Solved and unsolved problems in generalized notions of Connes amenability

Journal of Linear and Topological Algebra

مقاله 1، دوره 02، شماره 01، خرداد 2013، صفحه 1-7   

نوع مقاله: Research Paper

نویسنده

A. Mahmoodi Kebriya

Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Tehran, Iran

چکیده

We survey the recent investigations on (bounded, sequential) approximate Connes amenability and pseudo-Connes amenability for dual Banach algebras. We will discuss the core problems concerning these notions and address the signicance of any solutions to them to the development of the eld.

کلیدواژه‌ها

Connes amenability؛ pseudo-Connes amenability؛ approximate Connes amenability؛ injectivity

مراجع

[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.