# Flat Ultrafilters (Michigan Logic Seminar Sept 21, 2011)

Remember how our about page says that Booles’ Rings is about best practices for an acacdemic homepage? Ok, let’s try one: making notes to talks available.

## Some introductory remarks

Skip this section if you only want mathematics.

Wednesday, I gave a short talk about flat ultrafilters at our Logic Seminar here at the University of Michigan (as announced on Set Theory Talks, the talk was recorded and the video will be online eventually is now online).

When I visited Toronto in June, Ilijas Farah introduced me to this somewhat strange new type of utrafilter on $\omega$. My talk this week was mostly about the results from a 2009 paper by Ilijas Farah, N. Christopher Philips and Juris Steprans [cite source='doi']10.1007/s00208-009-0448-z[/cite], but some of what I am going to write are insights I gained while discussing this notion with François Dorais and Andreas Blass. So, actually, this post is attempting another best practice: notes from reading a paper.

I would like to explain something about the result that P-points are not flat. When I started looking at their paper with Francois Dorais, we first re-proved that selective ultrafilters are not flat — instead of the (possibly stronger) functional analytic result from the paper we used the combinatorial definition of flat ultrafilters. Then we worked on improving that proof and, with Andreas’s help, got it to work for P-points.

Only after all of this happened did we notice that at the very end, the paper had already announced that they have a proof that P-points are not flat. Last week, Ilijas kindly send me some slides with further results on flat ultrafilters; even though the proof for P-points isn’t in there I would guess from the formulation on the slides that our proofs are essentially the same.

Long story short, the proof “P-points are not flat” below is “our” proof even though the result (and most likely the proof) should be credited to at least Steprans (according to the slides).

I will focus on my own interpretation of these notions, i.e., my rephrasing of the definition of flat ultrafilters; they might look different from what you’ll find in their paper even though it is formally the same notion.

Alright, nuff said. My talk was designed to first sketch the main functional analysis result of their paper. So the first two sections will be void of much explanations or proofs. But they motivate why the notion of flat ultrafilter is as strange as it is — it simply came out of those considerations.

## Some functional analysis

Let $H$ be a (the) separable, infinite dimensional Hilbert space, i.e., $$H\cong l_2(\omega) = \{ s: \omega \rightarrow \mathbb{C}, \sum_{n\in \omega} |s_n|^2 < \infty \}.$$

Then $B(H)$ is the space of bounded, linear operators, i.e.,

$$B(H) = \{ T: H\rightarrow H: T \mbox{ linear }, (\exists M)(\forall v\in H) ||Tv|| \leq M||v|| \},$$

and $K(H)$ the space of compact operators

$$K(H) = cl( \{ T \in B(H): T[H] \mbox{ finite dimensional } \}).$$

Finally, the Calkin Algebra is the quotient $C(H) = B(H) / K(H)$.

Farah, Philips and Steprans were interested in the relative commutant of subalgebras $A\subseteq B$: $$F_A(B) = \{ a\in A: (\forall b\in B) ab=ba \}.$$

## The relative commutant in the ultrapower

If $p \in \beta \omega$ is an ultrafilter, we can consider the norm-ultrapowers of $B(H)$ and $C(H)$ which I’ll denote by $B(H)^p, C(H)^p$. That is $$B(H)^p = \ell_\infty(B(H) / \{c \in \ell_\infty(B(H)): \lim_p ||c|| = 0\}.$$ (Similarly for $C(H)^p$.)

Kirchberg had shown that $F_{C(H)^p}(C(H)) = \mathbb{C} \cdot 1$ regardless of $p$ and asked it the analogous statement might be true of $B(H)$.

As a response, Farah, Philips and Steprans showed that $F_{B(H)^p}(B(H))$ depends on the choice of $p$ (under CH); the results from their paper relevant to this talk are as follows.

Theorem (Farah, Philips, Steprans)

• If $p$ is a flat ultrafilter (see below), then $F_{B(H)^p}(B(H)) \neq \mathbb{C} \cdot 1$.

