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!



Examples of amenable Banach algebras which have non-amenable subalgebra
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.


