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Keywords:
amenability; Banach algebra; inner amenability; locally compact group
Summary:
Let $G$ be a locally compact group. We continue our work [A. Ghaffari: $\Gamma $-amenability of locally compact groups, Acta Math. Sinica, English Series, 26 (2010), 2313–2324] in the study of $\Gamma $-amenability of a locally compact group $G$ defined with respect to a closed subgroup $\Gamma $ of $G\times G$. In this paper, among other things, we introduce and study a closed subspace $A_\Gamma ^p(G)$ of $L^\infty (\Gamma )$ and then characterize the $\Gamma $-amenability of $G$ using $A_\Gamma ^p(G)$. Various necessary and sufficient conditions are found for a locally compact group to possess a $\Gamma $-invariant mean.
References:
[1] Bami, M. L., Mohammadzadeh, B.: Inner amenability of locally compact groups and their algebras. Stud. Sci. Math. Hung. 44 (2007), 265-274. MR 2325523 | Zbl 1174.43001
[2] Bratteli, O., Robinson, D. W.: Operator Algebras and Quantum Statistical Mechanics I. Springer New York-Heidelberg-Berlin (1979). MR 0611508
[3] Dales, H. G.: Banach Algebras and Automatic Continuity. London Math. Soc. Monographs. Clarendon Press Oxford (2000). MR 1816726
[4] Dunford, N., Schwartz, J. T.: Linear Operators. Part I. Interscience New York (1958).
[5] Edwards, R. E.: Functional Analysis. Holt, Rinehart and Winston New York (1965). MR 0221256 | Zbl 0182.16101
[6] Effros, E. G.: Property $\Gamma$ and inner amenability. Proc. Am. Math. Soc. 47 (1975), 483-486. MR 0355626 | Zbl 0321.22011
[7] Eymard, P.: L'algebre de Fourier d'un groupe localement compact. Bull. Soc. Math. Fr. 92 (1964), 181-236 French. MR 0228628 | Zbl 0169.46403
[8] Folland, G. B.: A Course in Abstract Harmonic Analysis. CRC Press Boca Raton (1995). MR 1397028 | Zbl 0857.43001
[9] Ghaffari, A.: $\Gamma$-amenability of locally compact groups. Acta Math. Sin., Engl. Ser. 26 (2010), 2313-2324. DOI 10.1007/s10114-010-9498-0 | MR 2737302 | Zbl 1219.43001
[10] Ghaffari, A.: Structural properties of inner amenable discrete groups. Bull. Iran. Math. Soc. 33 (2007), 25-35. MR 2338797 | Zbl 1155.22006
[11] Hewitt, E., Ross, K. A.: Abstract Harmonic Analysis. Springer Berlin (1963) Zbl 0213.40103
[12] Herz, C.: Harmonic synthesis for subgroups. Ann. Inst. Fourier (Grenoble) 23 (1973), 91-123. DOI 10.5802/aif.473 | MR 0355482 | Zbl 0257.43007
[13] Lau, A. T.-M.: Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups. Fundam. Math. 118 (1983), 161-175. MR 0736276 | Zbl 0545.46051
[14] Lau, A. T.-M., Paterson, A. L. T.: Inner amenable locally compact groups. Trans. Am. Math. Soc. 325 (1991), 155-169. DOI 10.1090/S0002-9947-1991-1010885-5 | MR 1010885 | Zbl 0718.43002
[15] Lau, A. T.-M., Paterson, A. L. T.: Operator theoretic characterizations of [IN]-groups and inner amenability. Proc. Am. Math. Soc. 102 (1988), 893-897. MR 0934862 | Zbl 0644.43002
[16] Li, B., Pier, J.-P.: Amenability with respect to a closed subgroup of a product group. Adv. Math. (Beijing) 21 (1992), 97-112. MR 1153929 | Zbl 0763.43002
[17] Memarbashi, R., Riazi, A.: Topological inner invariant means. Stud. Sci. Math. Hung. 40 (2003), 293-299. MR 2036960 | Zbl 1047.43001
[18] Paterson, A. L. T.: Amenability. Math. Survey and Monographs Vol. 29. Am. Math. Soc. Providence (1988). MR 0961261
[19] Pier, J.-P.: Amenable Banach Algebras. Pitman Research Notes in Mathematics Series, Vol. 172. Longman Scientific & Technical/John Wiley & Sons Harlow/New York (1988). MR 0942218
[20] Pier, J.-P.: Amenable Locally Compact Groups. John Wiley & Sons New York (1984). MR 0767264 | Zbl 0621.43001
[21] Rudin, W.: Functional Analysis, 2nd ed. McGraw Hill New York (1991). MR 1157815 | Zbl 0867.46001
[22] Stokke, R.: Quasi-central bounded approximate identities in group algebras of locally compact groups. Ill. J. Math. 48 (2004), 151-170. MR 2048220 | Zbl 1037.43004
[23] Yuan, C. K.: Conjugate convolutions and inner invariant means. J. Math. Anal. Appl. 157 (1991), 166-178. DOI 10.1016/0022-247X(91)90142-M | MR 1109449 | Zbl 0744.43004
[24] Yuan, C. K.: Structural properties of inner amenable groups. Acta Math. Sin., New Ser. 8 (1992), 236-242. DOI 10.1007/BF02582912 | MR 1192624 | Zbl 0766.43003
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