A Banach algebra 
is called contractible if 
for all Banach 
-bimodules 
(Helemskii 1989, 1997).
A 
-algebra
is contractible if and only if it is finite-dimensional (Selivanov 1976).
Contractible Banach Algebra
This entry contributed by Mohammad Sal Moslehian
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References
Helemskii, A. Ya. The Homology of Banach and Topological Algebras. Dordrecht, Netherlands: Kluwer, 1989.Helemskii, A. Ya. "The Homology in Algebra of Analysis." In Handbook of Algebra, Vol. 2. Amsterdam, Netherlands: Elsevier, 1997.Selivanov, Yu. V. "Banach Algebras of Small Global Dimension Zero." Uspekhi Mat. Nauk 31, 227-228, 1976.Referenced on Wolfram|Alpha
Contractible Banach AlgebraCite this as:
Moslehian, Mohammad Sal. "Contractible Banach Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ContractibleBanachAlgebra.html