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.

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