A Banach algebra for which for all Banach -bimodules is called amenable (or Johnson amenable; Helemskii 1989, 1997).
This notion was first introduced by Johnson (1972).
For example, the -algebra
of compact operators on a Hilbert
space
is amenable, but the algebra
of all bounded operators on
is not.
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.Johnson,
B. E. Cohomology
in Banach Algebras. Providence, RI: Amer. Math. Soc., 1972.
for which 
for all Banach 
-bimodules 
is called amenable (or Johnson amenable; Helemskii 1989, 1997).
This notion was first introduced by Johnson (1972).
-algebra
of compact operators on a Hilbert
space 
is amenable, but the algebra 
of all bounded operators on 
is not.
