On positive embeddings of \(C(K)\) spaces (Q2840505)

Let \(K\) and \(L\) be compact spaces. The paper under review deals with isomorphic embeddings \(T:C(K) \to C(L)\) between Banach spaces of continuous functions. The main objective of this paper is to determine how \(K\) is related to \(L\) whenever \(C(K)\) admits a positive embedding into \(C(L)\). The author shows that in that case there exist a natural number \(p\) and a function \(\varphi: L \to [K]^{\leq p}\) which is upper semicontinuous and onto (that is, \(K\) is the union of values of \(\varphi\)). As a consequence, some topological properties of \(L\), such as countable tightness, or Fréchetness, are inherited by \(K\). Moreover, he proves that \(K\) has a \(\pi\)-base of sets with closures being continuous images of subspaces of \(L\).NEWLINENEWLINENext, he considers the following open problem. Suppose that \(T: C(K) \to C(L)\) is an isomorphic embedding and \(L\) is Corson compact. Is \(K\) necessarily Corson compact? The author shows that the answer is affirmative whenever \(K\) is homogeneous, that is, for each pair \(x_{1},x_{2} \in K\), there is a homeomorphism \(\theta: K \to K\) such that \(\theta(x_{1})=x_{2}\).NEWLINENEWLINEFinally, it is also stated that some isomorphic embeddings \(T: C(K) \to C(L)\) can be, in a sense, reduced to a positive embedding.