J. H. CONWAY, Atlas of Finite Groups. Clarendon Press, 1985. MR 0827219 (88g:20025)
  • [E-R] E. G. EFFROS and Z.-J. RUAN, Operator Spaces. Clarendon Press, 2000. MR 1793753 (2002a:46082)
  • [E-S] M. ENOCK and J.-M. SCHWARTZ, Kac Algebras and Duality of Locally Compact Groups (with a preface by A. Connes and a postface by A. Ocneanu). Springer-Verlag, 1992. MR 1215933 (94e:46001)
  • [Eym] P. EYMARD, L'algèbre de Fourier d'un groupe localement compact. Bull. Soc. Math. France 92 (1964), 181-236. MR 0228628 (37:4208)
  • [For et al.] B. E. FORREST, E. KANIUTH, A. T.-M. LAU, and N. SPRONK, Ideals with bounded approximate identities in Fourier algebras. J. Funct. Anal. 203 (2003), 286-304.MR 1996874 (2004e:43002)
  • [F-R] B. E. FORREST and V. RUNDE, Amenability and weak amenability of the Fourier algebra. Math. Z. 250 (2005), 731-744.
  • [F-W] B. E. FORREST and P. J. WOOD, Cohomology and the operator space structure of the Fourier algebra and its second dual. Indiana Univ. Math. J. 50 (2001), 1217-1240. MR 1871354 (2003d:43007)
  • [Haa] U. HAAGERUP, All nuclear $ {C^\ast}$-algebras are amenable. Invent. Math. 74 (1983), 305-319. MR 0723220 (85g:46074)
  • [Hos] B. HOST, Le théorème des idempotents dans $ B(G)$. Bull. Soc. Math. France 114 (1986), 215-223. MR 0860817 (88b:43003)
  • [I-S] M. ILIE and N. SPRONK, Completely bounded homomorphisms of the Fourier algebras. J. Funct. Anal. 225 (2005), 480-499. MR 2152508
  • [Joh1] B. E. JOHNSON, Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972). MR 0374934 (51:11130)
  • [Joh2] B. E. JOHNSON, Approximate diagonals and cohomology of certain annihilator Banach algebras. Amer. J. Math. 94 (1972), 685-698. MR 0317050 (47:5598)
  • [Joh3] B. E. JOHNSON, Non-amenability of the Fourier algebra of a compact group. J. London Math. Soc. (2) 50 (1994), 361-374. MR 1291743 (95i:43001)
  • [L-L-W] A. T.-M. LAU, R. J. LOY, and G. A. WILLIS, Amenability of Banach and $ {C^\ast}$-algebras on locally compact groups. Studia Math. 119 (1996), 161-178. MR 1391474 (97d:46065)
  • [Lep] H. LEPTIN, Sur l'algèbre de Fourier d'un groupe localement compact. C. R. Acad. Sci. Paris, Sér. A 266 (1968), 1180-1182. MR 0239002 (39:362)
  • [Los] V. LOSERT, On tensor products of Fourier algebras. Arch. Math. (Basel) 43 (1984), 370-372. MR 0802314 (87c:43004)
  • [Moo] C. C. MOORE, Groups with finite dimensional irreducible representations. Trans. Amer. Math. Soc. 166 (1972), 401-410. MR 0302817 (46:1960)
  • [Pie] J. P. PIER, Amenable Locally Compact Groups. Wiley-Interscience, 1984.MR 0767264 (86a:43001)
  • [Rua] Z.-J. RUAN, The operator amenability of $ A(G)$. Amer. J. Math. 117 (1995), 1449-1474. MR 1363075 (96m:43001)
  • [Run] V. RUNDE, Lectures on Amenability. Lecture Notes in Mathematics 1774, Springer Verlag, 2002. MR 1874893 (2003h:46001)
  • [Sae] S. SAEKI, On norms of idempotent measures. Proc. Amer. Math. Soc. 19 (1968), 600-602. MR 0225102 (37:697)
  • [Sto] R. STOKKE, Approximate diagonals and Følner conditions for amenable group and semigroup algebras. Studia Math. 164 (2004), 139-159.MR 2079635
  • [Tho] E. THOMA, Eine Charakterisierung diskreter Gruppen vom Typ I. Invent. Math. 6 (1968), 190-196. MR 0248288 (40:1540)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46H20, 20B99, 22D05, 22D10, 43A40, 46J10, 46J40, 46L07, 47L25, 47L50

Retrieve articles in all journals with MSC (2000): 46H20, 20B99, 22D05, 22D10, 43A40, 46J10, 46J40, 46L07, 47L25, 47L50


Additional Information

Volker Runde
Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: vrunde@ualberta.ca

DOI: https://doi.org/10.1090/S0002-9939-05-08164-5
Keywords: Locally compact group, Fourier algebra, amenable Banach algebra, amenability constant, almost abelian group, completely bounded map
Received by editor(s): September 27, 2004
Received by editor(s) in revised form: December 21, 2004
Published electronically: October 18, 2005
Additional Notes: This research was supported by NSERC under grant no. 227043-04
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society