• If $p$ is selective, then $F_{B(H)^p}(B(H)) = \mathbb{C} \cdot 1$; in particular, selective ultrafilters are not flat.

• Flat ultrafilters exist under ZFC.

• It was announced that P-points are not flat.

Curious fact: under CH all ultrapowers of $B(H)$ are isomorphic (via model theoretic arguments). Yet, by the above, if we take a flat and a selective ultrafilter, no such isomoprhism can map $B(H)$ to $B(H)$ (or else the relative commutants would be isomorphic). This is an odd situation.

If you look at the recording you will, as usual, see lots of confusion on my part. In particular, I remembered that the first result in the theorem was an equivalence. It turned out that the paper does not say so and this is an open question. In my defence, the slides did give it as an equivalence.

I’ll only give the proofs of the last two statements as well as some further observations on flat ultrafilters.

## Flat ultrafilters

What are flat ultrafilters? Well, let’s first introduce the assisting structure of a flatness scale, a set of sequences in $[0,1]$.

Definition
A flatness scale H is a countable subset of $$\{ h:\omega \rightarrow [0,1]: h(0) =1, \lim_{i \rightarrow \infty} h(i)=0 \}.$$

Addendum Nothing spectacular so far — a flatness scale is just a bunch of sequences converging to $0$. As such they can have some wild behavior along the way — but flat ultrafilters tame them.

The original definition of flat ultrafilters is then phrased as follows.

Definition [Farah, Philips, Steprans]
An ultrafilter $p$ is flat if there is a flatness scale $H = (h_n: n\in \omega)$ such that for every increasing $f: \omega \rightarrow \omega$ with $f(0) > 0$
$$\lim_{n\to p} ||(h_n – h_n \circ f)||_\infty = 0.$$
In other words, for every $f$ as above and every $\varepsilon>0$, we have $$\{n\in \omega: \sup_i |h_n(i) – h_n(f(i))| < \varepsilon \} \in p .$$
We then say that $H$ is a flatness scale for p.

One of the goals was to find out how I can best think about the notion of flat ultrafilters.

Addendum What to make of this? The first observation is that flat ultrafilters have a flatness scale such that, given $\varepsilon$, almost all of the sequences (with respect to the ultrafilter) drop by at most $\varepsilon$. That’s slow, but it’s even slower! The functions $f$ in the definition should be thought of as vastly increasing (think: dominating family) and they code huge intervals of the form $[n,f(n)]$. But even given those vast intervals, still almost all sequences of the flatness scale drop only by $\varepsilon$ on such long intervals. In other words, they never really drop by much, yet they converge to $0$.

One of the things I found irritating was to think of them as ultrafilters on $\omega$. Thankfully, the enumeration of $H$ does not matter much. In other words, I don’t have to think of $p$ as an ultrafilter on $\omega$ that just happens to come with a map to a flatness scale $H$ — I can really think of $p$ to be an ultrafilter on $H$. So allow me to re-phrase.

Definition [reformulation] Given a flatness scale $H$ and an ultrafilter $p$ on $H$, we say that $p$ is flat, if for every increasing $f:\omega\to\omega$ with $f(0) > 0$ and every $\varepsilon>0$, $$\{ h \in H: \sup_i |h(i) – h (f(i))| < \varepsilon \} \in p .$$
In other words, $$\lim_{p} ||(h – h \circ f)||_\infty = 0.$$ Here $\lim_p$ is the usual limit along the ultrafilter $p$.

You might wonder if we’re not loosing too much information — after all, the original definition did not forbid repititions. We’ll see later that this is not a problem. For now, I will simply stick to the reformulation.

First there’s an obvious question about how complicated a flatness scale can be. So let me show you their construction of flat ultrafilters.

