Second cohomology group of group algebras with coefficients in iterated duals
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- by A. Pourabbas
- Proc. Amer. Math. Soc. 132 (2004), 1403-1410
- DOI: https://doi.org/10.1090/S0002-9939-03-07219-8
- Published electronically: August 28, 2003
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Abstract:
In this paper we show that the first cohomology group $\mathcal {H}^1(\ell ^1(G),(\ell ^1(S))^{(n)})$ is zero for every odd $n\in \mathbb {N}$ and for every $G$-set $S$. In the case when $G$ is a discrete group, this is a generalization of the following result of Dales et al.: for any locally compact group $G$, $L^1(G)$ is $(2n+1)$-weakly amenable. Next we show that the second cohomology group $\mathcal {H}^2(\ell ^1(G),(\ell ^1(S))^{(n)})$ is a Banach space. Finally, for every locally compact group $G$ we show that $\mathcal {H}^2(L^1(G),(L^1(G))^{(n)})$ is a Banach space for every odd $n\in \mathbb {N}$.References
- W. G. Bade, P. C. Curtis Jr., and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. (3) 55 (1987), no. 2, 359–377. MR 896225, DOI 10.1093/plms/s3-55_{2}.359
- H. G. Dales, F. Ghahramani, and N. Grønbæk, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1998), no. 1, 19–54. MR 1489459
- M. Despić and F. Ghahramani, Weak amenability of group algebras of locally compact groups, Canad. Math. Bull. 37 (1994), no. 2, 165–167. MR 1275699, DOI 10.4153/CMB-1994-024-4
- Frederick P. Greenleaf, Norm decreasing homomorphisms of group algebras, Pacific J. Math. 15 (1965), 1187–1219. MR 194911
- R. I. Grigorchuk, Some results on bounded cohomology, Combinatorial and geometric group theory (Edinburgh, 1993) London Math. Soc. Lecture Note Ser., vol. 204, Cambridge Univ. Press, Cambridge, 1995, pp. 111–163. MR 1320279
- Niels Grønbæk, Some concepts from group cohomology in the Banach algebra context, Banach algebras ’97 (Blaubeuren), de Gruyter, Berlin, 1998, pp. 205–222. MR 1656607
- A. Ya. Helemskii, The homology of Banach and topological algebras, Mathematics and its Applications (Soviet Series), vol. 41, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by Alan West. MR 1093462, DOI 10.1007/978-94-009-2354-6
- N. V. Ivanov, The second bounded cohomology group, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 167 (1988), no. Issled. Topol. 6, 117–120, 191 (Russian, with English summary); English transl., J. Soviet Math. 52 (1990), no. 1, 2822–2824. MR 964260, DOI 10.1007/BF01099246
- Barry Edward Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, No. 127, American Mathematical Society, Providence, R.I., 1972. MR 0374934
- B. E. Johnson, Derivations from $L^1(G)$ into $L^1(G)$ and $L^\infty (G)$, Harmonic analysis (Luxembourg, 1987) Lecture Notes in Math., vol. 1359, Springer, Berlin, 1988, pp. 191–198. MR 974315, DOI 10.1007/BFb0086599
- B. E. Johnson, Weak amenability of group algebras, Bull. London Math. Soc. 23 (1991), no. 3, 281–284. MR 1123339, DOI 10.1112/blms/23.3.281
- Shigenori Matsumoto and Shigeyuki Morita, Bounded cohomology of certain groups of homeomorphisms, Proc. Amer. Math. Soc. 94 (1985), no. 3, 539–544. MR 787909, DOI 10.1090/S0002-9939-1985-0787909-6
- A. Pourabbas and M. C. White, Second Bounded Group Cohomology of Group Algebras, to appear.
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
- Allan M. Sinclair and Roger R. Smith, Hochschild cohomology of von Neumann algebras, London Mathematical Society Lecture Note Series, vol. 203, Cambridge University Press, Cambridge, 1995. MR 1336825, DOI 10.1017/CBO9780511526190
Bibliographic Information
- A. Pourabbas
- Affiliation: Faculty of Mathematics and Computer Science, Amirkabir University, 424 Hafez Avenue, Tehran 15914, Iran
- Email: arpabbas@aut.ac.ir
- Received by editor(s): January 14, 2002
- Received by editor(s) in revised form: December 31, 2002
- Published electronically: August 28, 2003
- Additional Notes: This research was supported by a grant from Amir Kabir University. The author would like thank the Institute for their kind support.
- Communicated by: Joseph A. Ball
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1403-1410
- MSC (2000): Primary 43A20; Secondary 46M20
- DOI: https://doi.org/10.1090/S0002-9939-03-07219-8
- MathSciNet review: 2053346