Example

— Let $X_1$ be the set of points $(z_1, z_2) \in \mathbf{C}^2$ such that $|z_1|^2 + |z_2|^2 \leq 1$ (the unit ball in $\mathbf{R}^4$) and let $\mathscr{A}_1'$ be the set of restrictions to $X_1$ of the holomorphic functions, with values in $\mathbf{C}$, defined in a neighborhood of $X_1$ in $\mathbf{C}^2$ (the neighborhood depending on the function considered); let $\mathscr{A}_1$ be the closure of $\mathscr{A}_1'$ in $\mathscr{C}(X_1; \mathbf{C})$, which is obviously a closed complex subalgebra of $\mathscr{C}(X_1; \mathbf{C})$ and separates the points of $X_1$. Application of the 'maximum principle' for holomorphic functions shows that $\mathrm{Ch}_{\mathscr{A}_1}(X_1)$ is the sphere $S_3$.

In the preceding definition, let us replace $X_1$ by the 'polydisk' $X_2$ defined by the relations $|z_1| \leq 1$ and $|z_2| \leq 1$, which yields subalgebras $\mathscr{A}_2'$ and $\mathscr{A}_2$ (the closure of $\mathscr{A}_2'$ of $\mathscr{C}(X_2; \mathbf{C})$. Here, the maximum principle shows that $\mathrm{Ch}_{\mathscr{A}_2}(X_2)$ is the 'torus' defined by the relations $|z_1| = 1$ and $|z_2| = 1$.

From these results, one deduces that there does not exist an analytic isomorphism of an open neighborhood of $X_1$ onto an open neighborhood of $X_2$ that transforms $X_1$ into $X_2$; for, if $v$ were the restriction to $X_1$ of such a mapping, one would have $\mathscr{A}_2 = v \mathscr{A}_1 v^{-1}$ and so $v$ would transform $S_3$ into a space homeomorphic to $T^2$, which is absurd since $S_3$ is simply connected but $T^2$ is not. One will observe, however, that the spaces $X_1$ and $X_2$ are homeomorphic, both being bounded convex sets in $\mathbf{R}^4$ with nonempty interior.*