# Weak amenability hereditary properties

Let $$\mathcal{A}$$ be a commutative weakly amenable Banach algebra and $$\mathcal{B}$$ be a Banach algebra, let $$\theta:\mathcal{A} \to \mathcal{B}$$ be a continuous homomorphism with dense range; then it is known that $$\mathcal{B}$$ is weakly amenable (and commutative).

Now my question

Let $$\mathcal{A}$$ be a weakly amenable Banach algebra and $$\mathcal{B}$$ be a Banach algebra, let $$\theta:\mathcal{A} \to \mathcal{B}$$ be a continuous homomorphism with the dense range. Is $$\mathcal{B}$$ a weakly amenable Banach algebra?

I am grateful for any help.

• Didn't we have some version of this question on MO before? In any case, the answer is no: you can have noncommutative WA Banach algebras which quotient onto non-WA Banach algebras – Yemon Choi May 28 '19 at 18:32
• However, this is not an obvious fact unless one knows the right sources of examples - I will have to refresh my memory and get back to you. I think that one source of counter-examples arises by taking $A= E\hat\otimes E^*$ for certain Banach spaces $E$, with multiplication defined by $(a\otimes\phi)\cdot (b\otimes\psi)= \phi(b) a\otimes\psi$, but I must confess I've forgotten the details at the moment – Yemon Choi May 28 '19 at 18:37
• Ah yes, this (very natural and good!) question was asked independently mathoverflow.net/questions/260170/… – Yemon Choi May 28 '19 at 18:38
• @Yemon Choi thank you – user62498 May 28 '19 at 18:50