Commutative Gelfand duality

The object assignments are well defined: for nonempty compact Hausdorff $X$, the algebra $C(X,\mathbb{C})$ is a nonzero unital commutative complex C*-algebra with the supremum norm and pointwise conjugation; and for every nonzero unital commutative C*-algebra $A$, the character space $\mathrm{\Delta }(A)$ is a nonempty compact Hausdorff space by Maximal ideal space is compact Hausdorff.

For a continuous map $g:Y\to X$ between compact Hausdorff spaces, the pullback $g^{\ast }:C(X)\to C(Y)$, $g^{\ast }(f)=f\circ g$, is a unital $\ast$-homomorphism: it is complex-linear, multiplicative, preserves constants and conjugation, and is bounded with $\mathrm{\parallel }{g}^{\ast }\mathrm{\parallel }\le 1$; the identity map induces the identity pullback and $(g\circ h{)}^{\ast }=h^{\ast }\circ g^{\ast }$ for composable continuous maps.

For a unital $\ast$-homomorphism $\phi :A\to B$ of unital commutative C*-algebras, the transpose ${\phi }^{\ast }:\mathrm{\Delta }(B)\to \mathrm{\Delta }(A)$, ${\phi }^{\ast }(\psi )=\psi \circ \phi$, is well defined: $\psi \circ \phi$ is a nonzero complex-linear multiplicative map because $\psi$ is, and $(\psi \circ \phi )(1_A)=\psi (1_B)=1$ by unitality of $\psi$ and $\phi$; it is continuous for the evaluation topologies, since for $a\in A$ the composition $\psi \mapsto (\psi \circ \phi )(a)=\psi (\phi (a))$ is the evaluation at $\phi (a)$. Moreover $({\mathrm{i}\mathrm{d}}_A{)}^{\ast }={\mathrm{i}\mathrm{d}}_{\mathrm{\Delta }(A)}$ and $(\psi \circ \phi {)}^{\ast }={\phi }^{\ast }\circ {\psi }^{\ast }$ for composable unital $\ast$-homomorphisms.

The evaluation homeomorphisms of [L2] and the inverse Gelfand isomorphisms ${\mathrm{\Gamma }}_A^{-1}$ of [L1] are the components of natural isomorphisms: for a continuous $g:Y\to X$ and $y\in Y$ one has ${\mathrm{e}\mathrm{v}}_{g(y)}={\mathrm{e}\mathrm{v}}_y\circ g^{\ast }$, that is, ${\eta }_X\circ g=g^{\ast \ast }\circ {\eta }_Y$; and for a unital $\ast$-homomorphism $\phi :A\to B$, every $\psi \in \mathrm{\Delta }(B)$ satisfies ${\mathrm{\Gamma }}_B(\phi (a))(\psi )=\psi (\phi (a))=(\psi \circ \phi )(a)={\mathrm{\Gamma }}_A(a)({\phi }^{\ast }(\psi ))$, that is, ${\mathrm{\Gamma }}_B\circ \phi ={\phi }^{\ast \ast }\circ {\mathrm{\Gamma }}_A$.

The assignments are inverse equivalences on arrows: given a unital $\ast$-homomorphism $\phi :A\to B$, naturality in [step 2.1] gives $\phi ={\mathrm{\Gamma }}_B^{-1}\circ ({\phi }^{\ast }{)}^{\ast }\circ {\mathrm{\Gamma }}_A$, so $\phi$ is determined by ${\phi }^{\ast }$; given a continuous $g:Y\to X$, the same identity at the space level gives $g={\eta }_X^{-1}\circ (g^{\ast }{)}^{\ast }\circ {\eta }_Y$, so $g$ is determined by $g^{\ast }$; and both $\phi \mapsto {\phi }^{\ast }$ and $g\mapsto g^{\ast }$ preserve identities and composition in the reversed order by [step 1.2] and [step 1.3]. Hence the two contravariant functors are mutually inverse up to the natural isomorphisms $\eta$ and ${\mathrm{\Gamma }}^{-1}$.

The object-level identifications [L1], [L2] and the arrow-level bijections [step 3.1] define a contravariant equivalence between ${\mathsf{C}\mathsf{H}\mathsf{a}\mathsf{u}\mathsf{s}}_{\mathrm{\ne }\mathrm{\varnothing }}$ and ${\mathsf{u}\mathsf{C}}_{\mathrm{\ne }0}^{\ast }$, as claimed.