CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The unital algebra generated by a separating complex family and its conjugates is dense

Statement

Let $X$ be a compact Hausdorff space and let $S \subseteq C (X, C)$ separate points. The smallest unital complex function algebra containing $S \cup {\overline{f} : f \in S}$ is uniformly dense in $C (X, C)$ .

Facts & Assumptions

Given: A compact Hausdorff space $X$ , a point-separating family $S \subseteq C (X, C)$ , and the unital complex function algebra $A$ generated by $S$ and all pointwise conjugates of members of $S$ .

[L1]

Every unital point-separating self-adjoint complex function algebra on a compact Hausdorff space is uniformly dense in $C (X, C)$ (Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense).

[L2]

Complex conjugation respects sums and products and is involutive: $\overline{z + w} = \overline{z} + \overline{w}$ , $\overline{z w} = \overline{z} \overline{w}$ , and $\overline{\overline{z}} = z$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[L3]

A complex function algebra is self-adjoint when it contains $\overline{f}$ with every $f$ , unital when it contains all constants, and point-separating when it distinguishes every distinct pair (Self-adjoint complex function algebras, unitality, and point separation).

Proof

technique · direct

1.1

By construction, $A$ is unital and contains the point-separating family $S$ , so it is unital and point-separating in the sense of [L3].

L3given

1.2

Conjugation maps every generator to another generator, fixes the real constants and conjugates complex constants, and respects sums and products by [L2]; therefore the conjugate of every finite algebraic expression in the generators belongs to $A$ , so $A$ is self-adjoint.

L2L3given

2.1

Steps 1.1 and 1.2 make $A$ a unital point-separating self-adjoint complex function algebra, so [L1] gives exactly the asserted conclusion that $A$ is uniformly dense in $C (X, C)$ .

step 1.1step 1.2L1∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 72 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. M. Erdman, A Companion to Real Analysis, Theorem 21.2.14 (standard reference, not scraped)
E. Carlen, Notes on Topology and the Stone-Weierstrass Theorem, Theorem 1.29 (standard reference, not scraped)