analysis – Maths to Share

Total Variation and Marathon Running

Total variation is a way of measuring how much a function $f:[a,b] \to \mathbb{R}$ “wiggles”. In this post, I want to motivate the definition of total variation by talking about elevation in marathon running.

Comparing marathon courses

On July 24th I ran the 2022 San Francisco (SF) marathon. All marathons are the same distance, 42.2 kilometres (26.2 miles) but individual courses can vary greatly. Some marathons are on road and others are on trails. Some locations can be hot and others can be rainy. And some, such as the SF marathon, can be much hillier than others. Below is a plot comparing the elevation of the Canberra marathon I ran last year to the elevation of the SF marathon:

A plot showing the relative elevation over the course of the Canberra and San Francisco marathons. Try to spot the two times I ran over the Golden Gate Bridge during the SF marathon.

Immediately, you can see that the range of elevation during the San Francisco marathon was much higher than the range of elevation during the Canberra marathon. However, what made the SF marathon hard wasn’t any individual incline but rather the sheer number of ups and downs. For comparison, the plot below shows elevation during a 32 km training run and elevation during the SF marathon:

A plot showing the relative elevation over the course of a training run and the San Francisco marathon. The big climb during my training run is the Stanford dish.

You can see that my training run was mostly flat but had one big hill in the last 10 kilometres. The maximum relative elevation on my training run was about 50 meters higher than the maximum relative elevation of the marathon, but overall the training run graph is a lot less wiggly. This meant there were far more individual hills during the marathon and so the first 32 km of the marathon felt a lot tougher than the training run. By comparing these two runs, you can see that the elevation range can hide important information about the difficulty of a run. We also need to pay attention to how wiggly the elevation curve is.

Wiggliness Scores

So far our definition of wiggliness has been imprecise and has relied on looking at a graph of the elevation. This makes it hard to compare two runs and quickly decide which one is wigglier. It would be convenient if there was a “wiggliness score” – a single number we could assign to each run which measured the wiggliness of the run’s elevation. Bellow we’ll see that total variation does exactly this.

If we zoom in on one of the graphs above, we would see that it actually consists of tiny straight line segments. For example, let’s look at the 22nd kilometre of the SF marathon. In this plot it looks like elevation is a smooth function of distance:

The relative elevation of the 22nd kilometre during the SF marathon.

But if we zoom in on a 100 m stretch, then we see that the graph is actually a series of straight lines glued together:

The relative elevation over a 100 metres during the marathon.

This is because these graphs are made using my GPS watch which makes one recording per second. If we place dots at each of these times, then the straight lines become clearer:

The relative elevation over the same 100 metre stretch. Each blue dots marks a point when a measurement was made.

We can use these blue dots to define the graph’s wiggliness score. The wiggliness score should capture how much the graph varies across its domain. This suggests that wiggliness scores should be additive. By additive, I mean that if we split the domain into a finite number of pieces, then the wiggliness score across the whole domain should be the sum of the wiggliness score of each segment.

In particular, the wiggliness score for the SF marathon is equal to the sum of the wiggliness score of each section between two consecutive blue dots. This means we only need to quantify how much the graph varies between consecutive blue dots. Fortunately, between two such dots, the graph is a straight line. The amount that a straight line varies is simply the distance between the y-value at the start and the y-value at the end. Thus, by adding up all these little distances we can get a wiggliness score for the whole graph. This wiggliness score is used in mathematics, where it is called the total variation.

Here are the wiggliness scores for the three runs shown above:

Run	Wiggliness score
Canberra Marathon 2021	617 m
Training run	742 m
SF Marathon 2022	2140 m

The total variation or wiggliness score for the three graphs shown above.

Total Variation

We’ve seen that by breaking up a run into little pieces, we can calculate the total variation over the course of the run. But how can we calculate the total variation of an arbitrary function $f:[a,b] \to \mathbb{R}$ ?

Our previous approach won’t work because the function $f$ might not be made up of straight lines. But we can approximate $f$ with other functions that are made of straight lines. We can calculate the total variation of these approximations using the approach we used for the marathon runs. Then we define the total variation of $f$ as the limit of the total variation of each of these approximations.

To make this precise, we will work with partitions of $[a,b]$ . A partition of $[a,b]$ is a finite set of points $P = \{x_0, x_1,\ldots,x_n\}$ such that:

$a = x_0 < x_1 < \ldots < x_n = b$ .

