Isomorphisms of spaces of continuous affine functions on compact convex sets with Lindelöf boundaries

Abstract:

Let $X,Y$ be compact convex sets such that every extreme point of $X$ and $Y$ is a weak peak point and both $\operatorname {ext} X$ and $\operatorname {ext} Y$ are Lindelöf spaces. We prove that if there exists an isomorphism $T:\mathfrak {A}^c(X)\to \mathfrak {A}^c(Y)$ with $\|T\|\cdot \|T^{-1}\|<2$, then $\operatorname {ext} X$ is homeomorphic to $\operatorname {ext} Y$. This generalizes results of C. H. Chu and H. B. Cohen.