Density of products of continuous bounded functions

Question

Let $C_b(X)$ denote the space of continuous and bounded real functions in a topological space $X.$ $C_b(X)$ is a real Banach space with the $\infty$-norm $$\|f\|_{\infty} = \sup_{x \in X} |f(x)|.$$ Let $C_b(X) \otimes C_b(X)$ be the space of functions in $C_b(X \times X)$ of the form $$f(x, y) = \sum_{i = 0}^{N} g_i(x) h_i(y) \,\text{ for some } N \in \mathbb{N}, \ \,g_i, h_i \in C_b(X) \,\,\, (i = 1, \cdots, N).$$

For the particular case $X = (0, 1]$ with the subspace topology of $\mathbb{R},$ is it true that $C_b(0,1] \otimes C_b(0,1]$ is dense in $C_b((0,1]\times(0,1])$ in the $\infty$-norm?

Answer 1

The answer is no.

Depending on how much generality you want you can formulate different theorems giving this result. Two useful looking specifications are:

Theorem

1. If $X$ is locally compact and metrisable then $C_b(X)\otimes C_b(X)\overset\mu\to C_b(X\times X)$ has dense image if and only if $X$ is compact.
2. If $X,Y$ are locally compact (non-finite) Hausdorff then $C_b(X)\otimes C_b(Y)\overset\mu\to C_b(X\times Y)$ has dense image if and only if $X$ and $Y$ are pseudo-compact.

In order to show this we make use of 3 ingredients:

1. The relation $C_b(X)=C(\beta X)$ for locally compact Hausdorff $X$, where $\beta X$ is the Stone-Cech compactification of $X$.
2. The Stone-Weierstraß theorem.
3. Thereom 1 from this paper (Stone-Cech compactifications of Products, Irving Glicksberg), which in our case establishes that $\beta(X\times X)=\beta X\times \beta X$ if and only if $X\times X$ is pseudo-compact.

First if $X$ is compact then the image of $C(X)\otimes C(X)$ under multiplication $\mu:C(X)\otimes C(X)\to C(X\times X)$ separates points in $X\times X$, and thus this image (which is a $*$-subalgebra of $C(X\times X)$) must be dense in $C(X\times X)$ by the Stone-Weierstraß theorem.

Thus for compact spaces $\mu(C(X)\otimes C(X))$ is always dense in $C(X\times X)$. Now we suppose that $X$ is not compact, consider the following composition:

$$C_b(X)\times C_b(X) = C(\beta X)\otimes C(\beta X) \overset{\mu_{\beta X}}\to C(\beta X\times \beta X)\overset{r}\to C_b(X\times X),$$ where $r:C(\beta X\times \beta X)$ is the restriction map $f\mapsto f\lvert_{X\times X}$. This restriction obviously behaves well with the multplication: $\mu_{\beta X}(f,g)\lvert_{X\times X}=\mu_X(f\lvert_X, g\lvert _X)$, hence this composition is the same as the map $$C_b(X)\otimes C_b(X)\overset{\mu_X}\to C_b(X\times X).$$ The restriction map $r$ is clearly isometric (as $X\times X$ is dense in $\beta X\times \beta X$) and the map $\mu_{\beta X}:C(\beta X)\otimes C(\beta X)\to C(\beta X\times\beta X)$ has dense image by Stone-Weierstraß, so the question of when the image of $\mu_X$ is dense in $C_b(X\times X)$ is the same the question of when $r(C(\beta X\times \beta X))=C_b(X\times X)$ holds. Identifying $C_b(X\times X)$ with $C(\beta (X\times X))$ we get:

$\mu_X(C_b(X)\otimes C_b(X))$ is dense in $C_b(X\times X)$ if and only if $r(C(\beta X\times \beta X))=C(\beta (X\times X))$.

This means that $r$ must be an isomorphism of commutative $C^*$-algebras (we have noted that it is isometric, it is easy to check that it is an $*$-morphism, so the desired surjectivity is all thats needed to get isomorphism). This means that it must induce a homoemorphism $\beta X\times \beta X\to \beta (X\times X)$ and the compactification $\beta X\times \beta X$ of $X\times X$ must be the Stone-Cech compactification.

By Theorem 1 in the linked paper this is the case if and only if $X\times X$ is pseudo-compact. By other statements scattered throughout that paper you can note that on paracompact spaces pseudo-compact is the same as compact, hence for metrisable $X$ you have $\beta X\times \beta X\not\cong \beta (X\times X)$ whenever $X$ is not compact, giving the first statement above.

By another statement if $X, Y$ are locally compact Hausdorff then $X\times Y$ is pseudo-compact if an only if $X$ and $Y$ are pseudo-compact. Hence provided both are non-finite spaces we get $\beta X\times \beta Y = \beta (X\times Y)$ if and only if $X,Y$ are pseudo-compact.