mathtools: Birkhoff-Kakutani theorem

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A topological space ${X}$ is said to be metrisable if one can find a metric ${d: X \times X \rightarrow [0,+\infty)}$ on it whose open balls ${B(x,r) := \{ y \in X: d(x,y) generate the topology. There are some obvious necessary conditions on the space <IMG class=latex title={X} alt={X} src=$ in order for it to be metrisable. For instance, it must be Hausdorff, since all metric spaces are Hausdorff. It must also be first countable, because every point ${x}$ in a metric space has a countable neighbourhood base of balls ${B(x,1/n)}$ , ${n=1,2,\ldots}$ .

In the converse direction, being Hausdorff and first countable is not always enough to guarantee metrisability, for a variety of reasons. For instance the long line is not metrisable despite being both Hausdorff and first countable, due to a failure of paracompactness, which prevents one from gluing together the local metric structures on this line into a global one. Even after adding in paracompactness, this is still not enough; the real line with the lower limit topology (also known as the Sorgenfrey line) is Hausdorff, first countable, and paracompact, but still not metrisable (because of a failure of second countability despite being separable).

However, there is one important setting in which the Hausdorff and first countability axioms do suffice to give metrisability, and that is the setting of topological groups:

Theorem 1 (Birkhoff-Kakutani theorem) Let ${G}$ be a topological group (i.e. a topological space that is also a group, such that the group operations ${\cdot: G \times G \rightarrow G}$ and ${()^{-1}: G \rightarrow G}$ are continuous). Then ${G}$ is metrisable if and only if it is both Hausdorff and first countable.

Remark 1 It is not hard to show that a topological group is Hausdorff if and only if the singleton set ${\{\hbox{id}\}}$ is closed. More generally, in an arbitrary topological group, it is a good exercise to show that the closure of ${\{\hbox{id}\}}$ is always a closed normal subgroup ${H}$ of ${G}$ , whose quotient ${G/H}$ is then a Hausdorff topological group. Because of this, the study of topological groups can usually be reduced immediately to the study of Hausdorff topological groups. (Indeed, in many texts, topological groups are automatically understood to be an abbreviation for “Hausdorff topological group”.)

The standard proof of the Birkhoff-Kakutani theorem (which we have taken from this book of Montgomery and Zippin) relies on the following Urysohn-type lemma:

Lemma 2 (Urysohn-type lemma) Let ${G}$ be a Hausdorff first countable group. Then there exists a bounded continuous function ${f: G \rightarrow [0,1]}$ with the following properties:

Note that if ${G}$ had a left-invariant metric, then the function ${f(x) := \max( 1 - \hbox{dist}(x,\hbox{id}), 0)}$ would suffice for this lemma, which already gives some indication as to why this lemma is relevant to the Birkhoff-Kakutani theorem.

Let us assume Lemma 2 for now and finish the proof of the Birkhoff-Kakutani theorem. We only prove the difficult direction, namely that a Hausdorff first countable topological group ${G}$ is metrisable. We let ${f}$ be the function from Lemma 2, and define the function ${d_f := G \times G \rightarrow [0,+\infty)}$ by the formula $\displaystyle d_f( g, h ) := \| \tau_g f - \tau_h f \|_{BC(G)} = \sup_{x \in G} |f(g^{-1} x) - f(h^{-1} x)| \ \ \ \ \ (1)$

where ${BC(G)}$ is the space of bounded continuous functions on ${G}$ (with the supremum norm) and ${\tau_g}$ is the left-translation operator ${\tau_g f(x) := f(g^{-1} x)}$ .

Clearly ${d_f}$ obeys the the identity ${d_f(g,g) = 0}$ and symmetry ${d_f(g,h) = d_f(h,g)}$ axioms, and the triangle inequality ${d_f(g,k) \leq d_f(g,h) + d_f(h,k)}$ is also immediate. This already makes ${d_f}$ a pseudometric. In order for ${d_f}$ to be a genuine metric, what is needed is that ${f}$ have no non-trivial translation invariances, i.e. one has ${\tau_g f \neq f}$ for all ${g \neq \hbox{id}}$ . But this follows since ${f}$ attains its maximum at exactly one point, namely the group identity ${\hbox{id}}$ .

