Character Connes amenability of dual Banach algebras

Ramezanpour, Mohammad

Czechoslovak Mathematical Journal, Tome 68 (2018) no. 1, pp. 243-255

Abstract (VO)

We study the notion of character Connes amenability of dual Banach algebras and show that if $A$ is an Arens regular Banach algebra, then $A^{**}$ is character Connes amenable if and only if $A$ is character amenable, which will resolve positively Runde's problem for this concept of amenability. We then characterize character Connes amenability of various dual Banach algebras related to locally compact groups. We also investigate character Connes amenability of Lau product and module extension of Banach algebras. These help us to give examples of dual Banach algebras which are not Connes amenable.

Détail

DOI : 10.21136/CMJ.2018.0451-16

Classification : 22D15, 43A07, 46H20, 46H25
Keywords: dual Banach algebra; Connes amenability; character amenability; locally compact group

Bibliographie

[1] Bonsall, F. F., Duncan, J.: Complete Normed Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete 80, Springer, Berlin (1973). | DOI | MR | JFM

[2] Daws, M.: Connes-amenability of bidual and weighted semigroup algebras. Math. Scand. 99 (2006), 217-246. | DOI | MR | JFM

[3] Effros, E. G.: Amenability and virtual diagonals for von Neumann algebras. J. Funct. Anal. 78 (1988), 137-153. | DOI | MR | JFM

[4] Eymard, P.: L'algèbre de Fourier d'un groupe localement compact. Bull. Soc. Math. Fr. 92 (1964), 181-236 French. | DOI | MR | JFM

[5] Folland, G. B.: A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics, CRC Press, Boca Raton (1995). | MR | JFM

[6] Folland, G. B.: Real Analysis: Modern Techniques and Their Applications. Pure and Applied Mathematics, A Wiley-Interscience Series of Texts, Monographs, and Tracts. Wiley & Sons, New York (1999). | MR | JFM

[7] Hayati, B., Amini, M.: Connes-amenability of multiplier Banach algebras. Kyoto J. Math. 50 (2010), 41-50. | DOI | MR | JFM

[8] Helemskiĭ, A. Ya.: The homological essence of Connes amenability: Injectivity of the predual bimodule. Math. USSR, Sb. 68 (1990), 555-566. English. Russian original translation from Mat. Sb. 180 1989 1680-1690. | DOI | MR | JFM

[9] Hu, Z., Monfared, M. S., Traynor, T.: On character amenable Banach algebras. Stud. Math. 193 (2009), 53-78. | DOI | MR | JFM

[10] Johnson, B. E.: Cohomology in Banach Algebras. Memoirs of the American Mathematical Society 127, American Mathematical Society, Providence (1972). | DOI | MR | JFM

[11] Johnson, B. E., Kadison, R. V., Ringrose, J. R.: Cohomology of operator algebras. III: Reduction to normal cohomology. Bull. Soc. Math. Fr. 100 (1972), 73-96. | DOI | MR | JFM

[12] Kaniuth, E., Lau, A. T., Pym, J.: On character amenability of Banach algebras. J. Math. Anal. Appl. 344 (2008), 942-955. | DOI | MR | JFM

[13] Kaniuth, E., Lau, A. T., Pym, J.: On $\varphi$-amenability of Banach algebras. Math. Proc. Camb. Phil. Soc. 144 (2008), 85-96. | DOI | MR | JFM

[14] Lau, A. T.: Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups. Fundam. Math. 118 (1983), 161-175. | DOI | MR | JFM

[15] Mahmoodi, A.: On $\varphi$-Connes amenability of dual Banach algebras. J. Linear Topol. Algebra 3 (2014), 211-217. | MR

[16] Monfared, M. S.: On certain products of Banach algebras with applications to harmonic analysis. Stud. Math. 178 (2007), 277-294. | DOI | MR | JFM

[17] Monfared, M. S.: Character amenability of Banach algebras. Math. Proc. Camb. Phil. Soc. 144 (2008), 697-706. | DOI | MR | JFM

[18] Nasr-Isfahani, R., Renani, S. S.: Character contractibility of Banach algebras and homological properties of Banach modules. Stud. Math. 202 (2011), 205-225. | DOI | MR | JFM

[19] Runde, V.: Amenability for dual Banach algebras. Stud. Math. 148 (2001), 47-66. | DOI | MR | JFM

[20] Runde, V.: Lectures on Amenability. Lecture Notes in Mathematics 1774, Springer, Berlin (2002). | DOI | MR | JFM

[21] Runde, V.: Connes-amenability and normal, virtual diagonals for measure alebras I. J. Lond. Math. Soc., II. Ser. 67 (2003), 643-656. | DOI | MR | JFM

[22] Runde, V.: Connes-amenability and normal, virtual diagonals for measure algebras II. Bull Aust. Math. Soc. 68 (2003), 325-328. | DOI | MR | JFM

[23] Runde, V.: Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule. Math. Scand. 95 (2004), 124-144. | DOI | MR | JFM

[24] Runde, V., Uygul, F.: Connes-amenability of Fourier-Stieltjes algebras. Bull. Lond. Math. Soc. 47 (2015), 555-564. | DOI | MR | JFM

[25] Zhang, Y.: Weak amenability of module extensions of Banach algebras. Trans. Am. Math. Soc. 354 (2002), 4131-4151. | DOI | MR | JFM