A bound-two isomorphism between $C(X)$ Banach spaces

Abstract

Nonhomeomorphic compact Hausdorff spaces $X$ and $Y$ and an isomorphism $\phi :C(X) \to C(Y)$ (onto) are constructed such that $||\phi ||\;||{\phi ^{ - 1}}|| = 2$. Amir had asked if such a $\phi$ exists with $||\phi ||\;||{\phi ^{ - 1}}|| < 3$.