I would like to prove the following:
where are continuous, complex-valued functions on Hausdorff space . 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
Then I think showing something like the subalgebra separates , 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"?
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, with . Does anyone have any idea if
So not as Banach spaces but as sets? Maybe something involving an infinite sum of tensor products of functions, like
I guess using the continuity would be key, but any idea if something like this works?
Thanks very much!
I should have mentioned in my previous comment that in the isomorphism between and (for compact Hausdorff spaces X and Y), the tensor product 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 .