To put it another way: because ${f}$ has no non-trivial translation invariances, the left translation action ${\tau}$ gives an embedding ${g \mapsto \tau_g f}$ , and ${G}$ then inherits a metric ${d_f}$ from the metric structure on ${BC(G)}$ .

Now we have to check whether the metric ${d_f}$ actually generates the topology. This amounts to verifying two things. Firstly, that every ball ${B(x,r)}$ in this metric is open; and secondly, that every open neighbourhood of a point ${x \in G}$ contains a ball ${B(x,r)}$ .

To verify the former claim, it suffices to show that the map ${g \mapsto \tau_g f}$ from ${G}$ to ${BC(G)}$ is continuous, follows from the uniform continuity hypothesis. The second claim follows easily from the neighbourhood base hypothesis, since if ${d_f(g,h) then <IMG class=latex title=$ 1-1/n}" alt="{f(g^{-1} h) > 1-1/n}" src="http://s0.wp.com/latex.php?latex=%7Bf%28g%5E%7B-1%7D+h%29+%3E+1-1%2Fn%7D&bg=ffffff&fg=000000&s=0">.

Remark 2 The above argument in fact shows that if a group ${G}$ is metrisable, then it admits a left-invariant metric. The idea of using a suitable continuous function ${f}$ to generate a useful metric structure on a topological group is a powerful one, for instance underlying the Gleason lemmas which are fundamental to the solution of Hilbert’s fifth problem. I hope to return to this topic in a future post.

Now we prove Lemma 2. By first countability, we can find a countable neighbourhood base

$\displaystyle V_1 \supset V_2 \supset \ldots \supset \{\hbox{id}\}$

of the identity. As ${G}$ is Hausdorff, we must have $\displaystyle \bigcap_{n=1}^\infty V_n = \{\hbox{id}\}.$

Using the continuity of the group axioms, we can recursively find a sequence of nested open neighbourhoods of the identity $\displaystyle U_1 \supset U_{1/2} \supset U_{1/4} \supset \ldots \supset \{\hbox{id}\} \ \ \ \ \ (2)$

such that each ${U_{1/2^n}}$ is symmetric (i.e. ${g \in U_{1/2^n}}$ if and only if ${g^{-1} \in U_{1/2^n}}$ ), is contained in ${V_n}$ , and is such that ${U_{1/2^{n+1}} \cdot U_{1/2^{n+1}} \subset U_{1/2^n}}$ for each ${n \geq 0}$ . In particular the ${U_{1/2^n}}$ are also a neighbourhood base of the identity with $\displaystyle \bigcap_{n=1}^\infty U_{1/2^n} = \{\hbox{id}\}. \ \ \ \ \ (3)$

For every dyadic rational ${a/2^n}$ in ${(0,1)}$ , we can now define the open sets ${U_{a/2^n}}$ by setting

$\displaystyle U_{a/2^n} := U_{1/2^{n_k}} \cdot \ldots \cdot U_{1/2^{n_1}}$

where ${a/2^n = 2^{-n_1} + \ldots + 2^{-n_k}}$ is the binary expansion of ${a/2^n}$ with ${1 \leq n_1 . By repeated use of the hypothesis <IMG class=latex title=$ we see that the ${U_{a/2^n}}$ are increasing in ${a/2^n}$ ; indeed, we have the inclusion $\displaystyle U_{1/2^n} \cdot U_{a/2^n} \subset U_{(a+1)/2^n} \ \ \ \ \ (4)$

for all ${n \geq 1}$ and ${1 \leq a .We now <a href=$ set

$\displaystyle f(x) := \sup \{ 1 - \frac{a}{2^n}: n \geq 1; 1 \leq a with the understanding that <IMG class=latex title={f(x)=0} alt={f(x)=0} src=$ if the supremum is over the empty set. One easily verifies using (4) that ${f}$ is continuous, and furthermore obeys the uniform continuity property. The neighbourhood base property follows since the ${U_{1/2^n}}$ are a neighbourhood base of the identity, and the unique maximum property follows from (3). This proves Lemma 2.

Remark 3 A very similar argument to the one above also establishes that every topological group ${G}$ is completely regular.

