Is there an example of an unital commutative semi-simple Banach algebra which it is not amenable?



Semi-simple Banach algebra
Some examples off the top of my head:


1) the disc algebra
$$ A({\mathbb D}) = \{ f \in C(\overline{\mathbb D}) \mid \hbox{$f$ is analytic on $\mathbb{D}$} \} $$
equipped with the supremum norm.


2) $C^1{([0,1])}$, the space of continuously differentiable functions $[0,1]\to {\mathbb C}$ equipped with the norm
$$ \Vert f\Vert\_{(1)} = \Vert f\Vert\_\infty + \Vert f'\Vert\_{\infty} $$


3) $A(SO(3))$, the Fourier algebra of the (real, compact) Lie group $SO(3)$.


All of these examples are semisimple, commutative and unital. Examples 2) and 3) are *regular* Banach algebras; Example 1) is not regular. Examples 1) and 2) have non-zero continuous point-derivations, so are not weakly amenable, hence are not amenable. Example 3) has no non-zero continuous point derivations, but Johnson showed in 1994 that it is not weakly amenable (and hence not amenable).


