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.