Thanks for suggesting this question: Image of the tensor product of strict maps of Banach spaces I read the reference and realize that for a short exact sequence of Banach algebra: $0 \to J \to A \to A/J \to 0$ only makes sense if $J \to A$ has closed image. So in general $D \otimes_\pi$ will not preserve that.
So my new question is: It seems to me that it is hard enough to be a closed subspace. Is there any reference on the ideals of projective tensor product of Banach algebras? Better if examples of computation is available. Thanks!