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A closed unital real function algebra is $C (Y, R)$ on its indistinguishability quotient

Statement

Let $X$ be a compact Hausdorff space and let $A \subseteq C (X, R)$ be a uniformly closed unital real function algebra. Let $Y_{A} = X / \sim_{A}$ be its indistinguishability quotient and $q_{A} : X \to Y_{A}$ the canonical projection (The quotient that identifies points indistinguishable by a real function algebra). Then $Y_{A}$ is compact Hausdorff, every $f \in A$ descends uniquely to a continuous $\tilde{f} \in C (Y_{A}, R)$ with $f = \tilde{f} \circ q_{A}$ , and $A ⟶ C (Y_{A}, R), f ⟼ \tilde{f},$ is a unital algebra isomorphism. When $X$ is nonempty it is also isometric for the uniform metric, which For a nonempty set $X$ and a metric space $(Y, d)$ the uniform metric $\overset{ˉ}{ρ} (f, g) = \sup_{x} \min {d (f (x), g (x)), 1}$ is a metric on $Y^{X}$ defines only on a nonempty domain. Thus $A$ is canonically the full continuous real function algebra on the quotient whose points it separates.

Facts & Assumptions

Given: A compact Hausdorff space $X$ , a uniformly closed unital real function algebra $A \subseteq C (X, R)$ , its indistinguishability quotient $Y_{A}$ , and the canonical surjection $q_{A} : X \to Y_{A}$ .

[L1]

The relation $x \sim_{A} y$ means $f (x) = f (y)$ for every $f \in A$ , and $Y_{A} = X / \sim_{A}$ carries the quotient topology of $q_{A}$ (The quotient that identifies points indistinguishable by a real function algebra).

[L2]

For a quotient map $q : X \to Y$ , a set $V \subseteq Y$ is open exactly when $q^{- 1} [V]$ is open in $X$ (The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection).

[L3]

If $f : X \to Y$ is continuous and $X$ is compact, then $f [X]$ is a compact subset of $Y$ (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, clause 1).

[L4]

Every unital point-separating real function algebra on a compact Hausdorff space is uniformly dense in the full continuous real function space (Real Stone–Weierstrass theorem for compact Hausdorff spaces).

[L5]

A topological space is Hausdorff when any two distinct points have disjoint open neighbourhoods (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not).

[L6]

The function $d_{R} (s, t) = ∣ s - t ∣$ is a metric on $R$ , and its metric topology is the usual topology (The absolute value makes $R$ a metric space: $d (x, y) = ∣ x - y ∣$ is a metric, its open balls are the intervals $(x - r, x + r)$ , and it is unbounded).

[L7]

In every metric space, distinct points have disjoint open balls; hence every metric space is Hausdorff (Distinct points of a metric space have disjoint balls around them).

[L8]

For a nonempty set $X$ and a metric space $(Y, d)$ , the uniform metric on $Y^{X}$ is $\overset{ˉ}{ρ} (f, g) = \sup_{x \in X} \min {d (f (x), g (x)), 1}$ (For a nonempty set $X$ and a metric space $(Y, d)$ the uniform metric $\overset{ˉ}{ρ} (f, g) = \sup_{x} \min {d (f (x), g (x)), 1}$ is a metric on $Y^{X}$ ).

Proof

technique · direct

1.1

For $f \in A$ , [L1] makes $f$ constant on each fibre of $q_{A}$ , so there is a unique function $\tilde{f} : Y_{A} \to R$ satisfying $f = \tilde{f} \circ q_{A}$ .

L1given

1.2

The quotient map $q_{A}$ is continuous and surjective by [L2], so [L3] makes $Y_{A} = q_{A} [X]$ compact.

L2L3

2.1

For every open $U \subseteq R$ , one has $q_{A}^{- 1} [{\tilde{f}}^{- 1} [U]] = f^{- 1} [U]$ , which is open because $f$ is continuous; [L2] therefore makes $\tilde{f}$ continuous.

step 1.1L2

2.2

The descent map is an injective unital algebra homomorphism, because descent respects the pointwise operations and $f = \tilde{f} \circ q_{A}$ determines $\tilde{f}$ on the surjective image. When $X$ is nonempty it is moreover isometric: $q_{A}$ is onto, so the two families of values coincide and $\sup_{x \in X} ∣ g (q_{A} (x)) - h (q_{A} (x)) ∣ = \sup_{y \in Y_{A}} ∣ g (y) - h (y) ∣$ . For $X = \emptyset$ both function spaces have the unique empty function as their only member, so the map is a bijection; no isometry is asserted there, since [L8] defines the uniform metric only on a nonempty domain.

step 1.1L8algebra

3.1

If $[x]_{A} \neq [y]_{A}$ , then [L1] supplies $f \in A$ with $f (x) \neq f (y)$ . By [L6] and [L7], the distinct real values $\tilde{f} ([x]_{A})$ and $\tilde{f} ([y]_{A})$ have disjoint open neighbourhoods; their inverse images under the continuous $\tilde{f}$ are disjoint open neighbourhoods of the two classes, so $Y_{A}$ is Hausdorff by [L5].

L1step 2.1L5L6L7

3.2

The descended family $\tilde{A} : = {\tilde{f} : f \in A}$ is a unital real function algebra because descent respects the pointwise operations, and it separates points by the definition of $\sim_{A}$ in [L1].

step 1.1step 2.1L1algebra

4.1

Since $Y_{A}$ is compact Hausdorff by steps 1.2 and 3.1, [L4] makes $\tilde{A}$ uniformly dense in $C (Y_{A}, R)$ .

step 1.2step 3.1step 3.2L4

5.1

For $g \in C (Y_{A}, R)$ and $ε > 0$ , step 4.1 supplies $f \in A$ with $∣ \tilde{f} (y) - g (y) ∣ < ε$ for every $y \in Y_{A}$ ; since $f = \tilde{f} \circ q_{A}$ , the same bound reads $∣ f (x) - g (q_{A} (x)) ∣ < ε$ for every $x \in X$ , so $g \circ q_{A}$ is uniformly approximable by members of $A$ . As $A$ is uniformly closed, $g \circ q_{A} \in A$ , and its unique descent in step 1.1 is $g$ because $q_{A}$ is surjective. Hence the descent map is surjective, and with step 2.2 it is the claimed unital algebra isomorphism, isometric whenever $X \neq \emptyset$ .

step 1.1step 4.1step 2.2given∎

Depends on

Used by

Endpoint-duplicating functions on [0,1] become all continuous functions on the endpoint quotient Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 119 results over 19 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 21.2.15 (standard reference, not scraped)

A closed unital real function algebra is C(Y,R)C(Y,R) on its indistinguishability quotient