If $X$ is a compact topological space there is the isomorphism $ C(X\times X) \cong C(X) \otimes C(X)$, proved with, for example, Stone-Weierstrass. Is there some way of finding the isomorphic image on the right side of a function on the left? All proofs I know start with a map from right to left by mapping two functions to their product but is there some way to geht a concrete formula for the inverse?
Thanks a lot!
You can have the formula for the inverse for the isomorphism of polynomial rings $\mathbb C[x,y]\simeq\mathbb C[x]\otimes\mathbb C[y]$, namely $$\sum a_{mn}x^my^n\mapsto\sum (a_{mn}x^m)\otimes y^n.$$ But the isomorphism $C(X\times X) \cong C(X) \otimes C(X)$ involves the completion of the subspace of finite sums $\sum_{m,n} f_m(x)g_n(y)$. Hence you can have the explicit inverse only on the dense subspace of $C(X\times X)$.