$C^*$-homomorphisms and compact $T_2$ spaces

Question

Let $A,B$ be both $T_2$ and compact s.t. $A \underbrace{\cong}_{\text{ homeo.}} B$. Show that $C(A)$ and $C(B)$ are $star$-isomorphic.

Proof: set up $h: A \rightarrow B$ as a homeomorphism and claim that $C(h): C(B) \rightarrow C(A) : \ f \mapsto f \circ h$ is a unital $star$-homomorphism.

1. Linearity: $ ((f + \lambda g) \circ h)(x) = f((h(x)) + \lambda g (h(x))) = (f \circ h + \lambda g \circ h)(x),$
2. Multiplicativity: $(f \circ g) ( g \circ h) (x) = f(g \circ h)(x) = ((fg) \circ h) (x),$
3. Involution Preserving: $(\overline{f} \circ h)(x) = \overline{f}(x) = (\overline{f \circ h})(x)$.

So I am a bit confused about the unital part i.e. $1 \circ h = 1$ and the bijective part.

Answer 1

The unital part: note that the units of the $C^*$-algebras are the functions that are constantly equal to $1$.

The surjective part: if $f\in C(A)$, then $f\circ h^{-1}\in C(B)$ and $C(h)(f\circ h^{-1})=f$.