ABSTRACT
The Banach-Stone theorem tells us that every surjective isometry between C(Q) and C(K) must be of the form Tf(t) = h(t)f(ϕ(t)) where ϕ is a homeomorphism of the compact Hausdorff space K onto the compact Hausdorff space Q, and h is a unimodular function defined on K. The C(Q) spaces are more than just Banach spaces, of course, and are Banach algebras which are, in fact, commutative C∗-algebras. The mapping ϕ induces an algebra automorphism Φ : C(Q) → C(K) which is defined by Φ(f) = f ◦ ϕ. The map Φ is more than just an automorphism, it is an isometric-∗-automorphism in the sense that for every f ∈ C(Q) it is true that Φ(f∗) = (Φ(f))∗ (where f∗(s) ≡ f(s)) and ‖Φ(f)‖ = ‖f‖. In addition to the automorphism, there is multiplication by an element h of C(K) which satisfies the condition hh∗ = h∗h = 1. Hence, every linear isometry in this setting is the action of a ∗-automorphism followed by multiplication by a unitary element.