That is, $x_0, x_1,\ldots,x_n$ is a collection of increasing points in $[a,b]$ that start at $a$ and end at $b$ . For a given partition $P$ of $[a,b]$ , we calculate how much the function $f$ varies over the points in the partition $P$ . As with the blue dots above, we can simply add up the distance between consecutive $y$ values $f(x_i)$ and $f(x_{i-1})$ . In symbols, we define $V_P(f)$ (the variation over $f$ over the partition $P$ ) to be:

$V_P(f) = \sum\limits_{i=1}^n |f(x_i)-f(x_{i-1})|$ .

To define the variation of $f$ over the interval $[a,b]$ , we can imagine taking finer and finer partitions of $[a,b]$ . To do this, note that whenever we add more points to a partition, the total variation over that partition can only increase. Thus, we can think of the total variation of $f$ as the maximum total variation over any partition. We denote the total variation of $f$ by $V(f)$ and define it as:

$V(f) = \sup\{V_P(f) : P \text{ is a partition of } [a,b]\}$ .

Surprisingly, there exist continuous function for which the total variation is infinite. Sample paths of the Brownian motion are canonical examples of continuous functions with infinite total variation. Such functions would be very challenging runs.

Some Limitations

Total variation does a good job of measuring how wiggly a function is but it has some limitations when applied to course elevation. The biggest issue is that total variation treats inclines and declines symmetrically. A steep line sloping down increases the total variation by the same amount as a line with the same slope going upwards. This obviously isn’t true when running; an uphill is very different to a downhill.

To quantify how much a function wiggles upwards, we could use the same ideas but replace the absolute value $|f(x_i)-f(x_{i-1})|$ with the positive part $(f(x_i)-f(x_{i-1}))_+ = \max\{f(x_i)-f(x_{i-1}),0\}$ . This means that only the lines that slope upwards will count towards the wiggliness score. Lines that slope downwards get a wiggliness score of zero.

Another limitation of total variation is that it measures total wiggliness across the whole domain rather than average wiggliness. This isn’t much of a problem when comparing runs of a similar length, but when comparing runs of different lengths, total variation can give surprising results. Below is a comparison between the Australian Alpine Ascent and the SF marathon:

The Australian Alpine Ascent is a 25 km run that goes up Australia’s tallest mountain. Despite the huge climbs during the Australian Alpine Ascent, the SF marathon has a higher total variation. Since the Australian Alpine Ascent was shorter, it gets a lower wiggliness score (1674 m vs 2140 m). For this comparison it would be better to divide each wiggliness score by the runs’ distance.

Summary

Despite these limitations, I still think that total variation is a useful metric for comparing two runs. It doesn’t tell you exactly how tough a run will be but if you already know the run’s distance and starting/finishing elevation, then the total variation helps you know what to expect.

A Nifty Proof of the Cauchy-Schwarz Inequality

This blog post is entirely based on the start of this blog post by Terry Tao. I highly recommend reading the post. It gives an interesting insight into how Terry sometimes thinks about proving inequalities. He also gives a number of cool and more substantial examples.

The main idea in the blog post is that Terry likes to do “arbitrage” on an inequality to improve it. By starting with a weak inequality he exploits the symmetry of the environment he is working in to get better and better inequalities. He first illustrates this with a proof of the Cauchy-Schwarz inequality. The proof given is really nifty and much more memorable than previous proofs I’ve seen. I felt that just had to write it up and share it.

Let $(V,\langle, \rangle)$ be an inner product space. The Cauchy-Schwarz inequality states that for all $v,w \in V$ , $|\langle v, w \rangle | \le \|v\| \|w\|$ . It’s an important result that leads, among other things, to a proof that $\| \cdot \|$ satisfies the triangle inequality. There are many proofs of the Cauchy-Schwarz inequality but here is the one Terry presents.

Since $\langle \cdot, \cdot \rangle$ is positive definite we have $0 \le \langle v-w,v-w\rangle$ . Now using the fact that $\langle \cdot, \cdot \rangle$ is additive in each coordinate we have

$0 \le \langle v,v \rangle -\langle v, w \rangle -\langle w,v\rangle+ \langle w,w \rangle$ .

Since $\langle w,v\rangle = \overline{\langle v,w\rangle}$ , we can rearrange the above expression to get the inequality

$\text{Re}(\langle v,w \rangle) \le \frac{1}{2}\left(\| v \|^2 + \|w\|^2\right)$ .

