The Stone-Čech Compactification – Part 2

This is the second post in a series about the Stone-Čech compactification. In the previous post we discussed compactifications and defined the Stone-Čech compactification. In this blog post we will show the existence of the Stone-Čech compactification of an arbitrary space. To do this we will use a surprising tool, C*-algebras. In the final blog post we take a closer look at what’s going on when our space is $\mathbb{R}$.

The C*-algebra of operators on a Hilbert space

Before I define what a C*-algebra is, it is good to see a few examples of C*-algebras. If $H$ is a Hilbert space over the complex numbers, then we define $B(H)$ the space of bounded linear operators from $H$ to $H$. The space $B(H)$ is a Banach space under the operator norm. The space $B(H)$ is also a unital algebra since we can compose operators in $B(H)$ and the identity operator acts as a unit. This composition satisfies the inequality $\Vert ST \Vert \le \Vert S \Vert \Vert T \Vert$ for all $S,T \in B(H)$. Thus $B(H)$ is a Banach algebra. Finally we have an involution $* : B(H) \rightarrow B(H)$ given by the adjoint. That is if $T$ is a bounded operator on $H$, then $T^*$ is the unique bounded operator satisfying

$\langle T h, g \rangle = \langle h, T^* g \rangle$,

for every $h,g \in H$. This involution is conjugate linear and satisfies $(ST)^* = T^*S^*$ for all $S, T \in B(H)$. This involution also satisfies the C*-property that $\Vert T^*T\Vert = \Vert T \Vert^2$ for all $T \in B(H)$.

The C*-algebra of continuous functions on a compact set

If $K$ is a compact topological space, then the Banach space $C(K)$ of continuous functions from $K$ to $\mathbb{C}$ is a unital Banach algebra. The norm on this space is the supremum norm

$\Vert f \Vert = \sup_{x \in K} \vert f(x) \vert$

and multiplication is defined pointwise. This algebra has a unit which is the function that is constantly one. This space also has an involution $* : C(K) \rightarrow C(K)$ given by $f^*(x) = \overline{f(x)}$. This involution is also conjugate linear and it satisfies $(fg)^* = f^*g^* = g^*f^*$ and the C*-property $\Vert f^*f \Vert = \Vert f \Vert^2$.

Both $B(H)$ and $C(K)$ are examples of unital C*-algebras. We will define a unital C*-algebra to be a unital Banach algebra $A$ with an involution $* : A \rightarrow A$ such that

1. The involution is conjugate linear.
2. $(ab)^* = b^*a^*$ for all $a, b \in A$.
3. $\Vert a^*a \Vert = \Vert a \Vert^2$ for all $a \in A$.

Our two examples $B(H)$ and $C(K)$ are different in one important way. The C*-algebra $C(K)$ is commutative whereas in general $B(H)$ is not. In some sense the C*-algebras $C(K)$ are the only commutative unital C*-algebras. It is the precise statement of this fact that will let us define the Stone-Čech compactification of a space.

The Gelfand spectrum

If $A$ is any unital C*-algebra we can define it’s Gelfand spectrum $\sigma(A)$ to be the set of all continuous, non-zero C*-homomorphisms from $A$ to $\mathbb{C}$. That is every $\omega \in \sigma(A)$ is a non-zero continuous linear functional from $A$ to $\mathbb{C}$ such that $\omega (ab) = \omega (a) \omega (b)$ and $\omega (a^*) = \overline{ \omega (a)}$. It can be shown that $\sigma(A)$ is a weak*-closed subset of the unit ball in $A'$, the dual of $A$. Thus by the Banach-Alaoglu theorem, $\sigma(A)$ is compact in the relative weak*-topology.

For example, take $A = C(K)$ for some compact Hausdorff set $K$. In this case we have a map from $K$ to $\sigma(C(K))$ given by $p \mapsto \ \omega _p$ where $\omega _p : C(K) \rightarrow \mathbb{C}$ is the evaluation map given by $\omega _p(f) = f(p)$. This gives a continuous injection from $K$ into $\sigma(C(K))$. It turns out that this map is in fact also surjective and hence a homeomorphism between $K$ and $\sigma(C(K))$. Thus every continuous non-zero homeomorphism on $C(K)$ is of the form $\omega_p$ for some $p \in K$. Thus we may simply regard $\sigma(C(K))$ as being equal to $K$.

The Gelfand spectrum $\sigma(A)$ contains essentially all of the information about $A$ when $A$ is a commutative C*-algebra. This claim is made precise by the following theorem.

Theorem: If $A$ is a unital commutative C*-algebra, then $A$ is C*-isometric to $C(\sigma(A))$, the space of continuous functions $f : \sigma(A) \to \mathbb{C}$. This isomorphism is given by the map $a\in A \mapsto f_a$ where $f_a : \sigma(A) \rightarrow \mathbb{C}$ is given by $f_a( \omega) = \omega (a)$ for all $\omega \in \sigma(A)$.

