# Does the topological Varopoulos algebra consist of functions that are continuous and Varopoulos norm bounded?

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 $$V(X_1,\dots,X_n) = \mathbf{V}(X_1,\dots,X_n) \cap C(X_1 \times \dots \times X_n)?$$