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The real-valued part of a point-separating self-adjoint complex function algebra is separating and has the same common zeros

Statement

Let $X$ be a compact Hausdorff space and let $A \subseteq C (X, C)$ be a self-adjoint point-separating complex function algebra. Its real-valued part $A_{R} : = {f \in A : f (X) \subseteq R}$ is a point-separating real function algebra. The common-zero sets of $A_{R}$ and $A$ are equal. If $A$ is unital, then $A_{R}$ is unital.

Facts & Assumptions

Given: A compact Hausdorff space $X$ and a self-adjoint point-separating complex function algebra $A \subseteq C (X, C)$ .

[L1]

A complex function algebra is a complex vector subspace closed under pointwise multiplication; self-adjointness means $f \in A$ implies $\overline{f} \in A$ , and point separation supplies a member distinguishing each distinct pair (Self-adjoint complex function algebras, unitality, and point separation).

[L2]

Every complex number has a unique form $a + b i$ , with $(a + b i) + (u + v i) = (a + u) + (b + v) i$ and $(a + b i) (u + v i) = (a u - b v) + (a v + b u) i$ ( $C = R [x] / (x^{2} + 1)$ is a field, every element is uniquely $a + b i$ , and every nonzero element has inverse $(a - b i) / (a^{2} + b^{2})$ ).

[L3]

The map $Φ (a + b i) = (a, b)$ is a bijection $C \to R^{2}$ , and it carries complex addition to $(a + u, b + v)$ and multiplication to $(a u - b v, a v + b u)$ ( $C$ is the real coordinate plane, with coordinate arithmetic).

[L4]

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

[L5]

For $z = x + i y$ and $w = u + i v$ , $d_{C} (z, w) = ∣ z - w ∣ = \sqrt{(x - u)^{2} + (y - v)^{2}}$ , and continuity on subsets of $C$ uses this metric (The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane).

[L6]

A real function algebra is a real vector subspace closed under pointwise multiplication; unitality and point separation have their literal constant-function and distinct-pair meanings (Unital, point-separating, and nowhere-vanishing real function algebras on a compact Hausdorff space).

[L7]

For $z = a + b i$ , $Re z = a$ , $Im z = b$ , and $\overline{z} = a - b i$ (Real and imaginary parts, complex conjugation, and modulus).

Proof

technique · direct

1.1

For $f \in A$ , self-adjointness and complex linearity put $u : = (f + \overline{f}) / 2$ and $v : = (f - \overline{f}) / (2 i)$ in $A$ .

L1L2L4

1.2

Sums, real scalar multiples, and products of real-valued members of $A$ are again real-valued by the displayed coordinate formulas in [L2] and [L3], so $A_{R}$ is a real function algebra by [L1] and [L6]; if $A$ is unital, its real constant functions lie in $A_{R}$ .

L1L2L3L6

2.1

The coordinate formulas in [L2], [L3], and [L7] give $u (x) = Re f (x)$ and $v (x) = Im f (x)$ for every $x$ , so $u$ and $v$ are real-valued. They are continuous as maps into $R$ : each is continuous into $C$ as a member of $A$ , and by [L5] the distance $d_{C}$ restricted to the real values agrees with $∣ s - t ∣$ , so the corestriction of a real-valued continuous map to $R$ is again continuous. Hence $u, v \in C (X, R)$ , and [L5] also gives $∣ u (x) - u (y) ∣ \leq d_{C} (f (x), f (y))$ and $∣ v (x) - v (y) ∣ \leq d_{C} (f (x), f (y))$ .

step 1.1L2L3L5L7algebra

3.1

If $x \neq y$ , choose $f \in A$ with $f (x) \neq f (y)$ . Since $Φ$ in [L3] is injective, either $Re f (x) \neq Re f (y)$ or $Im f (x) \neq Im f (y)$ , and step 2.1 places the corresponding separator in $A_{R}$ .

L1L3step 2.1choose

4.1

If every member of $A$ vanishes at $x$ , then every member of $A_{R}$ does. Conversely, if every member of $A_{R}$ vanishes at $x$ , then step 2.1 makes both real and imaginary parts of every $f (x)$ zero, so coordinate uniqueness in [L2] gives $f (x) = 0$ ; hence the two common-zero sets are equal.

step 2.1L2∎

Depends on

Used by

Complex Stone–Weierstrass dichotomy for separating self-adjoint algebras; the unital case is dense Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 100 results over 15 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, proof of Theorem 21.2.14 (standard reference, not scraped)
E. Carlen, Notes on Topology and the Stone-Weierstrass Theorem, proof of Theorem 1.29 (standard reference, not scraped)