Isomorphisms of subspaces of vector-valued continuous functions

On the Banach-Mazur distance between continuous function spaces with scattered boundaries

“…It is well known that if two C(K) spaces are isomorphic, then the underlying compact spaces have the same cardinality, see [9] for the case of scalar functions and [15] for an analogous result for vector-valued functions. Generalizations of those results to the context of subspaces were given in [22] and [21].…”

“…If there exists an isomorphism T : H 1 → H 2 with T T −1 < 2, then their Choquet boundaries Ch Hi K i are homeomorphic (we recall that x ∈ K i is a weak peak point (with respect to H i ) if for a given ε ∈ (0, 1) and a neighborhood U of x there exists a function h ∈ B Hi such that h(x) > 1 − ε and |h| < ε on Ch Hi K i \ U ). Vector-valued extension of this result was given in [21].…”

“…If H is an extremely regular subspace of C(K), then it is clear that each point of K is a weak peak point with respect to H, and thus Ch H K = K, see [21,Lemma 2.5]. Thus extremely regular spaces serve as a simple example of spaces that have closed boundaries consisting of weak peak points.…”

“…It is natural to ask whether the assumption that the boundaries are closed is necessary in Theorem 1.2 (we remind that in the case of the Amir-Cambern theorem for subspaces, no topological assumptions need to be imposed on the boundaries, see [22,Theorem 1.1] or [21,Theorem 1.1]). In section 3, we present a simple example showing that Theorem 1.2 does not hold without the assumption of closed Choquet boundaries (see Example 3.2).…”

“…Notice that all the above results hold automatically also for subspaces H ⊆ C 0 (K, E), where K is locally compact. Indeed, by [21,Lemma 2.10], if J = K ∪ {α} is the one-point compactification of K, then H is isometric to a subspace H of J, defined by…”