Amenability properties of vector-valued function algebras

Document Type : Research Paper


School of Mathematics and Computer Sciences, Damghan University, Damghan, P.O.BOX 36715-364, Iran


Let $X$ be a compact Hausdorff space, $A$ be a (commutative) Banach algebra and $\mathcal{A}$ be a Banach $A$-valued function algebra on $X$. Let $\mathfrak{A}$ be the function algebra on $X$, consisting of scalar-valued functions in $\mathcal{A}$. We study and characterize various amenability properties of the algebra $\mathcal{A}$ in terms of cohomological properties of $\mathfrak{A}$ and $A$. Containing some well-known examples, such as $C(X,A)$ and $Lip(X,A)$, the class of vector-valued function algebras also includes, in some sense, the tensor products $\mathfrak{A} \hat \otimes_\gamma A$. As consequences, some known results in this area are covered.


[1] M. Abtahi, Vector-valued characters on vector-valued function algebras, Banach J. Math. Anal. 10 (2016), no. 3, 608–620.
[2] M. Abtahi, On the character space of Banach vector-valued function algebras, Bull. Iran. Math. Soc. 43 (2017), no. 5, 1195–1207.
[3] M. Abtahi and S. Farhangi. Vector-valued spectra of Banach algebra valued continuous functions, Rev. Real Acad. Cienc. Exactas F´ıs. Natur. Ser. A. Mat. 112 (2018), no. 1, 103–115.
[4] M. Cambern and P. Greim. The bidual of C(X, E), Proc. Amer. Math. Soc. 85 (1982), no. 1, 53–58.
[5] H.G. Dales, Banach Algebras and Automatic Continuity, LMS Monographs 24, Clarenden Press, Oxford, 2000.
[6] M. Dashti, R. Nasr-Isfahani, and S. Soltani Renani, Vector-valued invariant means on spaces of bounded linear maps, Colloq. Math. 1 (2013), no. 132, 1–11.
[7] R. Ghamarshoushtari and Y .Zhang, Amenability properties of Banach algebra valued continuous functions, J. Math. Anal. Appl., 422 (2014), 1335–1341.
[8] A. Hausner, Ideal in a certain Banach algebra, Proc. Amer Math. Soc. 8 (1957), 246–249.
[9] Z. Hu, M. S. Monfared, and T. Tranor. On Character amenability of Banach algebras, 344 (2008), no. 2, 942–955.
[10] B.E. Johnson, Cohomology in Banach Algebras, Amer. Math. Soc., 1970.
[11] E. Kaniuth, A. Lau, and J. Pym. On character amenability of Banach algebras, J. Math. Anal. App.344 (2008), 942–955.
[12] E. Kaniuth, A.T. Lau, and J. Pym. On ϕ-amenability of Banach algebras, Math. Proc. Cambridge Philos. Soc. 144 (2008), 85–96.
[13] E. Kaniuth, A Course in Commutative Banach Algebras, Graduate Texts in Mathematics, 246, Springer, 2009.
[14] A.T.-M. Lau, The second conjugate algebra of the Fourier algebra of a locally compact group. Trans. Amer. Math. Soc. 267 (1981), 53–63.
[15] 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.
[16] M.S. Monfared, Character amenability of Banach algebras,. Math. Proc. Cambridge Philos. Soc. 144 (2008), 697–706.
[17] R. Nasr-Isfahani and M. Nemati, Character pseudo-amenability of Banach algebras. Colloq. Math, 132 (2013),177–193.
[18] A. Nikou and A.G. O’Farrell. Banach algebras of vector-valued functions. Glasgow Math. J., 56 (2014), no. 2, 419–426.
[19] M. Ramezanpour, N. Tavallaei, and B. Olfatian Gillan. Character amenability and contractibility of some Banach algebras on left coset spaces. Annal. Func. Anal., 7 (2016), no. 4, 564–572.
[20] V. Runde, Lectures on amenability. Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002.
Volume 14, Issue 3
March 2023
Pages 369-377
  • Receive Date: 06 December 2021
  • Revise Date: 10 March 2022
  • Accept Date: 13 March 2022