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Let $X_1,\dots,X_n$ be compact Hausdorff spaces. Let's define the Varopoulos algebra as the projective tensor product: $$V(X_1,\dots,X_n) := C(X_1) \hat{\otimes} \dots \hat{\otimes} C(X_n),$$ i.e. the space of functions on $X_1 \times \dots \times X_n$ that can be represented as $$f = \sum_k f_{1k} \otimes \dots \otimes f_{nk},$$ where $f_{ik} \in C(X_i)$ and the series is absolutely convergent. The corresponding norm is of course the projective tensor norm: $$\Vert f \Vert_V := \inf_{f = \sum_k f_{1k} \otimes \dots \otimes f_{nk}} \Vert f_{1k} \Vert \dots \Vert f_{nk} \Vert.$$

There is a result of Saeki that implies that if $Y_i$ are factor spaces of $X_i$ via some continuous surjections $X_i \to Y_i$, then $$V(Y_1,\dots,Y_n) = V(X_1,\dots,X_n) \cap C(Y_1 \times \dots \times Y_n)$$ (the function spaces on $Y$ are understood to be embedded into function spaces on $X$ in the obvious way). This allows to do things like identifying Varopoulos functions with the measurable Varopoulos functions (defined analogously with $L^\infty$ in place of $C$) that happen to be continuous on the product...

... Which motivates the following question. Let's define the "largest possible" Varopoulos-type algebra. Let $f$ be any function on the product $X_1 \times \dots \times X_n$. Define $$\Vert f \Vert_\mathbf{V} := \sup_{Z_i \subset X_i, Z_i \text{ finite}} \Vert f \restriction_{Z_1 \times \dots \times Z_n} \Vert_{V(Z_1, \dots, Z_n)},$$ $$\mathbf{V}(X_1,\dots,X_n) := \{f:X_1 \times \dots \times X_n \to \mathbb{C} \, | \, \Vert f \Vert_\mathbf{V} < \infty\}.$$ It can be equivalently characterized as the algebra of functions that are representable as absolutely convergent integrals, as opposed to sums, of products of bounded functions (the relevant measurable structure on $\ell^\infty$ is product, not Borel).

Up to this point, $X_i$ were just sets. Now if we make them into compact spaces, is it true that $$V(X_1,\dots,X_n) = \mathbf{V}(X_1,\dots,X_n) \cap C(X_1 \times \dots \times X_n)?$$

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A negative answer follows from this paper of Graham. In fact, my naive intuition on Varopoulos and tilde algebras turned out to be completely incorrect. Hope to find more on that in Graham & McGehee "Essays in commutative harmonic analysis".

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