And now it is time to exploit the symmetry of the above expression and turn this inequality into the Cauchy-Schwarz inequality. The above inequality is worse than the Cauchy Schwarz inequality for two reasons. Firstly, unless $\langle v, w \rangle$ is a positive real number, the left hand side is smaller than $|\langle v,w \rangle|$ . Secondly, unless $\|v\|=\|w\|$ , the right hand side is larger than the quantity $\|v\|\|w\|$ that we want. Indeed we want the geometric mean of $\|v\|^2$ and $\|w\|^2$ whereas we currently have the arithmetic mean on the right.

Note that the right hand side is invariant under the symmetry $v \mapsto e^{i \theta} v$ for any real number $\theta$ . Thus choose $\theta$ to be the negative of the argument of $\langle v,w \rangle$ . This turns the left hand side into $|\langle v,w \rangle |$ while the right hand side remains invariant. Thus we have done our first bit of arbitrage and now have the improved inequality

$| \langle v,w \rangle | \le \frac{1}{2}\left(\|v\|^2 + \|w\|^2\right)$

We now turn our attention to the right hand side and observe that the left hand side is invariant under the map $(v,w) \mapsto \left(c\cdot v, \frac{1}{c} \cdot w\right)$ for any $c > 0$ . Thus by choosing $c$ we can minimize the right hand side. A little bit of calculus shows that the best choice is $c = \sqrt{\frac{\|w\|}{\|v\|}}$ (this is valid provided that $v,w \neq 0$ , the case when $v=0$ or $w=0$ is easy since we would then have $\langle v,w \rangle=0$ ). If we substitute in this optimal value of $c$ , the right hand side of the above inequality becomes

$\frac{1}{2}\left( \left\| \sqrt{\frac{\|w\|}{\|v\|}}\cdot v \right \|^2 +\left \| \sqrt{\frac{\|v\|}{\|w\|}} \cdot w\right \|^2 \right)=\frac{1}{2}\left(\frac{\|w\|}{\|v\|}\|v\|^2+\frac{\|v\|}{\|w\|}\|w\|^2 \right) = \|v\|\|w\|.$

Thus we have turned our weak starting inequality into the Cauchy-Schwarz inequality! Again I recommend reading Terry’s original post to see many more examples of this sort of arbitrage and symmetry exploitation.

The Stone-Čech Compactification – Part 3

In the first blog post of this series we discussed two compactifications of $\mathbb{R}$ . We had the circle $S^1$ and the interval $[-1,1]$ . In the second post of this series we saw that there is a correspondence between compactifications of $\mathbb{R}$ and sub algebras of $C_b(\mathbb{R})$ . In this blog post we will use this correspondence to uncover another compactification of $\mathbb{R}$ .

Since $S^1$ is the one point compactification of $\mathbb{R}$ we know that it corresponds to the subalgebra $C_\infty(\mathbb{R})$ . This can be seen by noting that a continuous function on $S^1$ is equivalent to a continuous function, $f$ , on $\mathbb{R}$ such that $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to -\infty} f(x)$ both exist and are equal. On the other hand the compactification $[-1,1]$ corresponds to the space of functions $f \in C_b(\mathbb{R})$ such that $\lim_{x \to +\infty} f(x)$ and $\lim_{x \to - \infty}f(x)$ both exist (but these limits need not be equal).

We can also play this game in reverse. We can start with an algebra $A \subseteq C_b(\mathbb{R})$ and ask what compactification of $\mathbb{R}$ it corresponds to. For example we may take $A$ to be the following sub algebra

$\{g+h \mid g,h \in C(\mathbb{R}), g(x+2\pi)=g(x) \text{ for all } x \in \mathbb{R} \text{ and } \lim_{x \to \pm \infty} h(x)=0 \}.$

That is $A$ contains precisely those functions in $C_b(\mathbb{R})$ that are the perturbation of a $2\pi$ periodic function by a function that vanishes at both $\infty$ and $- \infty$ . Since any constant function is $2 \pi$ periodic, we know that $C_\infty(\mathbb{R})$ is contained in $A$ . Thus, as explained in the previous blog post, we know that $\sigma(A)$ corresponds to a compactification of $\mathbb{R}$ .

Recall that $\sigma(A)$ consists of all the non-zero continuous C*-homeomorphisms from $A$ to $\mathbb{C}$ . The space $\sigma(A)$ contains a copy o $\mathbb{R}$ as a subspace. A point $t \in \mathbb{R}$ corresponds to the homeomorphism $\omega_t$ given by $\omega_t(f)=f(t)$ . There are also a circle’s worth of homeomorphisms $\nu_p$ given by $\nu_p(f) = \lim_{k \to \infty} f(2 \pi k+p)$ for $p \in [0,2\pi)$ . The homeomorphism $\nu_p$ isolates the value of the $2\pi$ periodic part of $f$ at the point $p$ . This is because if $f = g+h$ with $g$ a $2 \pi$ periodic function and $h$ a function that vanishes at infinity. Then

$\nu_p(f) = \lim\limits_{k \to \infty} (g(2 \pi k + p)+h(2 \pi k + p)) = g(p)+\lim\limits_{k \to \infty} h(2 \pi k + p) = g(p)$ .