This powerful theorem tells us that every unital commutative C*-algebra is of the form $C(K)$ for some compact space $K$. Furthermore this theorem tells us that we can take $K$ to be the Gelfand spectrum of our C*-algebra.

The Gelfand spectrum and compactifications

We will now turn back to our original goal of constructing compactifications. If $X$ is a locally compact Hausdorff space then we can define $C_\infty(X)$ to be the space of continuous functions $f : X \rightarrow \mathbb{C}$ that “have a limit at infinity”. By this we mean that for every $f \in C_\infty(X)$ there exists a constant $c \in \mathbb{C}$ such that for all $\varepsilon > 0$, there exists a compact set $K \subseteq X$ such that $|f(x)-c| < \varepsilon$ for all $x \in X \setminus K$. If we equip $C_\infty(X)$ with the supremum norm and define $f^*(x) = \overline{f(x)}$, then $C_\infty(X)$ becomes a commutative unital C*-algebra under point-wise addition and multiplication.

We have a map from $X$ to $\sigma(C_\infty(X))$ given by evaluation. This map is still an homeomorphism onto its image but the map is not surjective if $X$ is not compact. In the case when $X$ is not compact, we have an extra element of $\omega_\infty$ given by $\omega_\infty(f) = \lim f$. Thus we have that $\sigma(C_\infty(X)) \cong X \cup \{\omega_\infty\}$ and hence we have rediscovered the one-point compactification of $X$.

A similar approach can be used to construct the Stone-Čech compactification. Rather than using the C*-algebra $C_\infty(X)$, we will use the C*-algebra $C_b(X)$ of all continuous and bounded functions from $X$ to $\mathbb{C}$. This is a C*-algebra under the supremum norm. We will show that the space $\beta X := \sigma(C_b(X))$ satisfies the universal property of the Stone-Čech compactification. The map $\phi : X \rightarrow \beta X$ is the same one given above. For any $p \in X$, $\phi(p) \in \beta X = \sigma(C_b(X))$ is defined to be the evaluation at $p$ homomorphism $\omega_p$. This map is a homeomorphism between $X$ and an open dense subset of $\beta X$. As in the case of the one point compactification, this map is not surjective. There are heaps of elements of $\beta X \setminus \phi(X)$ as can be seen by the fact that $\beta X$ surjects onto any other compactification of $X$. However it is very hard to give an explicit definition of an element of $\beta X \setminus X$.

We will now show that $\beta X = \sigma(C_b(X))$ satisfies the universal property of the Stone-Čech compactification. Let $(K,\psi)$ be a compactification of $X$. We wish to construct a morphism from $(\beta X,\phi)$ to $(K,\psi)$. That is we wish to find a map $f : \beta X \rightarrow K$ such that $f \circ \phi = \psi$. Note that such a map is automatically surjective as are all morphisms between compactifications. We can embed $C(K)$ in $C_b(X)$ by the map $f \mapsto f \circ \psi$. Since $\psi(X)$ is dense in $K$, we have that this map is a C*-isometry from $C(K)$ to its image in $C_b(X)$. Above we argued that $\sigma(C(K)) \cong K$. The compactification $(K,\psi)$ is in fact isomorphic to $(\sigma(C(K)), \widetilde {\psi})$ where $\widetilde {\psi}(p)=\omega_{\psi(p)}$. Thus we will construct our morphism from $(\beta X, \phi)$ to $(\sigma(C(K)), \widetilde {\psi})$.

Now elements of $\beta X$ are homeomorphism on $C_b(X)$ and elements of $\sigma(C(K))$ are homeomorphism on $C(K)$. Since we can think of $C(K)$ as being a subspace of $C_b(K)$ we can define the map $f : \beta X \rightarrow \sigma(C(K))$ to be restriction to $C(K)$. That is $f(\omega) = \omega_{\mid C(K)}$. Note that since $C(K)$ contains the unit of $C_b(X)$, the above map is well defined (in particular $\omega \neq 0$ implies $\omega_{\mid C(K)} \neq 0$). One can check that the relation $f \circ \phi = \widetilde{\psi}$ does indeed hold since both $\phi$ and $\widetilde{\psi}$ correspond to point evaluation. Thus we have realised the Stone-Čech compactification of $X$ as the Gelfand spectrum of $C_b(X)$.

The above argument can be modified to give a correspondence between compactifications of $X$ and sub C*-algebras of $C_b(X)$ that contain $C_\infty(X)$. This correspondence is given by sending the sub C*-algebra $A$ to $\sigma(A)$ and the point evaluation map. This correspondence is order reversing in the sense that if we have $A_1 \subseteq A_2$ for two sub C*-algebras, then we have a morphism from $\sigma(A_2)$ to $\sigma(A_1)$.

In the final blog post of the series we will further explore this correspondence between compactifications and subalgebras in the case when $X = \mathbb{R}$. Part one of this series can be found here.