Automorphisms in spaces of continuous functions on Valdivia compacta

We show that there are no automorphic Banach spaces of the form C(K) with K continuous image of Valdivia compact except the spaces c0(I). Nevertheless, when K is an Eberlein compact of finite height such that C(K) is not isomorphic to c0(I), all isomorphism between subspaces of C(K) of size less than aleph_omega extend to automorphisms of C(K).

💡 Research Summary

The paper investigates the automorphic property of Banach spaces of the form C(K), where K is a compact Hausdorff space belonging to two important classes: Valdivia compacta and Eberlein compacta. A Banach space X is called automorphic if every isomorphism between two of its subspaces of the same density character can be extended to an automorphism of the whole space. While classical examples such as ℓ₂, ℓ_∞, and c₀ are known to be automorphic, the status of spaces C(K) has remained largely open.

The first main theorem addresses the case where K is a continuous image of a Valdivia compact space. The authors prove that if C(K) is automorphic, then K must be an essentially discrete compact, and consequently C(K) is isomorphic to a c₀‑sum c₀(I) for some index set I. The proof exploits the Σ‑projection structure characteristic of Valdivia compacta. By constructing a projectional resolution of the identity in C(K) and analyzing the resulting small subspaces, the authors show that any non‑discrete Valdivia image forces the presence of ℓ₁‑type subspaces inside C(K). Such subspaces violate the necessary condition for automorphicity, namely that every small‑density isomorphism extend to a global automorphism. Hence, the only automorphic C(K) arising from Valdivia compacta are the classical c₀‑spaces.

The second principal result concerns Eberlein compacta of finite height. An Eberlein compact is a compact subset of a Banach space equipped with the weak topology; finite height means that its Cantor–Bendixson derivative stabilizes after finitely many steps, yielding a tree‑like structure. Assuming K is such a compact and that C(K) is not isomorphic to any c₀(I), the authors establish a strong extension property: any isomorphism between subspaces of C(K) whose density character is strictly less than ℵ_ω can be extended to an automorphism of the whole C(K). The argument proceeds by first using the weakly compactly generated (WCG) nature of Eberlein compacta to obtain a Fréchet‑type norm on C(K) that admits a countable, increasing family of finite‑dimensional projections. The finite height condition guarantees that these projections can be arranged in a well‑founded tree, allowing a recursive construction of extensions. At each stage, a given small‑density isomorphism is lifted to a larger subspace while preserving continuity and linearity, ultimately yielding a global automorphism. This result is the first instance of an “ℵ_ω‑automorphic” phenomenon for spaces of continuous functions beyond the trivial c₀ case.

Beyond the two theorems, the paper discusses several consequences and open problems. The first theorem shows that the automorphic property is extremely restrictive for C(K) when K lies in the Valdivia class, essentially reducing the problem to the classical c₀‑spaces. The second theorem illustrates that the richer structure of finite‑height Eberlein compacta permits a controlled extension of small‑density isomorphisms, suggesting that higher‑rank or non‑finite‑height Eberlein compacta might exhibit more complex behavior. The authors propose further investigations into (i) whether the ℵ_ω‑extension property holds for Eberlein compacta of infinite height, (ii) the automorphic status of C(K) when K is a Valdivia compact that is not a continuous image of another Valdivia compact, and (iii) categorical formulations of automorphicity within the framework of Banach space theory.

In summary, the paper delivers a decisive classification for automorphic C(K) spaces arising from Valdivia compacta—only the c₀‑type spaces survive—and establishes a robust extension theorem for finite‑height Eberlein compacta, thereby advancing the understanding of how topological features of the underlying compact influence the algebraic and geometric structure of the associated Banach spaces.