Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

(${\tilde{\varphi}}$, ${\tilde{\psi}}$)-AMENABILITY OF L1(G)

  • Received : 2018.12.08
  • Accepted : 2019.01.21
  • Published : 2019.09.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper we introduce and study the concept of of (${\varphi}$, ${\psi}$)-am-enability of a locally compact group G, where ${\varphi}$ is a continuous homomorphism on G and ${\psi}:G{\rightarrow}{\mathbb{C}}$ multiplicative linear function. We prove that if the group algebra $L^1$ (G) is (${\tilde{\varphi}}$, ${\tilde{\psi}}$)-amenable then G is (${\varphi}$, ${\psi}$)-amenable, where ${\tilde{\varphi}}$ is the extension of ${\varphi}$ to M(G). In the case where ${\varphi}$ is an isomorphism on G it is shown that the converse is also valid.

Keywords

References

  1. M. Ashraf and N.Rehman, On (${\sigma}-{\tau}$) derivations in prime rings, Arch. Math. (BRNO) 38 (2002), 259 - 264.
  2. F. F. Bonsall and J. Duncan, Complete Normed Algebra, Springer-Verlag, 1973.
  3. H. G. Dales, Banach algebras and automatic continuity, London Mathematical Society Monographs 24 (Clarendon Press, Oxford), 2000.
  4. Z. Ghorbani and M. Lashkarizadeh Bami, ${\varphi}$- amenable and ${\varphi}$- biflat Banach algebras, Bull. Iranian Math. Soc. 39 (2013), 507-515.
  5. Z. Ghorbani and M. Lashkarizadeh Bami, ${\varphi}$-approximate biflat and ${\varphi}$- amenable Banach algebras, Proc. Ro. Acad. Series A. 13 (2012), 3-10.
  6. A. Ya. Helemskii, Banach and locally convex algebras, Clarendon Press, Oxford University Press, New York, 1993.
  7. A. Ya. Helemskii, Flat Banach modules and amenable algebras, (translated from the Russian). Trans. Moscow Math. Soc. 47 (1985), 199-224.
  8. A. Ya. Helemskii, The Homology of Banach and Topological Algebras, 41 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers Group, Dordrecht, 1989.
  9. B. E. Johnson, Cohomology in Banach algebras, Mem. Amer. Math. Soc. 127 1972.
  10. E. Kaniuth, A. Lau, and J. Pym, On ${\varphi}$-amenability of Banach algebras, Math. Proc. Camb. Phil. Soc. 144 (2008), 85-96. https://doi.org/10.1017/S0305004107000874
  11. J. L. Kelley, General topology, D. Van Nostrand Company, Inc, New Yprk, 1955.
  12. E. Kotzmann and H. Rindler, Segal algebras on non-abelian groups, Trans. Amer. Math. Soc. 237 (1978) 271-281. https://doi.org/10.1090/S0002-9947-1978-0487277-4
  13. M. Mirzavaziri and M.S. Moslehian, ${\sigma}$-derivations in Banach algebras, Bull. Iranian Math. Soc. (2006), 65-78.
  14. M.S. Moslehian and A.N. Motlagh, Some notes on (${\sigma},{\tau}$)-amenability of Banach algebras, Stud. Univ. Babes-Bolyai Math. 53 (2008), 57-68.
  15. H. Reiter, $L^1$-algebras and Segal Algebras, Lecture Notes in Mathematics 231, Springer, Berlin, 1971.
  16. V. Runde, Lectures on Amenability, Lecture Notes in Mathematics 1774, Springer, 2002.