I understand that you can take the tensor product of Banach spaces in many different ways by specifying different norms; of particular interest to me are the cross-norms. The projective and injective norms are respectively the largest and smallest cross-norms. The projective tensor product of two Banach algebras is itself a Banach algebra. Is the same true for the injective tensor product of two Banach algebras? Further, is the multiplication map contractive in this case?

## 1 Answer

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Lamadrid shows the counterexample of the Banach property of the injective tensor norm:

- Jesús Gil de Lamadrid,
*Uniform cross norms and tensor products of Banach algebras*, Bull. Amer. Math. Soc.**69**(1963), 797-803 https://doi.org/10.1090/S0002-9904-1963-11037-X

I wonder there is some kind of "minimal tensor products" of Banach algebra and generalized nuclearity like $C^{*}$-algebras, operator spaces or locally convex spaces; that is the equivalence of having unique tensor products and having nice approximation property.

Tensor products and Banach algebras, Journal of the London Mathematical Society, Volume s2-17, Issue 3, June 1978, Pages 480–488, doi.org/10.1112/jlms/s2-17.3.480 $\endgroup$