It is shown that the existence of a biseparating map between a large class of spaces of vector-valued continuous functions A(X,E) and A(Y,F) implies that some compactifications of X and Y are homeomorphic. In some cases, conditions are given to warrant the existence of a homeomorphism between the realcompactifications of X and Y; in particular we find remarkable differences with respect to the scalar context: namely, if E and F are infinite-dimensional and T: C*(X,E) → C*(Y,F) is a biseparating...

Let $C(X,\mathbb{Z})$, $C(X,\mathbb{Q})$ and $C\left(X\right)$ denote the $\ell$-groups of integer-valued, rational-valued and real-valued continuous functions on a topological space $X$, respectively. Characterizations are given for the extensions $C(X,\mathbb{Z})\le C(X,\mathbb{Q})\le C\left(X\right)$ to be rigid, major, and dense.

We prove that a Hausdorff space $X$ is locally compact if and only if its topology coincides with the weak topology induced by $C_{\infty }\left(X\right)$. It is shown that for a Hausdorff space $X$, there exists a locally compact Hausdorff space $Y$ such that $C_{\infty }\left(X\right)\cong C_{\infty }\left(Y\right)$. It is also shown that for locally compact spaces $X$ and $Y$, $C_{\infty }\left(X\right)\cong C_{\infty }\left(Y\right)$ if and only if $X\cong Y$. Prime ideals in $C_{\infty }\left(X\right)$ are uniquely represented by a class of prime ideals in $C^{*}\left(X\right)$. $\infty$-compact spaces are introduced and it turns out that a locally compact space $X$ is $\infty$-compact if and only if every...