Thus we know that the topological space $\sigma(A)$ is the union of a line and a circle. We now just need to work out the topology of how these to spaces are put together to make $\sigma(A)$ . We need to work out which points on the line are “close” to our points on the circle. Suppose we have a sequence of real numbers $(t_n)_{n \in \mathbb{N}}$ and a point $p \in [0,2\pi)$ such that the following two conditions are satisfied. Firstly $| t_n | \rightarrow \infty$ and secondly there exists a sequence of integers $(k_n)_{n \in \mathbb{N}}$ such that $\lim\limits_{n \to \infty} (t_n - 2\pi k_n) = p$ . Then we would have that $\omega_{t_n}$ converges to $\nu_p$ in $\sigma(A)$ .

Thus we know that the copy of $\mathbb{R}$ in must spiral towards the copy of $S^1$ in $\sigma(A)$ and that this spiraling must happen as we approach either positive infinity or negative infinity. Thus we can realise $\sigma(A)$ as the following subset of $\mathbb{R}^3$ that looks a bit like Christmas tree decoration:

Here we have the black circle sitting in the x,y plane of $\mathbb{R}^3$ . The red line is a copy of $\mathbb{R}$ that spirals towards the circle. Negative numbers sit below the circle and positive numbers sit above. On left is a sideways view of this space and on the right is the view from above. I made these images in geogebra. If you follow this link, you can see the equations that define the above space and move the space around.

This example shows just how complicated the Stone-Čech compactification $\beta \mathbb{R}$ of $\mathbb{R}$ must be. Our relatively simple algebra $A$ gave this quite complicated compactification shown above. The Stone-Čech compactification surjects onto the above compactification and corresponds to the huge algebra of all bounded and continuous function from $\mathbb{R}$ to $\mathbb{C}$ .

References

The Wikipedia page on the Stone-Čech compactification and these notes by Terrence Tao were where I first learned of the Stone-Čech compactification. I learnt about C*-algebras in a great course on operator algebras run by James Tenner at ANU. We used the textbook A Short Course on Spectral Theory by William Averson which has some exercises at the end of chapter 2 about the Stone-Čech compactification of $\mathbb{R}$ . The example of the algebra $A \subseteq C_b(\mathbb{R})$ used in this blog post came from an assignment question set by James.

Parts one and two of this series can be found here.

The Stone-Čech Compactification – Part 2

This is the second post in a series about the Stone-Čech compactification. In the previo us 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

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

The Stone-Čech Compactification – Part 1

Mathematics is full of surprising connections between two seemingly unrelated topics. It’s one of the things I like most about maths. Over the next few blog posts I hope to explain one such connection which I’ve been thinking about a lot recently.

The Stone-Čech compactification connects the study of C*-algebras with topology. This first blog post will set the scene by explaining the topological notion of a compactification. In the next blog post, I’ll define and discuss C*-algebras and we’ll see how they can be used to study compactifications. In the final post we will look at a particular example.

Compactifications

Let $X$ be a locally compact Hausdorff space. A compactification of $X$ is a compact Hausdorff space $K$ and a continuous function $\phi : X \rightarrow K$ such that $\phi$ is a homeomorphism between $X$ and $\phi(X)$ and $\phi(X)$ is an open, dense subset of $K$ . Thus a compactification is a way of nicely embedding the space $X$ into a compact space $K$ .

For example if $X$ is the real line $\mathbb{R}$ , then the circle $S^1$ and the stereographic projection is a compactification of $X$ . In this case the image of $X$ is all of the circle apart from one point. Since the circle is compact, this is indeed a compactification. This is an example of a one-point compactification. An idea which we will return to later.

Comparing Compactifications

A space $X$ will in general have many compactifications and we would like to compare these different compactifications. Suppose that $(K_1,\phi_1)$ and $(K_2, \phi_2)$ are two compactifications. Then a morphism from $(K_1,\phi_1)$ to $(K_2,\phi_2)$ is a continuous map $f : K_1 \rightarrow K_2$ such that $\phi_2 = f \circ \phi_1$ . That is, the below diagram commutes:

Hence we have a morphism from $(K_1,\phi_1)$ to $(K_2,\phi_2)$ precisely when the embedding $\phi_2 : X \rightarrow K_2$ extends to a map $f : K_1 \rightarrow K_2$ . Since $\phi_1(X)$ is dense in $K_1$ , if such a function $f$ exists, it is unique.

