Biamenability of Banach algebras and its applications

Document Type : Original Article

Authors

Faculty of Basic Sciences, University of Bojnord, P.O. Box 1339, Bojnord, Iran

Abstract

In this paper, we introduce the concept of biamenability of Banach algebras and we show that despite the apparent similarities between amenability and biamenability of Banach algebras, they lead to very different, and somewhat opposed, theories. In this regard, we show that commutative Banach algebras and the group algebra $L^1(G)$, for each locally compact group $G$, tend to lack biamenability, while they may be amenable and highly non-commutative Banach algebras such as $B(H)$ for an infinite-dimensional Hilbert space $H$ tend to be biamenable, while they are not amenable. Also, we show that although the unconditional unitization of an amenable Banach algebra is amenable but in general unconditional unitization of a Banach algebra is not biamenable. This concept may be applied for studying the character space of some Banach algebras and also for studying some spansion or density problems.

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References

[1] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc., 2 (1951), pp. 839–848.
[2] , Operations induced in function classes, Monatsh. Math., 55 (1951), pp. 1–19.
[3] D. Benkovicˇ, Biderivations of triangular algebras, Linear Algebra Appl., 431 (2009), pp. 1587–1602.
[4] M. Breˇsar, On generalized biderivations and related maps, J. Algebra, 172 (1995), pp. 764–786.
[5] , Commuting maps: a survey, Taiwanese J. Math., 8 (2004), pp. 361–397.
[6] H. G. Dales, Banach algebras and automatic continuity, vol. 24 of London Mathematical Society Monographs. New Series, The Clarendon Press, Oxford University Press, New York, 2000. Oxford Science Publications.
[7] Y. Du and Y. Wang, Biderivations of generalized matrix algebras, Linear Algebra Appl., 438 (2013), pp. 4483– 4499.
[8] G. J. Murphy, C ∗ -algebras and operator theory, Academic Press, Inc., Boston, MA, 1990.
[9] W. Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-D¨usseldorf-Johannesburg, 1973.
[10] V. Runde, Lectures on amenability, vol. 1774 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002.
[11] Y. Zhang, Weak amenability of module extensions of Banach algebras, Trans. Amer. Math. Soc., 354 (2002), pp. 4131–4151.
AUT Journal of Mathematics and Computing
Volume 4, Issue 2
March 2023
Pages 129-136

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