Self-adjoint complex function algebras, unitality, and point separation

Let $X$ be a compact Hausdorff space (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not), and let $\mathbb{C}$ be the published complex field (The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$, $\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$) with conjugation and modulus as in Real and imaginary parts, complex conjugation, and modulus and Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive. The space $C(X,\mathbb{C})$ consists of the continuous maps from $X$ to $\mathbb{C}$ (Continuity of a map of topological spaces at a point and globally), where $\mathbb{C}$ carries the metric $d_{\mathbb{C}}(z,w)=\mathrm{\mid }z-w\mathrm{\mid }$ of The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane. That metric is a metric on $\mathbb{C}$, not on $C(X,\mathbb{C})$.

Uniform approximation on this page. For $F,G\subseteq C(X,\mathbb{C})$ and $f\in C(X,\mathbb{C})$, $f$ is uniformly approximable by members of $F$ means that for every $\epsilon >0$ there is $g\in F$ with $d_{\mathbb{C}}(f(x),g(x))<\epsilon$ for every $x\in X$; the uniform closure of $F$ is the set of members of $C(X,\mathbb{C})$ uniformly approximable by members of $F$, and $F$ is uniformly dense when that closure is all of $C(X,\mathbb{C})$. This reading is stated in terms of $d_{\mathbb{C}}$ alone and is therefore available for every $X$, the empty space included. For nonempty $X$ it is exactly density for the topology of uniform convergence of Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on $Y^X$ and on $C(X,Y)$ applied to the metric $d_{\mathbb{C}}$, whose uniform metric $\bar{\rho }$ that item defines only on a nonempty domain.

A subset $A\subseteq C(X,\mathbb{C})$ is a complex function algebra when it is a complex vector subspace under the pointwise operations of The vector space $F^X$ of all functions $X\to F$ with pointwise operations, and $F^n$ as the case $X=n=\{0,1,\dots ,n-1\}$ and is closed under the pointwise multiplication of The ring $R^X$ of all functions from a set $X$ into a ring, with pointwise operations. It is self-adjoint when $f\in A⟹\bar{f}\in A,$ where $\bar{f}(x):=\overline{f(x)}$.

The algebra is unital when it contains every constant complex-valued function, point-separating when every distinct $x,y\in X$ admit $f\in A$ with $f(x)\mathrm{\ne }f(y)$, and nowhere-vanishing when every $x\in X$ admits $f\in A$ with $f(x)\mathrm{\ne }0$.