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Let $H^{(j)}$ and $G^{(j)}$ be Banach spaces for $j\in\{1,\dots,n\}$. Call norms $\|\cdot\|_{H}$ and $\|\cdot\|_{G}$ on the algebraic tensor products $H:=\bigotimes_{j=1}^n H^{(j)}$ and $G:=\bigotimes_{j=1}^n G^{(j)}$ uniform if the operator norm satisfies $$ \|\bigotimes_{j=1}^n A^{(j)}\|_{H\to G}=\prod_{j=1}^n \|A^{(j)}\|_{H^{(j)}\to G^{(j)}}. $$

Is there an extensive list of pairs of spaces with uniform crossnorms? (Preferably with a focus on function spaces; Lebesgue, Sobolev, Hoelder would already be great)

Of course, the results for tensor products of well-known spaces depend on the norms that we equip these tensor products with. For example, it would be good to know if $C(\Omega_1)\otimes C(\Omega_2)$ equipped with the $C(\Omega_1\times\Omega_2)$ norm and $H^1(\Omega_3)\otimes H^1(\Omega_4)$ equipped with the $H^{1}_{\text{mix}}(\Omega_3\times\Omega_4)$ norm are uniform, and if these norms actually turn the algebraic tensor products into the spaces $C(\Omega_1\times\Omega_2)$ and $H^{1}_{\text{mix}}(\Omega_3\times\Omega_4)$, respectively (by closure).

Posting any specific results instead of a reference would be appreciated too; I will keep track in the list below:

  • If $H^{(j)}$ and $G^{(j)}$ are Hilbert spaces, then equipping $H$ and $G$ with the induced Hilbert space structure yields uniform crossnorms. The induced scalar product on $H$ is the unique bilinear extension of $$\langle \otimes_{j=1}^n f^{(j)}_1 ,\otimes_{j=1}^n f^{(j)}_2\rangle_{H}=\prod_{j=1}^n \langle f^{(j)}_1,f^{(j)}_2\rangle_{H^{(j)}}$$ (Proposition 4.127 in W. Hackbusch, "Tensor spaces and numerical tensor calculus". Springer, 2012)
  • Equipping $H$ with the projective norm and $G$ with any crossnorm (that is, a norm that is multiplicative w.r.t the tensor product) yields uniform crossnorms. (Have no reference)

  • Equipping $G$ with the injective norm and $H$ with any crossnorm yields uniform crossnorms (Have no reference)

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The book

  • MR0582655 Pietsch, Albrecht: Operator ideals. Translated from German by the author. North-Holland Mathematical Library, 20. North-Holland Publishing Co., Amsterdam-New York, 1980. 451 pp. ISBN: 0-444-85293-X 47-02 (47D30)

is a comprehensive review of operator ideals and list many of them. Many results have been carried over to tensor products in

  • MR0496883 Michor, Peter W.: Functors and categories of Banach spaces. Tensor products, operator ideals and functors on categories of Banach spaces. Lecture Notes in Mathematics, 651. Springer, Berlin, 1978. iii+99pp. ISBN: 3-540-08764-8 (Reviewer: J. Wick Pelletier) 46M15 (46B20 46M05 47D30). Available here.
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  • $\begingroup$ Could you maybe, for a pair of spaces of functions of your choice, explain how the fact that the norms are uniform crossnorms can be retrieved from the provided references? $\endgroup$ – Bananach Mar 30 '16 at 9:52
  • $\begingroup$ Operator ideal property is what you want, but with $\le$ instead of $=$. Equality is easy on decomposable tensors. In the second reference it is the functorial property. $\endgroup$ – Peter Michor Mar 30 '16 at 9:55
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As an addendum to Peter Michor's reply: the term "crossnorm" goes back to Schatten but this theme was absorbed into modern functional analysis by Grothendieck in his thesis and his celebrated "Resumé", both readily available, the former as a Memoir of the A.M.S., the latter online---just google "Grothendieck resumé". Most recent books on Banach spaces have chapters on this subject and there is a fairly recent monograph by Ryan, all of which have copious material on the themes you mention.

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