Automatic continuity of biseparating homomorphisms defined between groups of continuous functions
The Banach-Stone theorem associates to every linear isometry \(T: C(X,\mathbb R)\to C(Y,\mathbb R)\), \(X\) and \( Y\) compact Hausdorff spaces, a homeomorphism \(\phi: Y\to X\) in such a way that \(Tf= (T1) \cdot (f\circ \phi)\) for every \(f \in C(X,\mathbb R)\). It follows that the Banach space structure (linear and metric structure) of \(C(X,\mathbb R)\) completely determines the topological space \(X\). The metric structure here can be replaced by some additional algebraic structure. One of the most efficient ways to do this is by so-called separating maps. A linear operator \(T\) as above is said to be separating provided \(Tf\cdot Tg=0\) when \(f\cdot g=0\) and a linear bijection \(T\) is said to be biseparating provided both \(T\) and \(T^{-1}\) are separating. The present paper addresses similar questions for the group \(C(X,\mathbb T)\) of continuous functions of \(X\) into the unit circle \(\mathbb T\) of the complex plane. In order to investigate which group isomorphisms \(H: C(X,\mathbb T)\to C(Y,\mathbb T)\) are associated to homeomorphisms between \(X\) and \(Y\), the authors say that a map \(H: C(X,\mathbb T)\to C(Y,\mathbb T)\) is separating if whenever \(f,g \in C(X,\mathbb T)\) are such that \((f(x)-1)(g(x)-1)=0\) for every \(x\in X\), then \((Hf(y)-1)(Hg(y)-1)=0\) for every \(y\in Y\). This concept, that adds a pinch of the additive structure of \(C(X,\mathbb C)\) in the multiplicative group \(C(X,\mathbb T)\), is shown to be a good analog in \(C(X,\mathbb T)\) of the concept of separating map. Automatic continuity theorems and representations as composition operators that are typical of the linear theory (see, for instance, \textit{S. Hernández, E. Beckenstein} and \textit{L. Narici} [Manuscr. Math. 86, 409--416 (1995; Zbl 0827.46032)] or \textit{J. Araujo, E. Beckenstein} and \textit{L. Narici} [J. Math. Anal. Appl. 192, 258--265 (1995; Zbl 0828.47024)], the paper's list of references contains many more examples) are proved in this paper for biseparating group isomorphisms between groups of continuous functions. A brief summary of the results proved in the paper follows. Denote the by \(C^0(X,\mathbb T)\) the subgroup of \(C(X,\mathbb T)\) consisting of those functions of the form \(e^{2\pi i f}\) with \(f\in C(X,\mathbb R)\) (this is the connected component of the identity of \(C(X,\mathbb T)\)). It is first shown that a biseparating group isomorphism \(H: C^0(X,\mathbb T)\to C^0(Y,\mathbb T)\) that is continuous on the subgroup of all constant functions is automatically continuous and induces a homeomorphism between \(X\) and \(Y\). When \(X\) is first countable, this leads to the main representation theorem: a biseparating group isomorphism \(H: C(X,\mathbb T)\to C(Y,\mathbb T)\) that is continuous on the constant functions and maps \(C^0(X,\mathbb T)\) onto \(C^0(Y,\mathbb T)\) is automatically continuous with respect to both the uniform and pointwise topologies and can be represented as \[ H(f)(y)=f\bigl(h(y)\bigr)^{\beta(y)},\quad \text{for all } f\in C(X,\mathbb T) \text{ and } y\in Y, \] with \(h: Y\to X\) being a homeomorphism and \(\beta : Y\to \{1,-1\}\) being a continuous mapping. These results are then restated in terms of how unitary groups of commutative \(C^\ast\)-algebras preserve the structure of the algebra. It is also shown that, contrary to what happens in the linear case, topological group isomorphisms that are isometric need not be representable in this way.
scientific article; zbMATH DE number 5710129