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?
3
-
1$\begingroup$ Reference suggestion: “Tensor products and Banach algebras” by T.K. Carne, 1978 (the easy to trace on the net). $\endgroup$– user131781Commented Nov 29, 2020 at 17:56
-
1$\begingroup$ T.K. Carne, 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 $\endgroup$– David Roberts ♦Commented Mar 5, 2023 at 1:10
-
$\begingroup$ Please see Definiton 4.2.25 & Theorem 4.2.26 in Palmer's book books.google.com/books?id=3k0lKh6QY-QC&&pg=PA473 $\endgroup$– Onur OktayCommented Mar 8, 2023 at 20:02
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
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.
