Ideals and exactness of projective tensor product of Banach spaces / algebras

Question

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!

Answer 1

General description of ideals of tensor products of Banach algebras is hopeless, so the only thing is consideration of some important cases. Here is what I found

Maximal ideals in tensor products of Banach algebras Arnold Lebow

Maximal two-sided ideals in tensor products of Banach algebras Kjeld. B. Laursen

Primitive ideals in tensor product of Banach algebras Jun Tomiyama

If you will not restrict yourself only with projective tensor product, you can take a look at the

Ideals in Operator Space Projective Tensor Product of $C^*$-algebras. Ranjana Jain, Ajay Kumar

It contains a lot of references to the similar question for Haagerup and minimal tensor product of $C^*$-algebras. In fact minimal tensor product of $C^*$-algebras is much more important than others becuase it relates different nice properties such as exactness, nuclearity and ameanability. For details see section IV.3 in

Operator algebras. Theory of C-star-algebras and von Neumann algebras. B. Blackadar