| Title:
|
Some algebraic and homological properties of Lipschitz algebras and their second duals (English) |
| Author:
|
Abtahi, F. |
| Author:
|
Byabani, E. |
| Author:
|
Rejali, A. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
55 |
| Issue:
|
4 |
| Year:
|
2019 |
| Pages:
|
211-224 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $(X,d)$ be a metric space and $\alpha >0$. We study homological properties and different types of amenability of Lipschitz algebras $\operatorname{Lip}_\alpha X$ and their second duals. Precisely, we first provide some basic properties of Lipschitz algebras, which are important for metric geometry to know how metric properties are reflected in simple properties of Lipschitz functions. Then we show that all of these properties are equivalent to either uniform discreteness or finiteness of $X$. Finally, some results concerning the character space and Arens regularity of Lipschitz algebras are provided. (English) |
| Keyword:
|
amenability |
| Keyword:
|
Arens regularity |
| Keyword:
|
biprojectivity |
| Keyword:
|
biflatness |
| Keyword:
|
Lipschitz algebra |
| Keyword:
|
metric space |
| MSC:
|
11J83 |
| MSC:
|
46H05 |
| MSC:
|
46J10 |
| idZBL:
|
Zbl 07144736 |
| idMR:
|
MR40383556 |
| DOI:
|
10.5817/AM2019-4-211 |
| . |
| Date available:
|
2019-10-30T08:51:18Z |
| Last updated:
|
2020-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147874 |
| . |
| Reference:
|
[1] Abtahi, F., Azizi, M., Rejali, A.: Character amenability of some intersections of Lipschitz algebras.Canad. Math. Bull. 60 (4) (2017), 673–689. MR 3710653, 10.4153/CMB-2017-039-8 |
| Reference:
|
[2] Alaghmandan, M., Nasr Isfahani, R., Nemati, M.: On $\phi -$contractibility of the Lebesgue-Fourier algebra of a locally compact group.Arch. Math. (Basel) 95 (2010), 373–379. MR 2727314, 10.1007/s00013-010-0177-2 |
| Reference:
|
[3] Bade, W.G., Curtis, Jr., P.C., Dales, H.G.: Amenability and weak amenability for Beurling and Lipschitz algebras.Proc. London Math. Soc. 55 (2) (1987), 359–377. Zbl 0634.46042, MR 0896225 |
| Reference:
|
[4] Dales, H.G.: Banach algebras and automatic continuity.London Math. Soc. Mono-graphs, vol. 24, Clarendon Press, Oxford, 2000. Zbl 0981.46043, MR 1816726 |
| Reference:
|
[5] Dashti, M., Nasr Isfahani, R., Soltani Renani, S.: Character amenability of Lipschitz algebras.Canad. Math. Bull. 57 (1) (2014), 37–41. MR 3150714, 10.4153/CMB-2012-015-3 |
| Reference:
|
[6] Ghahramani, F., Zhang, Y.: Pseudo-amenable and pseudo-contractible Banach algebras.Math. Proc. Cambridge Philos. Soc. 142 (1) (2007), 111–123. MR 2296395, 10.1017/S0305004106009649 |
| Reference:
|
[7] Gourdeau, F.: Amenability of Banach algebras.Math. Proc. Cambridge Philos. Soc. 105 (2) (1989), 351–355. MR 0974991, 10.1017/S0305004100067840 |
| Reference:
|
[8] Helemskii, A.Ya.: The homology of Banach and topological algebras.Kluwer Academic Publishers Group, Dordrecht, 1989. MR 1093462 |
| Reference:
|
[9] Hu, Z., Monfared, M.S., Traynor, T.: On character amenable Banach algebras.Studia Math. 193 (1) (2009), 53–78. MR 2506414, 10.4064/sm193-1-3 |
| Reference:
|
[10] Johnson, B.E.: Cohomology in Banach algebras.Mem. Amer. Math. Soc., vol. 127, 1972, pp. iii+96 pp. Zbl 0256.18014, MR 0374934 |
| Reference:
|
[11] Johnson, J.A.: Banach spaces of Lipschitz functions and vector-valued Lipschitz functions.Trans. Amer. Math. Soc. 148 (1970), 147–169. MR 0415289, 10.1090/S0002-9947-1970-0415289-8 |
| Reference:
|
[12] Kaniuth, E.: A course in commutative Banach algebras.Graduate Texts in Mathematics, Springer, New York, 2009. MR 2458901 |
| Reference:
|
[13] Kaniuth, E., Lau, A.T., Pym, J.: On $\varphi -$amenability of Banach algebras.Math. Proc. Cambridge Philos. Soc. 144 (1) (2008), 85–96. MR 2388235, 10.1017/S0305004107000874 |
| Reference:
|
[14] Loomis, L.H.: An introduction to abstract harmonic analysi.D. Van Nostrand Company, Inc., Toronto-New York-London, 1953. MR 0054173 |
| Reference:
|
[15] Monfared, M.S.: Character amenability of Banach algebras.Math. Proc. Cambridge Philos. Soc. 144 (3) (2008), 697–706. MR 2418712, 10.1017/S0305004108001126 |
| Reference:
|
[16] Runde, V.: Lectures on amenability.Lecture Notes in Mathematics, vol. 1774, Springer-Verlag, Berlin, 2002. Zbl 0999.46022, MR 1874893 |
| Reference:
|
[17] Samei, E., Spronk, N., Stokke, R.: Biflatness and pseudo-amenability of Segal algebras.Canad. J. Math. 62 4) (2010), 845–869. MR 2674704, 10.4153/CJM-2010-044-4 |
| Reference:
|
[18] Sherbert, D.R.: Banach algebras of Lipschitz functions.Pacific J. Math. 13 (1963), 1387–1399. Zbl 0121.10203, MR 0156214, 10.2140/pjm.1963.13.1387 |
| Reference:
|
[19] Sherbert, D.R.: The structure of ideals and point derivations in Banach algebras of Lipschitz functions.Trans. Amer. Math. Soc. 111 (1964), 240–272. Zbl 0121.10204, MR 0161177, 10.1090/S0002-9947-1964-0161177-1 |
| Reference:
|
[20] Zhang, Y.: Weak amenability of a class of Banach algebras.Canad. Math. Bull. 44 (4) (2001), 504–508. Zbl 1156.46306, MR 1863642, 10.4153/CMB-2001-050-7 |
| . |