Notice that the function ${f}$ constructed in the above argument was localised to the set ${V_1}$ . As such, it is not difficult to localise the Birkhoff-Kakutani theorem to local groups. A local group is a topological space ${G}$ equipped with an identity ${\hbox{id}}$ , a partially defined inversion operation ${()^{-1}: \Lambda \rightarrow G}$ , and a partially defined product operation ${\cdot: \Omega \rightarrow G}$ , where ${\Lambda}$ , ${\Omega}$ are open subsets of ${G}$ and ${G \times G}$ , obeying the following restricted versions of the group axioms:

(Continuity) ${\cdot}$ and ${()^{-1}}$ are continuous on their domains of definition. (Identity) For any ${g \in G}$ , ${\hbox{id} \cdot g}$ and ${g \cdot \hbox{id}}$ are well-defined and equal to ${g}$ . (Inverse) For any ${g \in \Lambda}$ , ${g \cdot g^{-1}}$ and ${g^{-1} \cdot g}$ are well-defined and equal to ${\hbox{id}}$ . ${\hbox{id}^{-1}}$ is well-defined and equal to ${\hbox{id}}$ . (Local associativity) If ${g, h, k \in G}$ are such that ${g \cdot h}$ , ${(g \cdot h) \cdot k}$ , ${h \cdot k}$ , and ${g \cdot (h \cdot k)}$ are all well-defined, then ${(g \cdot h) \cdot k = g \cdot (h \cdot k)}$ .

Informally, one can view a local group as a topological group in which the closure axiom has been almost completely dropped, but with all the other axioms retained. A basic way to generate a local group is to start with an ordinary topological group ${G}$ and restrict it to an open neighbourhood ${U}$ of the identity, with ${\Lambda := \{ g \in U: g^{-1} \in U \}}$ and ${\Omega := \{ (g,h) \in U \times U: gh \in U \}}$ . However, this is not quite the only way to generate local groups (ultimately because the local associativity axiom does not necessarily imply a (stronger) global associativity axiom in which one considers two different ways to multiply more than three group elements together).

Remark 4 Another important example of a local group is that of a group chunk, in which the sets ${\Lambda}$ and ${\Omega}$ are somehow “generic”; for instance, ${G}$ could be an algebraic variety, ${\Lambda, \Omega}$ Zariski-open, and the group operations birational on their domains of definition. This is somewhat analogous to the notion of a “ ${99\%}$ group” in additive combinatorics. There are a number of group chunk theorems, starting with a theorem of Weil in the algebraic setting, which roughly speaking assert that a generic portion of a group chunk can be identified with the generic portion of a genuine group.

We then have

Theorem 3 (Birkhoff-Kakutani theorem for local groups) Let ${G}$ be a local group which is Hausdorff and first countable. Then there exists an open neighbourhood ${V_0}$ of the identity which is metrisable.

Proof: (Sketch) It is not difficult to see that in a local group ${G}$ , one can find a symmetric neighbourhood ${V_0}$ of the identity such that the product of any ${100}$ (say) elements of ${V_0}$ (multiplied together in any order) are well-defined, which effectively allows us to treat elements of ${V_0}$ as if they belonged to a group for the purposes of simple algebraic manipulation, such as applying the cancellation laws ${gh=gk \implies h=k}$ for ${g,h,k \in V_0}$ . Inside this ${V_0}$ , one can then repeat the previous arguments and eventually end up with a continuous function ${f \in BC(G)}$ supported in ${V_0}$ obeying the conclusions of Lemma 2 (but in the uniform continuity conclusion, one has to restrict ${x}$ to, say, ${V_0^{10}}$ , to avoid issues of ill-definedness). The definition (1) then gives a metric on ${V_0}$ with the required properties, where we make the convention that ${\tau_g f(x)}$ vanishes for ${x \not \in V_0^{10}}$ (say) and ${g \in V_0}$ . $\Box$

My motivation for studying local groups is that it turns out that there is a correspondence (first observed by Hrushovski) between the concept of an approximate group in additive combinatorics, and a locally compact local group in topological group theory; I hope to discuss this correspondence further in a subsequent post.

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Wednesday, 15 June 2011

Birkhoff-Kakutani theorem

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