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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?

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    $\begingroup$ Reference suggestion: “Tensor products and Banach algebras” by T.K. Carne, 1978 (the easy to trace on the net). $\endgroup$
    – user131781
    Commented Nov 29, 2020 at 17:56
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    $\begingroup$ 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 Oktay
    Commented Mar 8, 2023 at 20:02

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

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.

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