Function spaces satisfying $\mathcal{F}(M\times N)\simeq\mathcal{F}(M)\otimes\mathcal{F}(N)$

Question

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.

Answer 1

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.)