Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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WEAK AMENABILITY OF THE LAU PRODUCT OF BANACH ALGEBRAS DEFINED BY A BANACH ALGEBRA MORPHISM

  • Received : 2016.08.18
  • Accepted : 2017.02.13
  • Published : 2017.11.30
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Abstract

Let A and B be two Banach algebras and $T:B{\rightarrow}A$ be a bounded homomorphism, with ${\parallel}T{\parallel}{\leq}1$. Recently, Dabhi, Jabbari and Haghnejad Azar (Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 9, 1461-1474) obtained some results about the n-weak amenability of $A{\times}_TB$. In the present paper, we address a gap in the proof of these results and extend and improve them by discussing general necessary and sufficient conditions for $A{\times}_TB$ to be n-weakly amenable, for an integer $n{\geq}0$.

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References

  1. G. W. Bade, P. C. Curtis, and H. G. Dales, Amenability and weak amenability for Beurling and Lipschitz algebras, Proc. London Math. Soc. 55 (1987), no. 2, 359-377.
  2. S. J. Bhatt and P. A. Dabhi, Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism, Bull. Aust. Math. Soc. 87 (2013), no. 2, 195-206. https://doi.org/10.1017/S000497271200055X
  3. P. A. Dabhi, A. Jabbari, and K. Haghnejad Azar, Some notes on amenability and weak amenability of Lau product of Banach algebras defined by a Banach algebra morphism, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 9, 1461-1474. https://doi.org/10.1007/s10114-015-4429-8
  4. H. G. Dales, F. Ghahramani, and N. Grnbk, Derivations into iterated duals of Banach algebras, Studia Math. 128 (1998), no. 1, 19-54.
  5. H. Javanshiri and M. Nemati, On a certain product of Banach algebras and some of its properties, Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. sci. 15 (2014), no. 3, 219-227.
  6. B. E. Johnson, Derivations from $L^1$(G) into $L^1$(G) and $L^{\infty}$(G), Harmonic analysis (Luxembourg, 1987), 191198, Lecture Notes in Math., 1359, Springer, Berlin, 1988.
  7. A. R. Khoddami, n-weak amenability of T-Lau product of Banach algebras, Chamchuri J. Math. 5 (2013), 57-65.
  8. A. T.-M. Lau, Analysis on a class of Banach algebras with applications to harmonic analysis on locally compact groups and semigroups, Fund. Math. 118 (1983), no. 3, 161-175. https://doi.org/10.4064/fm-118-3-161-175
  9. M. S. Monfared, On certain products of Banach algebras with applications to harmonic analysis, Studia Math. 178 (2007), no. 3, 277-294. https://doi.org/10.4064/sm178-3-4
  10. M. Nemati and H. Javanshiri, Some homological and cohomological notions on T-Lau product of Banach algebras, Banach J. Math. Anal. 9 (2015), no. 2, 183-195. https://doi.org/10.15352/bjma/09-2-13