Locally compact Gelfand duality

The objects. For any LCH $X$, extension by zero at infinity identifies $C_0(X)$ with $J=\{h\in C(X^{+}):h(\mathrm{\infty })=0\}$. Indeed continuity at infinity of the extension is exactly compactness of the closed sets $\{\mathrm{\mid }f\mathrm{\mid }\ge ϵ\}$: their complements provide neighborhoods; conversely each such level set for a continuous function vanishing at infinity is closed in compact $X^{+}$ and omits infinity. Thus $f$ is bounded, and extension preserves the supremum norm, including the empty space with norm zero. The ideal $J$ is star-closed and norm-closed because evaluation is bounded. It is complete: a norm-Cauchy sequence converges in the Banach space $C(X^{+})$ of [F5], and its limit still vanishes at infinity. The pointwise operations and C*-identity restrict to $J$, making $C_0(X)$ a commutative C*-algebra. The space $X^{+}$ is nonempty even when $X$ is empty, so [F5] applies. By [F2], $\mathrm{\Delta }(A)$ is LCH.

The transpose is defined and continuous. For proper $\phi :A\to B$ and $\psi \in \mathrm{\Delta }(B)$, choose an approximate unit $(e_i)$ of $A$ by [F2] and $b$ with $\psi (b)\mathrm{\ne }0$. From the evaluation formula and isometry in [F2], $\mathrm{\mid }\psi (b^{\mathrm{\prime }})\mathrm{\mid }\le \mathrm{\parallel }{b}^{\mathrm{\prime }}\mathrm{\parallel }$ for every $b^{\mathrm{\prime }}$, so $\psi$ is continuous, without any unit hypothesis. Properness gives $\phi (e_i)b\to b$, hence $\psi (\phi (e_i))\psi (b)\to \psi (b)$ and $\psi (\phi (e_i))\to 1$. Thus $\psi \circ \phi$ is nonzero and is a character by the algebraic properties. Its evaluation at $a$ is $\psi \mapsto \psi (\phi (a))$, continuous by [F3], so ${\phi }^{\ast }$ is continuous. Transposition reverses composition and preserves identities directly.

Pullbacks are bounded star-homomorphisms. For proper continuous $g:Y\to X$, the equality $\{\mathrm{\mid }f\circ g\mathrm{\mid }\ge ϵ\}=g^{-1}(\{\mathrm{\mid }f\mathrm{\mid }\ge ϵ\})$ proves $f\circ g\in C_0(Y)$. Pointwise operations prove linearity, multiplication and star preservation, and $\mathrm{\parallel }{g}^{\ast }f\mathrm{\parallel }\le \mathrm{\parallel }f\mathrm{\parallel }$. Identity and composition reverse as $(g\circ h{)}^{\ast }=h^{\ast }{g}^{\ast }$.

The transpose is proper by compact level sets. Given compact $K\subseteq \mathrm{\Delta }(A)$, choose $u\in C_c(\mathrm{\Delta }(A))$ with $u=1$ on $K$ and $0\le u\le 1$ by [F4], and let $a={\mathrm{\Gamma }}_A^{-1}(u)$. Whenever ${\phi }^{\ast }(\psi )\in K$, the function $v={\mathrm{\Gamma }}_B(\phi (a))\in C_0(\mathrm{\Delta }(B))$ satisfies $v(\psi )=u({\phi }^{\ast }(\psi ))=1$. Hence $({\phi }^{\ast }{)}^{-1}K$ lies in the compact level set $\{\mathrm{\mid }v\mathrm{\mid }\ge 1\}$. It is closed since $K$ is closed in the Hausdorff space $\mathrm{\Delta }(A)$ and ${\phi }^{\ast }$ is continuous. By [F7] it is compact. The argument includes empty $K$ and empty character spaces. It needs no asserted bounded extension of $\phi$ to unitizations.

