Ideal Connes-amenability of Lau product of Banach algebras

A. Minapoor ; A. Bodaghi ; O. T. Mewomo

Eurasian mathematical journal, Tome 12 (2021) no. 4, pp. 74-81.

Résumé

Let $\mathcal{A}$ and $\mathcal{B}$ be Banach algebras and $\theta$ be a non-zero character on $\mathcal{B}$. In the current paper, we study the ideal Connes-amenability of the algebra $\mathcal{A}\times_\theta\mathcal{B}$ so-called the $\tau$-Lau product algebra. We also prove that if $\mathcal{A}\times_\theta\mathcal{B}$ is ideally Connes-amenable, then both $\mathcal{A}$ and $\mathcal{B}$ are ideally Connes-amenable. As a result, we show that $l^1(S)\times_\theta l^1(S)$ is ideally Connes-amenable, where $S$ is a Rees matrix semigroup.

Bibliographie

[1] H. G. Dales, Banach algebras, automatic continuity, London Mathematical Society Monographs, New Series, 24, The Clarendon Press, Oxford, 2000

[2] H. G. Dales, A. T.-M. Lau, D. Strauss, “Banach algebras on semigroups and their compacti cations”, Memoirs Amer. Math. Soc., 205 (2010), 1–165

[3] M. E. Gordji, A. Jabbari, “Approximate ideal amenability of Banach algebras”, U. P.B. Sci. Bull. Ser. A, 74:2 (2012), 57–64

[4] M. E. Gorgi, T. Yazdanpanah, “Derivations into duals of ideals of Banach algebras”, Proc. Indian Acad. Sci. (Math. Sci.), 114:4 (2004), 399–408

[5] A. Ya. Helemskii, “Homological essence of amenability in the sense of A. Connes: the injectivity of the predual bimodule”, Math. USSR. Sb., 68 (1991), 555–566

[6] H. Hosseinzadeh, A. Jabbari, “Permanently weak amenability of Rees semigroup algebras”, Int. J. Anal. Appl., 16 (2018), 117–124

[7] J. M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs, 12, The Clarendon Press, Oxford, 1995

[8] B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc., 127, 1972

[9] B. E. Johnson, R. V. Kadison, J. R. Ringrose, “Cohomology of operator algebras III”, Bull. Soc. Math. France, 100 (1972), 73–96

[10] A. T.-M. Lau, “Analysis on a class of Banach algebras with application to harmonic analysis on locally compact groups and semigroups”, Fund. Math., 118 (1983), 161–175

[11] O. T. Mewomo, “On approximate ideal amenability in Banach algebras”, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (S.N.), 56:1 (2010), 199–208

[12] O. T. Mewomo, “On ideal amenability of Banach algebras”, An. Stiinf. Univ. Al. I. Cuzaiasi. Mat. (S.N.), 56:2 (2010), 273–278

[13] O. T. Mewomo, “Various notions of amenability in Banach algebras”, Expo. Math., 29 (2011), 283–299

[14] A. Minapoor, “Approximate ideal Connes amenability of dual Banach algebras and ideal Connes amenability of discrete Beurling algebras”, J. Eurasian Math., 11:2 (2020), 72–85

[15] A. Minapoor, “Ideal Connes amenability of $l^1$-Munn algebras and its application to semigroup algebras”, Semigroup Forum, 102 (2021), 756–764

[16] A. Minapoor, A. Bodaghi, D. Ebrahimi Bagha, “Ideal Connes-amenability of dual Banach Algebras”, Mediterr. J. Math., 14 (2017), 174 | DOI

[17] M. Ramezanpour, “Weak amenability of the Lau product of Banach algebras de ned by a Banach algebra morphism”, Bull. Korean Math. Soc., 54 (2017), 1991–1999

[18] N. Razi, A. Pourabbas, “Some homological properties of Lau product algebras”, Iran J. Sci. Technol. Trans. Sci., 43 (2019), 1671–1678

[19] V. Runde, “A Connes-amenable, dual Banach algebra need not have a normal, virtual diagonal”, Trans. Amer. Math. Soc., 358:1 (2005), 391–402

[20] V. Runde, “Amenability for dual Banach algebras”, Studia Math., 148 (2001), 47–66

[21] V. Runde, “Connes-amenability and normal, virtual diagonals for measure algebras (II)”, Bull. Austral. Math. Soc., 68 (2003), 325–328

[22] M. Sangani Monfared, “On certain product of Banach algebras with application to harmonic analysis”, Studia Math., 178:3 (2007), 277–294

[23] J. Von Neumann, “Zur allgemeinen theories des maßes”, Fund. Math., 13 (1929), 73–116