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Arens regularity of module actions and weak amenability of Banach algebras. (English) Zbl 1363.46036

The authors give some results concerning criteria of Arens regularity of Banach algebras in the sense of R. Arens [Proc. Am. Math. Soc. 2, 839–848 (1951; Zbl 0044.32601)]. A Banach algebra \(A\) is said to be weakly amenable if every derivation from \(A\) into its dual \(A^\ast\) is inner. Similarly, the notion of \(n\)-weakly amenable Banach algebras is introduced, following the paper of Y. Zhang [Trans. Am. Math. Soc. 354, No. 10, 4131–4151 (2002; Zbl 1008.46019)]. The authors also study \(n\)-weak amenability of module extensions of Banach algebras and weak\(^\ast\)-continuous derivations on Banach algebras.

MSC:

46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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[1] R. Arens, The adjoint of a bilinear operation. Proc. Am. Math. Soc. 2, 839-848 (1951) · Zbl 0044.32601 · doi:10.1090/S0002-9939-1951-0045941-1
[2] W. Bade, P. Curtis, H.G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. Lond. Math. Soc. 50, 359-377 (1987) · Zbl 0634.46042 · doi:10.1093/plms/s3-55_2.359
[3] J. Baker, A.T.-M. Lau, J. Pym, Module homomorphism and topological centers associated with weakly sequentially compact Banach algebras. J. Funct. Anal. 158, 186-208 (1998) · Zbl 0911.46030 · doi:10.1006/jfan.1998.3280
[4] S. Barootkoob, H.R. Ebrahimi Vishki, Lifting derivations and \[n\] n-weak amenability of the second dual of Banach algebras. Bull. Aust. Math. Soc. 83, 122-129 (2011) · Zbl 1270.46042 · doi:10.1017/S0004972710001838
[5] H.G. Dales, Banach Algebras and Automatic Continuity, vol. 24, London Math Society Monographs (Clarendon Press, Oxford, 2000) · Zbl 0981.46043
[6] H.G. Dales, F. Ghahramani, N. Grønbæk, Derivation into iterated duals of Banach algebras. Studia Math. 128(1), 19-53 (1998) · Zbl 0903.46045
[7] H.G. Dales, A.T.-M. Lau, The second duals of Beurling algebras. Mem. Am. Math. Soc. 177, 836 (2005) · Zbl 1075.43003
[8] H.G. Dales, A. Rodrigues-Palacios, M. Velasco, The second transpose of a derivation. J. Lond. Math. Soc. 64(2), 707-721 (2001) · Zbl 1023.46051 · doi:10.1112/S0024610701002496
[9] M. Eshaghi Gordji, A. Ebadian, F. Habibian, B. Hayati, Weak \[^*\]∗-continuous derivations in dual Banach algebras. Arch. Math. (Brno) Tomus 48, 39-44 (2012) · Zbl 1274.46098 · doi:10.5817/AM2012-1-39
[10] M. Eshaghi Gordji, M. Filali, Arens regularity of module actions. Studia Math. 181(3), 237-254 (2007) · Zbl 1165.46024 · doi:10.4064/sm181-3-3
[11] M. Eshaghi Gordji, F. Habibian, A. Rejali, Module extension of dual Banach algebras. Bull. Korean Math. Soc. 47(4), 663-673 (2010) · Zbl 1204.46031 · doi:10.4134/BKMS.2010.47.4.663
[12] H. Farhadi, F. Ghahramani, Involutions on the second duals of group algebras and a multiplier problem. Proc. Edinb. Math. Soc. 50, 153-161 (2007) · Zbl 1121.43001 · doi:10.1017/S0013091505000660
[13] B.E. Forrest, L.W. Marcoux, Derivations of triangular Banach algebras. Indiana Univ. Math. J. 45, 441-462 (1996) · Zbl 0890.46035 · doi:10.1512/iumj.1996.45.1147
[14] B.E. Forrest, L.W. Marcoux, Weak amenability of triangular Banach algebras. Trans. Am. Math. Soc. 354, 1435-1452 (2001) · Zbl 1014.46017 · doi:10.1090/S0002-9947-01-02957-9
[15] B.E. Johoson, Weak amenability of group algebras. Bull. Lond. Math. Soc. 23, 281-284 (1991) · Zbl 0757.43002 · doi:10.1112/blms/23.3.281
[16] A.T.-M. Lau, Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups. Math. Proc. Camb. Philos. Soc. 99(2), 273-283 (1986) · Zbl 0591.43003 · doi:10.1017/S0305004100064197
[17] A.T.-M. Lau, V. Losert, On the second conjugate algebra of locally compact groups. J. Lond. Math. Soc. 37(2), 464-480 (1988) · Zbl 0608.43002 · doi:10.1112/jlms/s2-37.3.464
[18] A.T.-M. Lau, A. Ülger, Topological center of certain dual algebras. Trans. Am. Math. Soc. 348, 1191-1212 (1996) · Zbl 0859.43001 · doi:10.1090/S0002-9947-96-01499-7
[19] A.R. Medghalchi, M.H. Sattari, T. Yazdanpanah, Amenability and weak amenability of triangular Banach algebras. Bull. Iran. Math. Soc. 31(2), 57-69 (2005) · Zbl 1120.46030
[20] S. Mohamadzadih, H.M. Vishki, Arens regularity of module actions and the second adjoint of a derivation. Bull. Aust. Math. Soc. 77, 465-476 (2008) · Zbl 1159.46027
[21] M.S. Monfared, On certain products of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups. Studia Math. 178(3), 277-294 (2007) · Zbl 1121.46041 · doi:10.4064/sm178-3-4
[22] V. Runde, Lectures on the Amenability (Springer, Berlin, 2002) · Zbl 0999.46022 · doi:10.1007/b82937
[23] Y. Zhang, Weak amenability of module extentions of Banach algebras. Trans. Am. Math. Soc. 354(10), 4131-4151 (2002) · Zbl 1008.46019 · doi:10.1090/S0002-9947-02-03039-8
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