Let $A$ be a weak amenable Banach algebra and $I$ be a closed (twosided) ideal of $A$. In general $\frac{A}{I}$ is not weakly amenable. Is there an example of this type of weak amenable Banach algebra?(Can you give me an example of a weak amenable Banach algebra with a non weak amenable quotient?)
1 Answer
Let $E$ be a Banach space, let $E\hat{\otimes}E^*$ be the projective tensor product of $E$ with it's dual $E^*$. It can be considered as a Banach algebra and $E\hat{\otimes}E^*$ is weakly amenable for every Banach space $E$ (see "Derivations iterated duals of Banach algebras" by Dales, Ghahramani and Grønbaek ($\star$), Theorem 5.1).
Let $\tau :E\hat{\otimes}E^*\to B(E)$ given by $\tau(x\otimes y)(\xi)=\langle \xi, y\rangle_{E} x$ (on simple tensors). The range of this map is typically denoted with $N(E)$, the nuclear operators $E\to E$, which can be considered as the quotient $E\hat{\otimes}E^*/\mathrm{ker}(\tau)$ endowed with the quotient norm. One can show that if $E$ satisfies the approximation property (https://en.wikipedia.org/wiki/Approximation_property), then $\tau$ is injective so that $E\hat{\otimes}E^*\cong N(E)$ as Banach algebras (see again $\star$, in the beginning of Section 5.)
Now you need to know some conditions when $N(E)$ is weakly amenable or not (some are stated in "Amenability and weak amenability of tensor algebras and algebras of nuclear operators" by Grønbaek), so that you can construct desired examples of such quotients. One possibility is to take $E=M\oplus M$ where $M$ is a Banach space which does not have the approximation property.

3$\begingroup$ Nice example. I mention just for interest that $E\hat\otimes E^*$ satisfies a much stronger property: it is biflat. All biflat algebras are weakly amenable; this is folklore, but I believe a direct proof is given in Dales's book $\endgroup$ Jan 23, 2017 at 12:04

$\begingroup$ @Yemon Choi thank you! Oh yes, you are right $\endgroup$ Jan 23, 2017 at 13:10