[Lamadrid 63-1] shows the counterexample of the Banach property of the injective tensor norm.

https://www.ams.org/journals/bull/1963-69-06/S0002-9904-1963-11037-X/S0002-9904-1963-11037-X.pdf

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.

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.