Thread: Tensor Products of Space of Continuous Functions

1. Tensor Products of Space of Continuous Functions

Hey all,
I would like to prove the following:

$C(X \times Y)=C(X)\otimes C(Y),$

where $C(X)$ are continuous, complex-valued functions on Hausdorff space $X$. I don't want to specify them as bounded, but I think my solution requires that.

The first question is what '=' means here, so it at least means isomorphic, and probably isometric too. So if someone wants to just comment about that part it would be appreciated (like are isomorphic Banach spaces always isometric or something).

So my thought for the isomorphism is something like the map taking

$f\otimes g \to F\in C(X\otimes Y)\mbox{ where } F(x,y)=f(x)g(y).$

Then I think showing something like the subalgebra $f\otimes g\subset C(X)\otimes C(Y)$ separates $C(X\otimes Y)$, and then invoke the Stone-Weierstrass theorem.

Does this make sense to anyone? And is it common convention to assume "isomorphic Banach algebras" are "isomorphic, isometric Banach algebras"?

2. Originally Posted by cduston
Hey all,
I would like to prove the following:

$C(X \times Y)=C(X)\otimes C(Y),$

where $C(X)$ are continuous, complex-valued functions on Hausdorff space $X$. I don't want to specify them as bounded, but I think my solution requires that.

The first question is what '=' means here, so it at least means isomorphic, and probably isometric too. So if someone wants to just comment about that part it would be appreciated (like are isomorphic Banach spaces always isometric or something).

So my thought for the isomorphism is something like the map taking

$f\otimes g \to F\in C(X\otimes Y)\mbox{ where } F(x,y)=f(x)g(y).$

Then I think showing something like the subalgebra $f\otimes g\subset C(X)\otimes C(Y)$ separates $C(X\otimes Y)$, and then invoke the Stone-Weierstrass theorem.

Does this make sense to anyone? And is it common convention to assume "isomorphic Banach algebras" are "isomorphic, isometric Banach algebras"?
Yes, that makes very good sense. You have the correct isomorphism, and the correct tool for proving the result (namely the Stone–Weierstrass theorem). The only snag is that the S–W theorem requires that the spaces X and Y should be compact, or at least locally compact (in which case you must restrict the functions to being those that vanish at infinity). You certainly won't be able to deal with arbitrary (unbounded) continuous functions, because they don't form a Banach space.

3. Ok great, good to know I'm on the right track. And I did not mention it but I am assuming the spaces X and Y to be compact. What you then say about unbounded functions brings to mind another question I suppose; if C(X) is to be Banach space then certainly the functions must be bounded. But one could also consider just the set of all possible continuous functions on X, $C'(X)$ with $C(X)\subset C'(X)$. Does anyone have any idea if

$C'(X\times Y)=C'(X)\otimes C'(Y)$?

So not as Banach spaces but as sets? Maybe something involving an infinite sum of tensor products of functions, like

$F(x,y)=\sum f_i(x)\otimes g_i(y).$

I guess using the continuity would be key, but any idea if something like this works?

Thanks very much!

4. Originally Posted by cduston
Ok great, good to know I'm on the right track. And I did not mention it but I am assuming the spaces X and Y to be compact. What you then say about unbounded functions brings to mind another question I suppose; if C(X) is to be Banach space then certainly the functions must be bounded. But one could also consider just the set of all possible continuous functions on X, $C'(X)$ with $C(X)\subset C'(X)$. Does anyone have any idea if

$C'(X\times Y)=C'(X)\otimes C'(Y)$?

So not as Banach spaces but as sets? Maybe something involving an infinite sum of tensor products of functions, like

$F(x,y)=\sum f_i(x)\otimes g_i(y).$

I guess using the continuity would be key, but any idea if something like this works?
On a compact space every continuous function is automatically bounded, so I don't think you need to worry about boundedness.

I should have mentioned in my previous comment that in the isomorphism between $C(X)\otimes C(Y)$ and $C(X\times Y)$ (for compact Hausdorff spaces X and Y), the tensor product $C(X)\otimes C(Y)$ is not the algebraic tensor product but the completion of the algebraic tensor product with respect to a particular norm called the injective cross-norm. To distinguish it from the algebraic tensor product, it is sometimes denoted $C(X)\mathbin{\check{\otimes}}C(Y)$.