TITLE: Examples of amenable Banach algebras which have non-amenable subalgebra
QUESTION [5 upvotes]: I am looking for examples of amenable Banach algebras  which have  non-amenable subalgebra
I know

1: Each amenable Banach algebra has a bounded approximate identity


2: If $I$ be a closed ideal in an amenable Banach algebra, then


$I$ amenable if and only if  $I$ has a bounded approximate identity

Does anybody have an example?
Thank you for any suggestions!

REPLY [12 votes]: Mateusz's answer mentions lots of good mathematics but I feel obliged to point out that the fundamental example which answers your original question in the negative is $M_2({\bf C})$. (Banach algebras behave very differently from ${\rm C}^*$-algebras and $L^1$-group algebras.)
The point is that the algebra
$$
{\bf C}[x] / (x^2) \cong 
\left\{ 
\begin{pmatrix} a & b \cr 0 & a \end{pmatrix} 
\colon a,b \in {\bf C} \right\}
\subset M_2({\bf C})
$$
is not amenable, and should not be "generalized amenable" in any sensible version of "generalized amenability".
Non-amenability can be seen in many ways, but the most direct one -- if you use the definition of amenability in terms of derivations -- is to note that the mapping
$$
\begin{pmatrix} a & b \cr 0 & a \end{pmatrix} \to b
$$
is a non-zero derivation. In some sense this is the philosophical idea behind derivations on associative algebras, they are the 1st-order terms that arise when one perturbs a homomorphism.