Evaluation is a homeomorphism ${\eta }_X:X\to \mathrm{\Delta }(C_0(X))$. A cutoff on a singleton makes each evaluation nonzero, and a cutoff supported in an open set separating two points distinguishes their evaluations. Each $h\in C(X^{+})$ decomposes uniquely as $\tilde{f}+\lambda 1$, with $\lambda =h(\mathrm{\infty })$ and $f\in C_0(X)$ by step 1.1. For a character $\chi$ on $C_0(X)$, define $\widehat{\chi }(h)=\chi (f)+\lambda$. Expanding products proves it a nonzero unital character on $C(X^{+})$. By [F5] it is evaluation at a unique point; this point cannot be infinity, since the restriction there is zero. Thus it lies in $X$, proving surjectivity. Evaluation is continuous by [F3]. For an open $U\subseteq X$ and $x\in U$, a cutoff $u$ with $u(x)=1$ and $u=0$ off $U$ gives the character neighborhood $\{\chi :\mathrm{\mid }\chi (u)\mathrm{\mid }>1\mathrm{/}2\}$ of ${\eta }_X(x)$ contained in ${\eta }_X(U)$. Hence the inverse is continuous. For $X=\mathrm{\varnothing }$, there are no characters on the zero algebra, so the same conclusion holds.

Pullbacks preserve every approximate unit. Let $(e_i)$ be any approximate unit of $C_0(X)$. Its positivity factorization gives pointwise $e_i\ge 0$, and its norm bound gives $e_i\le 1$. By step 2.1 each $e_i\circ g$ belongs to $C_0(Y)$, is a contraction and has the pulled-back positive factorization and self-adjointness. Fix $f\in C_0(Y)$ and $ϵ>0$. Put $\delta =ϵ\mathrm{/}(2(1+\mathrm{\parallel }f\mathrm{\parallel }))$ and $K=\{\mathrm{\mid }f\mathrm{\mid }\ge \delta \}$. The image $g[K]$ is compact by [F7]. A cutoff $u$ on it satisfies $u=1$ there and $0\le u\le 1$. Since $\mathrm{\parallel }u{e}_i-u\mathrm{\parallel }\to 0$, eventually $\mathrm{\mid }1-e_i(g(y))\mathrm{\mid }<\delta$ on $K$. Off $K$, $\mathrm{\mid }f\mathrm{\mid }<\delta$ and $\mathrm{\mid }1-e_i\circ g\mathrm{\mid }\le 1$. Consequently $\mathrm{\parallel }f-f(e_i\circ g)\mathrm{\parallel }\le \delta \max (1,\mathrm{\parallel }f\mathrm{\parallel })<ϵ$. Commutativity gives the other product. This proves properness for every approximate unit, including zero functions and empty spaces.

Naturality. For $y\in Y$, ${\mathrm{e}\mathrm{v}}_y\circ g^{\ast }={\mathrm{e}\mathrm{v}}_{g(y)}$, so double transposition recovers $g$ through step 2.3. For $a\in A$ and $\psi \in \mathrm{\Delta }(B)$, ${\mathrm{\Gamma }}_B(\phi (a))(\psi )=\psi (\phi (a))={\mathrm{\Gamma }}_A(a)({\phi }^{\ast }(\psi ))$, whence ${\mathrm{\Gamma }}_B\phi =({\phi }^{\ast }{)}^{\ast }{\mathrm{\Gamma }}_A$. The isometric star-isomorphisms ${\mathrm{\Gamma }}_A$ and their inverses preserve every approximate unit by the explicit b*b factorization, norms and convergence, and hence are proper arrows. Homeomorphisms and their inverses are proper because they carry compact sets to compact sets. Thus both object identifications are isomorphisms in the stated categories. Identities and composites are proper on each side by their definitions. The object and arrow constructions above and these natural identifications prove the contravariant equivalence.

Compact and zero boundaries. Between nonzero unital algebras, a proper morphism takes the constant approximate unit $1_A$ to a constant approximate unit, so $\phi (1_A)b=b=b\phi (1_A)$ for all $b$; thus $\phi (1_A)=1_B$. Conversely, a bounded unital star-homomorphism between commutative unital algebras preserves every approximate unit: $e_i\to 1_A$ in norm, so its images tend to $1_B$; positivity is preserved, and contractivity follows from [F2] and step 1.2's nonzero character composition (here nonzero follows directly from unitality). Nonempty compact spaces correspond exactly to these objects by [F5] and step 2.3. For $B=0$ the unique map $A\to 0$ is proper but not unital under the nonzero-unit convention. If $A=0$ and $B\mathrm{\ne }0$, the zero net cannot approximate a nonzero $b$, so no proper arrow exists. These correspond exactly to the unique map $\mathrm{\varnothing }\to X$ and the absence of maps from nonempty $X$ to $\mathrm{\varnothing }$.