Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-18T21:22:18.334Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximate characterizations of projectivity and injectivity for Banach modules

Published online by Cambridge University Press:  01 September 2007

A. YU. PIRKOVSKII
A. YU. PIRKOVSKII*
Affiliation:
Department of Nonlinear Analysis and Optimization, Faculty of Science, Peoples' Friendship University of Russia, Mikluho-Maklaya 6, 117198 Moscow, Russia. email: pirkosha@sci.pfu.edu.ru, pirkosha@online.ru
*
†Address for correspondence: Krupskoi 8-3-89, 119311 Moscow, Russia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We characterize projective and injective Banach modules in approximate terms, generalizing thereby a characterization of contractible Banach algebras given by F. Ghahramani and R. J. Loy. As a corollary, we show that each uniformly approximately amenable Banach algebra is amenable. Some applications to homological dimensions of Banach modules and algebras are also given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 143 , Issue 2 , September 2007 , pp. 375 - 385
Copyright
Copyright © Cambridge Philosophical Society 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Yu Aristov, O.. On the approximation of flat Banach modules by free Banach modules. Math. Sbornik 196 (11) (2005), (in Russian).CrossRefGoogle Scholar
[2]Dales, H. G.. Banach Algebras and Automatic Continuity (London Mathematical Society Monographs, New Series, 24. Oxford Science Publications (The Clarendon Press, Oxford University Press, 2000).Google Scholar
[3]Dales, H. G., Loy, R. J. and Zhang, Y.. Approximate amenability for Bunach sequence algebras. Studia Math 177 (1) (2006), 81–96.CrossRefGoogle Scholar
[4]Ghahramani, F. and Loy, R. J.. Generalized notions of amenability. J. Funct. Anal. 208 (1) (2004), 229–260.CrossRefGoogle Scholar
[5]Ghahramani, F., Loy, R. J. and Zhang, Y.. Generalized notions of amenability, II. In preparation.Google Scholar
[6]Helemskii, A. Ya.. The Homology of Banach and Topological Algebras (Moscow University Press, 1986 (Russian); English transl.: Kluwer Academic Publishers, 1989).Google Scholar
[7]Helemskii, A. Ya.. Banach and Polynormed Algebras: General Theory, Representations, Homology (Nauka, Moscow, 1989 (Russian); English transl.: Oxford University Press, 1993).Google Scholar
[8]Johnson, B. E.. Cohomology in Banach algebras. Mem. Amer. Math. Soc. 127 (1972).Google Scholar
[9]Johnson, B. E.. Approximate diagonals and cohomology of certain annihilator Banach algebras. Amer. J. Math. 94 (3) (1972), 685–698.CrossRefGoogle Scholar
[10]Pirkovskii, A. Yu.. On certain homological properties of Stein algebras. J. Math. Sci. (New York) 95 (6) (1999), 2690–2702.CrossRefGoogle Scholar
[11]Ramis, J.-P. and Ruget, G.. Complexe dualisant et théorèmes de dualité en géométrie analytique complexe. Publ. Math. I.H.E.S. 38 (2) (1970), 77–91.CrossRefGoogle Scholar
[12]Zhang, Y.. Nilpotent ideals in a class of Banach algebras. Proc. Amer. Math. Soc. 127 (11) (1999), 3237–3242.CrossRefGoogle Scholar