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Let $M \mapsto\mathcal{F}(M)$ be a map associating topological vector spaces of some type (that I will call "function spaces") to geometric spaces $M$ of some type.

For $M$, I'm mostly thinking of manifolds with some additional structure, or locally compact topological spaces. $\mathcal F$ may or may not be a functor in some way, though it's better if it's a contravariant functor. I'm mostly interested in the case where $\mathcal{F}(M)$ is a usual function space such as $L^p(M)$, $W^{k,p}(M)$, $\mathrm{Meas}(M)$, like in this question, and this one. I want the function spaces of the form $\mathcal{F}(M)$ to have some completed tensor product $\otimes$.

Question 1: When does it happen that $\mathcal{F}(M\times N)\simeq\mathcal{F}(M)\otimes\mathcal{F}(N)$ and when does it fail and how badly?

The above tensor property, when $\mathcal F$ is a functor, would be better intended to hold naturally, i.e. $\mathcal F$ is to be a monoidal functor from spaces with their Cartesian product $\times$ to function spaces with $\otimes$, but the emphasis is not on the categorical aspect.

Edit: I'm aware that, as Nik Weaver points out in the comments, I can't expect to get a completely general answer. Rather, the question (which I find very natural) should be intended in "community wiki" style, i.e. partial contributions are ok.

Qfwfq
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    The question is impossible to answer at this level of generality. – Nik Weaver Dec 28 '20 at 02:22
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    I'm not surprised that giving a completely general answer is nearly impossible. Still, I think some users may know the answer for many well known, occurring-in-real-life, function spaces or classes thereof. – Qfwfq Dec 28 '20 at 02:26
  • Oh, you just want some examples where this happens? – Nik Weaver Dec 28 '20 at 14:14
  • I had received a notice from a reader that you want this to be CW, but that's not too clear to me. But please alert me if that's what you intend. – Todd Trimble Dec 28 '20 at 17:29
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    Spaces of distributions and test functions typically behave well with respect to tensor products. – Abdelmalek Abdesselam Dec 28 '20 at 17:50
  • @Todd Trimble: partial answers like the one by Sergei Arbakov are ok, like answers like those in the linked questions. If you deem it should be cw, yes please go ahead and hit the button; for me, honestly, in this case it's indifferent if it's cw or not. Thank you for the attention – Qfwfq Dec 28 '20 at 18:11
  • Spaces of test functions typically behave well with respect to tensor products -- that's right, but one has to take the correct tensor products. Even though $\mathscr D(\mathbb R)$ is a nuclear locally convex space, $\mathscr D(\mathbb R) \tilde\otimes \mathscr D(\mathbb R)=\mathscr D(\mathbb R^2)$ does NOT hold for the projective or injective tensor product but only for the so-called inductive tensor product. – Jochen Wengenroth Dec 28 '20 at 18:29
  • Qfwfq: I won't take any immediate action on this. – Todd Trimble Dec 28 '20 at 22:34

1 Answers1

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In the theory of stereotype spaces there is a series of natural functors that satisfy this identity with one of the two main tensor products (the so-called ``injective stereotype tensor product'' $\odot$): $$ {\mathcal F}(M\times N)\cong {\mathcal F}(M)\odot {\mathcal F}(M) $$ In particular, this holds for

  • the stereotype algebras ${\mathcal C}$ of continuous functions on paracompact locally compact spaces: $$ {\mathcal C}(M\times N)\cong {\mathcal C}(M)\odot {\mathcal C}(M) $$

  • the stereotype algebras ${\mathcal E}$ of smooth functions on smooth manifolds: $$ {\mathcal E}(M\times N)\cong {\mathcal E}(M)\odot {\mathcal E}(M) $$

  • the stereotype algebras ${\mathcal O}$ of holomorphic functions on Stein manifolds: $$ {\mathcal O}(M\times N)\cong {\mathcal O}(M)\odot {\mathcal O}(M) $$

  • the stereotype algebras ${\mathcal P}$ of polynomials (= regular functions) on affine algebraic manifolds: $$ {\mathcal P}(M\times N)\cong {\mathcal P}(M)\odot {\mathcal P}(M) $$

(The cases of ${\mathcal E}$ and ${\mathcal O}$ are just reformulations of the classical results of functional analysis, and the whole picture is stated here. This is closely connected with the constructions of group algebras in analysis.)

Sergei Akbarov
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