Note that the map $f$ must be surjective since $\phi_2(X)$ is dense in $K_2$ and $g(K_1)$ is a closed set containing $\phi_2(X)$ . We can think of the compactification $(K_1,\phi_1)$ as being “bigger” or “more general” than the compactification $(K_2, \phi_2)$ as $f$ is a surjection onto $K_2$ . More formally we will say that $(K_1,\phi_1)$ is finer than $(K_2, \phi _2)$ and equivalently that $(K_2, \phi _2)$ is coarser than $(K_1, \phi _1)$ whenever there is a morphism from $(K_1, \phi_1)$ to $(K_2,\phi_2)$ . Note that the composition of two morphism of compactifications is again a morphism of compactifications. Thus we can talk about the category of compactifications of a space. The compactifications $(K_1, \phi _1)$ and $(K_2, \phi _2)$ are isomorphic if there exists a morphism $g$ between $(K_1, \phi _1)$ and $(K_2, \phi _2)$ such that $g$ is a homeomorphism between $K_1$ and $K_2$ .

For example again let $X = \mathbb{R}$ . Then the closed interval $[-1,1]$ is again a compactification of $\mathbb{R}$ with the map $(x,y) \mapsto \frac{x}{1+|x|}$ which maps $\mathbb{R}$ onto the open interval $(-1,1)$ . We can then create a morphism from $[-1,1]$ to the circle by sending the endpoints of $[-1,1]$ to the top of the circle and sending the interior of $[-1,1]$ to the rest of the circle. We can perform this map in such a way that the following diagram commutes:

Thus we have a morphism from the compactification $[-1,1]$ to the compactification $S^1$ . Thus the compactification $[-1,1]$ is finer than the compactification $S^1$ .

Now that we have a way of comparing compactifications of $X$ we can ask about the existence of extremal compactifications of $X$ . Does there exists a compactification of $X$ that is the coarser than any other compactification? Or one which is finer than any other? From a category-theory perspective, we are interested in the existence of terminal and initial objects in the category of compactifications of $X$ . We will first show the existence of a coarsest or “least general” compactification.

The one point compactification

A coarsest compactification would be a terminal object in the category of compactifications. That is a compactification $(\alpha X, i)$ with the property that for all compactification $(K, \phi)$ there is a unique extension $g : K \rightarrow \alpha X$ of $i : X \rightarrow \alpha X$ . If such a coarsest compactification exists, then it is unique up to isomorphism. Thus we can safely refer to the coarsest compactification.

The one point compactification of $X$ is constructed by adding a single point denoted by $\infty$ to $X$ . It is defined to be the set $\alpha X = X \sqcup \{ \infty\}$ with the topology given by the collection of open sets in $X$ and sets of the form $\alpha X \setminus K$ for $K \subseteq X$ a compact subset. The map $i : X \rightarrow \alpha X$ is given by simply including $X$ into $\alpha X = X \sqcup \{\infty\}$ .

The one point compactification is the coarsest compactification of $X$ . Let $(K,\phi)$ be another compactification of $X$ . Then the map $g : K \rightarrow \alpha X$ given by

$g(y) = \begin{cases} i(\phi^{-1}(y)) & \text{if } y \in \phi(X),\\ \infty & \text{if } y \notin \phi(X), \end{cases}$

is the unique morphism from $(K,f)$ to $(\alpha X, i)$ .

The Stone-Čech compactification

A Stone-Čech compactification of $X$ is a compactification $(\beta X,j)$ which is the finest compactification of $X$ . That is $(\beta X,j)$ is an initial object in the category of a compactifications of $X$ and so for every compactification $(K,f)$ there exists a unique morphism from $(\beta X,j)$ to $(K,f)$ . Thus any embedding $f : X \rightarrow K$ , has a unique extension $g : \beta X \rightarrow K$ . As with coarest compactification of $X$ , the Stone-Čech compactification of $X$ is unique up to isomorphism and thus we will talk of the Stone-Čech compactification.

Unlike the one point compactification of $X$ , there is no simple description of $\beta X$ even when $X$ is a very simple space such as $\mathbb{N}$ or $\mathbb{R}$ . To show the existence of a Stone-Čech compactification of any space $X$ we will need to make a detour and develop some tools from the study of C*-algebras which will be the topic of the next blog post.