# Density of products of continuous bounded functions

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?

• For non-compact $X$ it should be false (note that if you replace $C_b$ with $C_0$ it right). The simplest counterexample should be in $C_b(\Bbb N\times \Bbb N)$, take a look at $f(n,m) = \delta_{n,m}$ and try to see why it cant be approximated by products. – s.harp Jun 22 at 16:49

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.