Let $ \mathcal{A} $ and $ \mathcal{B} $ be Banach algebras, let $ \sigma:\mathcal{A}\twoheadrightarrow\mathcal{B} $ bea Banach algebra epimorphism from $ \mathcal{A} $ to $ \mathcal{B} $ , and let $ \Lambda $ be a nonzero character on $ \mathcal{B} $ .As is known if $ \mathcal{A} $ is $ \Lambda\circ\sigma $ -contractible (amenable)then $ \mathcal{B} $ is $ \Lambda $ -contractible (amenable).We prove that the converse is true under some conditions.As an important application, we study the $ \operatorname{Tr} $ -contractibility and amenability of the convolution algebra oftrace class operators $ \mathcal{T}(L^{2}(𝔾)) $ , where $ 𝔾 $ is a locally compact quantum group,and $ \operatorname{Tr} $ is a nonzero character on $ \mathcal{T}(L^{2}(𝔾)) $ .
