I want to know if $C(X)$ and $C(Y)$ are Riesz isomorphic is it true that $BC(X)$ (the space of bounded continuous functions on $X$) and $BC(Y)$ also have to be Riesz isomorphic.
I already found the contradiction for the inverse because $BC(N)$ and $BC(aN)$ are Riesz isomorphic while $C(N)$ and $C(aN)$ are definitely not!
I can probably use this hint but this only helps me when $X$ and $Y$ are real-compact.
Let $X$ and $Y$ be real-compact topological spaces. Suppose $T$ is a linear bijection $C(X)\to C(Y)$ that is multiplicative, i.e., $T(fg) = T(f) \cdot T(g)\,\,\, (f, g \in C(X))$. Then $X$ and $Y$ are homeomorphic. More explicitly: There exists a homeomorphism $\tau: Y \to X$ for which $T(f) ≔ f \circ \tau\,\,\, (f \in C(X))$.
Can anyone help me if there is a counter example or if this theorem is true!
Thanks a lot,,