A characterization for contractible Banach algebras

Document Type : Original Paper


Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan, Iran


In this short note, the following new characterization of contractibility is stated and proved: A Banach algebra is contractible if and only if for any pair of Banach bimodules over that algebra, the closed linear subspace of all continuous bimodule morphisms between them, in the space of all bounded linear operators between them, is naturally topological complemented. Here, the phrase ``natural'' has been used with its meaning in Category Theory.


Main Subjects

 B.E. Johnson, Cohomology in Banach algebras, Memoirs of the American Mathematical Society, vol. 127, 1972.
 O.T. Mewomo, Various notions of amenability in Banach algebras, Expositiones Mathematicae 29 (2011) 283–299.
 V. Runde, Lectures on amenability, Springer-Verlag, Berlin, Heidelberg, 2002.
 V. Runde, Amenable Banach algebras, Springer Monographs in Mathematics, Berlin, 2020.