TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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Nonunital commutative Gelfand Naimark

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $A$ be a commutative C*-algebra (C star algebra), possibly nonunital and possibly the zero algebra. Then the Gelfand transform

$Γ : A \to C_{0} (Δ (A)), Γ (a) (φ) : = φ (a),$

is an isometric $*$ -isomorphism onto $C_{0} (Δ (A))$ (Compact support, $C_{c} (X)$ , and $C_{0} (X)$ ), where $Δ (A)$ carries the weak-star topology and is locally compact Hausdorff; for a unital $A$ this is the statement $Γ : A ≅ C (Δ (A))$ of Commutative Gelfand Naimark, and for a nonunital nonzero $A$ the transform is the restriction of the unital transform $A^{+} \to C (Δ (A^{+}))$ to the ideal of functions vanishing at the point at infinity $χ_{\infty}$ , under the identification $Δ (A^{+}) = Δ (A) \cup {χ_{\infty}}$ of Character space of the unitization is one-point compactification.

Facts & Assumptions

Given: The Axiom of Choice, a commutative C*-algebra $A$ , its character space $Δ (A)$ , and the Gelfand transform $Γ$ .

[L1]

If $A$ is unital and nonzero, $Γ : A \to C (Δ (A))$ is an isometric unital $*$ -isomorphism onto $C (Δ (A))$ (Commutative Gelfand Naimark, The Axiom of Choice).

[L2]

If $A$ is nonzero and genuinely nonunital, then $A^{+}$ is a unital commutative C*-algebra containing $A$ as a closed two-sided $*$ -ideal of codimension one, and $Δ (A^{+}) = Δ (A) \cup {χ_{\infty}}$ with $Δ (A)$ an open dense locally compact Hausdorff subspace of the compact Hausdorff space $Δ (A^{+})$ (Minimal C star unitization, Character space of the unitization is one-point compactification, The Axiom of Choice).

[L3]

For a locally compact Hausdorff space $X$ , $C_{0} (X)$ is the set of continuous functions $f$ such that ${∣ f ∣ \geq ϵ}$ is compact for every $ϵ > 0$ , and $C_{0} (X) = C (X)$ when $X$ is compact; the one-point compactification topology on $X^{+} = X \cup {\infty}$ has as neighbourhoods of $\infty$ the complements of compact subsets of $X$ , so a continuous function $g$ on $X$ extends continuously to $X^{+}$ with value $0$ at $\infty$ if and only if $g \in C_{0} (X)$ (Compact support, $C_{c} (X)$ , and $C_{0} (X)$ , The one-point (Alexandroff) compactification $X^{*} = X \cup {\infty}$ , whose open sets are the open sets of $X$ together with the complements in $X^{*}$ of the closed compact subsets of $X$ , $X^{*}$ is compact and contains $X$ as an open subspace; $X$ is dense in $X^{*}$ exactly when $X$ is not compact; and $X^{*}$ is Hausdorff exactly when $X$ is locally compact and Hausdorff).

[L4]

For the zero algebra $A = {0}$ one has $Δ (A) = \emptyset$ and $C_{0} (\emptyset) = {0}$ , and the only map ${0} \to {0}$ is an isometric $*$ -isomorphism. [algebra]

Proof

technique · direct

1.1

If $A$ is unital and nonzero, the claim is [L1] together with $C_{0} (Δ (A)) = C (Δ (A))$ from [L3], since $Δ (A)$ is compact; if $A = {0}$ the claim is [L4].

L1L3L4

1.2

Assume now that $A$ is nonzero and genuinely nonunital. Under the unital Gelfand transform $Γ^{+} : A^{+} \to C (Δ (A^{+}))$ of [L1], the ideal $A$ maps onto the ideal $I_{\infty} : = {f \in C (Δ (A^{+})) : f (χ_{\infty}) = 0}$ : indeed $Γ^{+} (A) \subseteq I_{\infty}$ because $χ_{\infty}$ vanishes on $A$ by definition, and both $Γ^{+} (A)$ and $I_{\infty}$ are linear subspaces of codimension one, $Γ^{+} (A)$ being the image of a codimension-one subspace and $I_{\infty}$ the kernel of the evaluation at $χ_{\infty}$ .

