Let [Ufr ] be an associative Banach algebra. Given a set S, we write l∞(S, [Ufr ]) for the
Banach algebra of all bounded functions f: S→[Ufr ] with the usual norm ∥f∥∞ =
sups∈S∥f(s)∥[Ufr ] and pointwise multiplication. When S is countable, we simply write
l∈([Ufr ]).
In this short note, we exhibit examples of amenable (resp. weakly amenable)
Banach algebras [Ufr ] for which l∈(S, [Ufr ]) fails to be amenable (resp. weakly amenable),
thus solving a problem raised by Gourdeau in [7] and [8]. We refer the reader to
[4, 9, 10] for background on amenability and weak amenability. For basic information
about the Arens product in the second dual of a Banach algebra the reader
can consult [5, 6].