Abstract
Let G be a locally compact group. We continue our work [A. Ghaffari: Γ-amenability of locally compact groups, Acta Math. Sinica, English Series, 26 (2010), 2313–2324] in the study of Γ-amenability of a locally compact group G defined with respect to a closed subgroup Γ of G × G. In this paper, among other things, we introduce and study a closed subspace A pΓ (G) of L ∞(Γ) and then characterize the Γ-amenability of G using A pΓ (G). Various necessary and sufficient conditions are found for a locally compact group to possess a Γ-invariant mean.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. L. Bami, B. Mohammadzadeh: Inner amenability of locally compact groups and their algebras. Stud. Sci. Math. Hung. 44 (2007), 265–274.
O. Bratteli, D.W. Robinson: Operator Algebras and Quantum Statistical Mechanics I. Springer, New York-Heidelberg-Berlin, 1979.
H.G. Dales: Banach Algebras and Automatic Continuity. London Math. Soc. Monographs. Clarendon Press, Oxford, 2000.
N. Dunford, J.T. Schwartz: Linear Operators. Part I. Interscience, New York, 1958.
R. E. Edwards: Functional Analysis. Holt, Rinehart and Winston, New York, 1965.
E.G. Effros: Property Γ and inner amenability. Proc. Am. Math. Soc. 47 (1975), 483–486.
P. Eymard: L’algebre de Fourier d’un groupe localement compact. Bull. Soc. Math. Fr. 92 (1964), 181–236. (In French.)
G.B. Folland: A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton, 1995.
A. Ghaffari: Γ-amenability of locally compact groups. Acta Math. Sin., Engl. Ser. 26 (2010), 2313–2324.
A. Ghaffari: Structural properties of inner amenable discrete groups. Bull. Iran. Math. Soc. 33 (2007), 25–35.
E. Hewitt, K.A. Ross: Abstract Harmonic Analysis. Vol. I. Springer, Berlin, 1963; Vol. II. Springer, Berlin, 1970.
C. Herz: Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23 (1973), 91–123.
A.T.-M. Lau: Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups. Fundam.Math. 118 (1983), 161–175.
A.T.-M. Lau, A. L.T. Paterson: Inner amenable locally compact groups. Trans. Am. Math. Soc. 325 (1991), 155–169.
A.T.-M. Lau, A. L.T. Paterson: Operator theoretic characterizations of [IN]-groups and inner amenability. Proc. Am. Math. Soc. 102 (1988), 893–897.
B. Li, J.-P. Pier: Amenability with respect to a closed subgroup of a product group. Adv. Math. (Beijing) 21 (1992), 97–112.
R. Memarbashi, A. Riazi: Topological inner invariant means. Stud. Sci. Math. Hung. 40 (2003), 293–299.
A. L.T. Paterson: Amenability. Math. Survey and Monographs Vol. 29. Am. Math. Soc., Providence, 1988.
J.-P. Pier: Amenable Banach Algebras. Pitman Research Notes in Mathematics Series, Vol. 172. Longman Scientific & Technical/John Wiley & Sons, Harlow/New York, 1988.
J.-P. Pier: Amenable Locally Compact Groups. John Wiley & Sons, New York, 1984.
W. Rudin: Functional Analysis, 2nd ed. McGraw Hill, New York, 1991.
R. Stokke: Quasi-central bounded approximate identities in group algebras of locally compact groups. Ill. J. Math. 48 (2004), 151–170.
C.K. Yuan: Conjugate convolutions and inner invariant means. J. Math. Anal. Appl. 157 (1991), 166–178.
C.K. Yuan: Structural properties of inner amenable groups. Acta Math. Sin., New Ser. 8 (1992), 236–242.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ghaffari, A. A generalization of amenability and inner amenability of groups. Czech Math J 62, 729–742 (2012). https://doi.org/10.1007/s10587-012-0043-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10587-012-0043-4