L1L2algebra

1.3

For every $g \in C_{0} (Δ (A))$ the extension $\tilde{g}$ of $g$ by $\tilde{g} (χ_{\infty}) : = 0$ is continuous on $Δ (A^{+}) = Δ (A) \cup {χ_{\infty}}$ : for $ϵ > 0$ the set ${∣ g ∣ \geq ϵ}$ is compact in $Δ (A)$ , so its complement is an open neighbourhood of $χ_{\infty}$ on which $∣ \tilde{g} ∣ < ϵ$ ; conversely the restriction of an $f \in I_{\infty}$ to $Δ (A)$ lies in $C_{0} (Δ (A))$ , because for $ϵ > 0$ the set ${x \in Δ (A) : ∣ f (x) ∣ \geq ϵ}$ is a closed subset of the compact space $Δ (A^{+})$ (it is the intersection of ${∣ f ∣ \geq ϵ}$ with $Δ (A)$ ) that omits $χ_{\infty}$ , hence a compact subset of $Δ (A)$ ; hence restriction and extension are mutually inverse bijections between $I_{\infty}$ and $C_{0} (Δ (A))$ .

L2L3algebra

2.1

The restriction map of [step 1.3] is an isometry: for $f \in I_{\infty}$ one has $\sup_{Δ (A)} ∣ f ∣ = \sup_{Δ (A^{+})} ∣ f ∣$ because $Δ (A)$ is dense in $Δ (A^{+})$ and $∣ f ∣$ is continuous, and $f (χ_{\infty}) = 0$ ; it is also a $*$ -homomorphism for the pointwise operations and conjugation.

step 1.3L2algebra

3.1

Composing the isometric $*$ -isomorphism $Γ^{+} : A \to I_{\infty}$ of [step 1.2] with the isometric $*$ -isomorphism $I_{\infty} \to C_{0} (Δ (A))$ of [step 2.1] gives an isometric $*$ -isomorphism $A \to C_{0} (Δ (A))$ , and unwinding the definitions it sends $a$ to the function $φ \mapsto φ (a)$ on $Δ (A)$ ; this is the Gelfand transform, so the claim is proved in the nonunital nonzero case as well.

step 1.2step 2.1algebra

4.1

Together with [step 1.1] this proves the theorem for every commutative C*-algebra, including the unital and zero cases.

step 1.1step 3.1∎

Remarks

Positivity is pointwise. Under the isomorphism, $a = b^{*} b$ corresponds to $∣ c ∣^{2}$ for $c = Γ (b)$ , hence to a nonnegative function; conversely a nonnegative $h \in C_{0} (Δ (A))$ has a continuous square root $\sqrt{h} \in C_{0} (Δ (A))$ (the compact set ${\sqrt{h} \geq ϵ}$ is ${h \geq ϵ^{2}}$ ), so $h = ∣ \sqrt{h} ∣^{2}$ is the transform of a positive element. This description of positivity is used in Every commutative C star algebra has an approximate unit.
The unitization bookkeeping is not optional. The proof genuinely passes through $A^{+}$ ; the space $Δ (A)$ alone is only locally compact, and its one-point compactification is supplied by the homeomorphism of Character space of the unitization is one-point compactification.

Depends on

Used by

C zero of a locally compact space Example
Every commutative C star algebra has an approximate unit Theorem
Locally compact Gelfand duality Theorem

Dependency tree · two levels

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Sources

Vahid Shirbisheh, Lectures on C-star Algebras, v2 — Theorem 3.1.34 and Proposition 3.1.35, printed pp. 65–66 (standard reference, not scraped)
Dana P. Williams, Lecture Notes on the Spectral Theorem — §4, printed pp. 9–11 (standard reference, not scraped)