Weak amenability of Beurling algebra free products

Subject Areas : Statistics

Elham  Gheisari ¹ (Department of Mathematics‎, ‎Faculty of Science‎, Central Tehran Branch‎, ‎Islamic Azad University‎,)
Akram  Yousofzadeh ² (Department of Mathematics, Faculty of science, Mobarakeh Branch, Islamic Azad University, Mobarakeh, Isfahan, Iran)
Mohammad Sadegh  Asagri ³ (Department of Mathematics‎, ‎Faculty of Science‎, ‎Central Tehran Branch‎, ‎Islamic Azad University‎, Tehran‎, ‎Iran.)

Received: 2019-05-28 Accepted : 2020-02-27 Published : 2021-02-19

Keywords: وزن, میانگین‌پذیرضعیف, حاصلضرب آزاد, مشتق, گروه فشرده موضعی,

Abstract :

‎In this paper, for a discrete group ‎$G=mathbb{Z}‎astmathbb{Z}_n‎‎‎$‎ and a weight function of polynomial‎$‎‎‎omega_‎alpha‎‎$‎,‎ we show that the Burling algebra ‎$‎ell^1(G‎‎, ‎‎‎omega_‎alpha)‎$‎ is not weakly amenable ‎and ‎dihedral group‎ ‎$D_‎infty=mathbb{Z}_2astmathbb{Z}_2‎‎‎$ ‎is‎ amenable. We also show that for a continuous weight function ‎$‎‎‎ ‎‎omega‎$ ‎under certain conditions ‎on group ‎$‎‎‎‎‎ ‎G‎$‎, if the Burling algebra $‎ell^1(G‎‎, ‎‎‎omega‎)‎$‎ is weakly amenable‎ then ‎$‎‎omega‎‎$‎ is bounded.‎In this paper, for a discrete group ‎$G=mathbb{Z}‎astmathbb{Z}_n‎‎‎$‎ and a weight function of polynomial‎$‎‎‎omega_‎alpha‎‎$‎,‎ we show that the Burling algebra ‎$‎ell^1(G‎‎, ‎‎‎omega_‎alpha)‎$‎ is not weakly amenable ‎and ‎dihedral group‎ ‎$D_‎infty=mathbb{Z}_2astmathbb{Z}_2‎‎‎$ ‎is‎ amenable. We also show that for a continuous weight function ‎$‎‎‎ ‎‎omega‎$ ‎under certain conditions ‎on group ‎$‎‎‎‎‎ ‎G‎$‎, if the Burling algebra $‎ell^1(G‎‎, ‎‎‎omega‎)‎$‎ is weakly amenable‎ then ‎$‎‎omega‎‎$‎ is bounded.‎In this paper, for a discrete group ‎$G=mathbb{Z}‎astmathbb{Z}_n‎‎‎$‎ and a weight function of polynomial‎$‎‎‎omega_‎alpha‎‎$‎,‎ we show that the Burling algebra ‎$‎ell^1(G‎‎, ‎‎‎omega_‎alpha)‎$‎ is not weakly amenable ‎and ‎dihedral group‎ ‎$D_‎infty=mathbb{Z}_2astmathbb{Z}_2‎‎‎$ ‎is‎ amenable. We also show that for a continuous weight function ‎$‎‎‎ ‎‎omega‎$ ‎under certain conditions ‎on group ‎$‎‎‎‎‎ ‎G‎$‎, if the Burling algebra $‎ell^1(G‎‎, ‎‎‎omega‎)‎$‎ is weakly amenable‎ then ‎$‎‎omega‎‎$‎ is bounded.

References:

[1] W. G. Bade, P. C. Curtis Jr, H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987) 359-377.

 

[2] C. R. Borwick, The Johnson-Hochschild cohomology of weighted group algebras and augmentation ideals, PhD thesis, Univ. of Newcastle upon Tyne, 2003.

 

[3] N. Grønbæk, A characterization of weakly amenable Banach algebras, Studia Math. 94 (2) (1989) 149-162.

 

[4] N. Grønbæk, Amenability of weighted convolution algebras on locally compact groups, Trans. Amer. Math. Soc. 319 (2) (1990) 765-775.

 

[5] W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Presentations of groups in terms of generators and relations. Dover publications, New York, 1976.

 

[6] A. Pourabbas, The cohomology of some Banach algebras, PhD thesis, Univ. Of Newcastle upon Tyne, 1996.

 

[7] A. Pourabbas, Weak amenability of weighted group algebras, Atti sem. Mat. Fis. Univ. Modena 48 (2000), 299-316.

 

[8] H-R. Rahimi, A. Soltani, A note on approximately amenable modulo an ideal of Banach algebras, J. New Resea. Math., Vol. 1, No. 2, Summer 2015 5-12.

 

[9] E. Samei, Weak amenability and 2-weak amenability of Beurling algebras, J. Math. Anal. Appl. 346 (2) (2008) 451-467.

 

[10] V. Shepelska, Weak amenability of weighted group algebras on some discrete groups, Studia. Math. 230 (3) (2015) 189-214.

 

[11] Y. Zhang, Weak amenability of commutative Beurling algebras, Proc. Amer. Math. Soc. 142 (2014), 1649-1661.

 

[12] جواد سعادتمندان، علیرضا باقری ثالث، بررسی ارتباط بین وجود بردارهای پذیرفتنی با میانگین­پذیری و فشردگی یک گروه موضعاً فشرده، مجله پژوهش­های نوین در ریاضی، سال چهارم، شماره چهاردهم، تابستان 1397، 103-112.

 

[13] مرتضی قربانی، حمیدرضا رحیمی، دو تصویری بودن جبرهای باناخ نسبت به ایده­آل، مجله پژوهش­های نوین در ریاضی، سال پنجم، شماره هجدهم، تابستان 1398، 21-30.