Solved and unsolved problems in generalized notions of Connes amenability
الموضوعات :
A. Mahmoodi Kebriya
1
(Department of Mathematics, Faculty of Science, Islamic Azad University,
Central Tehran Branch, Tehran, Iran)
الکلمات المفتاحية: Connes amenability, pseudo-Connes amenability, injectivity, approximate Connes amenability,
ملخص المقالة :
We survey the recent investigations on (bounded, sequential) approximate Connesamenability and pseudo-Connes amenability for dual Banach algebras. We will discuss thecore problems concerning these notions and address the signi cance of any solutions to themto the development of the eld.
[1] W. G. Bade, P. C. Curtis, H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. 55 (1987), 359-377.
[2] H. G. Dales, Banach algebras and automatic continuity, Clarendon Press, Oxford, 2000.
[3] H. G. Dales, A. T.-M. Lau, The second duals of Beurling algebras. Mem. Amer. Math. Soc. 177 (2005), no 836.
[4] H. G. Dales, A. T.-M. Lau, D. Strauss, Banach algebras on semigroups and on their compacti cations. Memoirs of the Amer. Math. Soc. 205 (2010), no 966.
[5] H. G. Dales, R.J. Loy, Y. Zang, Approximate amenability for Banach sequence algebras. Studia Math. 177 (2006), 81 96.
[6] M. Daws, Connes-amenability of bidual and weighted semigroup algebras, Math. Scand. 99 (2006), 217-246.
[7] M. Daws, Dual Banach algebras: representations and injectivity, Studia Math. 178 (2007), 231-275.
[8] G. H. Esslamzadeh, B. Shojaee, A. Mahmoodi, Approximate Connes-amenability of dual Banach algebras, Bull. Belgian Math. Soc. Simon Stevin. 19 (2012), 193-213.
[9] F. Ghahramani, R.J. Loy, Generalized notions of amenability, J. Funct. Anal. 208 (2004), 229-260.
[10] F. Ghahramani, R.J. Loy, Y. Zang, Generalized notions of amenability, II. J. Funct. Anal. 254 (2008), 1776-1810.
[11] F. Ghahramani, Y. Zang, Pseudo-amenable and pseudo-contractible Banach algebras. Math. Proc. Camb. Phil. Soc. 142 (2007), 111-123.
[12] A. Ya. Helemskii, Homological essence of amenability in the sence of A. Connes: the injectivity of the predual bimodule (translated from the Russian), Math. USSR-Sb. 68 (1991) 555-566.
[13] B.E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 (1972).
[14] B.E. Johnson, Approximate diagonals and cohomology of certain annihilator Banach algebras, Amer. J. Math. 94 (1972) 685-698.
[15] B.E. Johnson, R. V. Kadison, J. Ringrose, Cohomology of operator algebras III, Bull. Soc. Math. France 100 (1972) 73-79.
[16] A. Mahmoodi, Connes amenability-like properties, Studia Math. (preprint).
[17] A. Mahmoodi, Bounded approximate Connes amenability of dual Banach algebras, Bull. Iranian Math. Soc. (preprint).
[18] A. Mahmoodi, Approximate injectivity of dual Banach algebras, Bull. Belgian Math. Soc. Simon Stevin. 20 (2013) 1-12.
[19] A. Yu. Pirkovskii, Approximate characterizations of projectivity and injectivity for Banach modules, Math. Proc. Cambridge. Philos. Soc. 143 (2007), 375-385.
[20] Z. J. Ruan, The operator amenability of A(G), Amer. J. Math. 117 (1995), 1449-1476.
[21] V. Runde, Amenability for dual Banach algebras, Studia Math. 148 (2001), 47-66.
[22] V. Runde, Lectures on amenability, Lecture Notes in Mathematics 1774, Springer Verlag, Berlin, 2002.
[23] V. Runde, Connes-amenability and normal, virtual diagonal for measure algebra I, J. London Math. Soc. 67 (2003), 643-656.
[24] V. Runde, Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule, Math. Scand. 95 (2004), 124-144.