## Constructing Flat Ultrafilters

### A simple observation

For the construction of flat ultrafilters, the key observation is as follows: all we have to do is find a flatness scale $H$ such that the sets $$X_{f,\varepsilon} = \{h \in H : \sup_i|h(i) – h (f(i)| < \varepsilon \}$$
are infinite for every increasing $f:\omega\to\omega$ with $f(0) > 0$ and every $\varepsilon > 0$.

Once we have this, we get the finite intersection property for free since $$X_{\max(f_1,f_2),\min(\varepsilon_1,\varepsilon_2)} \subseteq X_{f_1,\varepsilon_1} \cap X_{f_2,\varepsilon_2}.$$

In other words, any ultrafilter $p$ containing all the sets $X_{f,\varepsilon}$ is flat — witnessed by the flatness scale $H$.

Luckily, It turns out that there is an extremely simple flatness scale with this property.

Proposition [Farah, Philips, Steprans] There is a flatness scale $H$ such that the set
$$X_{f,\varepsilon} = \{h \in H : \sup_i |h(i) – h(f(i)| < \varepsilon \}$$
is infinite for every increasing $f:\omega\to\omega$ with $f(0) > 0$ and every $\varepsilon > 0$.

Proof

• Consider the finite subsets of $\omega$, $[\omega]^{<\omega}$.
• We can think of $s\in [\omega]^{<\omega}$ as encoding a simple step function, dropping by $1/|s|$ at each of the elements of $s$; in other words, we define $h_s : \omega \rightarrow [0,1]$ by
$$h_s(i) = 1- \frac{|s\cap i|}{|s|}.$$

• Clearly, $H_S := \{ h_s: s \in [\omega]^{<\omega} \}$ is a flatness scale (starting at $1$ and converging to zero, in fact being eventually zero).

• Now imagine we’re given some increasing $f:\omega\to\omega$ with $f(0) > 0$ as well as some $\varepsilon >0$.

• Then we can easily find infinitely many $s \in [\omega]^{<\omega}$ with the following properties:
• $1/|s| < \varepsilon$
• With the natural enumeration of $(s_i)_{i\in |s|}$ of $s$, $$s(i+1) \geq f(s(i))$$ for each $i < |s|$.
• But this means that for all $i$, $$h_s(f(i)) \geq h_s(i+1) = \frac{1-|s\cap (i+1)|}{|s|}.$$

• It follows that $\sup_i |h_s(i) – h_s(f(i))| \leq 1/|s| < \varepsilon$.

• Since we can find infinitely many such $s$, the sets $X_{f,\varepsilon}$ are all infinite.

Combining this with the previous observation we have.

Corollary (Farah, Philips, Steprans)
There is a flat ultrafilter via the above flatness scale $H_S$. (Let’s keep that name.)

## Some easy observations

Let’s try to get a feel for what we can actually expect from a flat ultrafilter.

For example, a flat ultrafilter must be non-principal — otherwise the flatness scale is just one map and it’s easy to find $f:\omega \rightarrow \omega$ that drops faster than a single sequence $h$!

### Flatness scales are simple

The above results from Farah, Philips and Steprans is surprising: at first since the flatness scale is extremely simple! It’s only natural to ask how complicated flatness scales can really be. Luckily, it’s always this easy!

Proposition If $p$ is flat, then $H_S$ as above is a flatness scale for $p$.

Proof

• Assume we have a flatness scale for $p$, say $H = (h_n: n\in \omega)$ with some fixed enumeration of $H$.

• We can define another flatness scale as follows: simply choose $h’_n(i)$ to be a multiple of $1/n$ closest to $h_n(i)$, i.e., minimize $|h_n(i) – j/n|$.

• Clearly, $h’_n \in H_S$.

• Then $(h’_n : n\in \omega)$ (and hence $H_S$) is a flatness scale for $p$!
• For $f,\varepsilon$ as in the defintion for flatness, we know $$\{ n: \sup_i |h_n(i) – h_n(f(i)| < \varepsilon / 3 \} \in p.$$
• But $|h’_n(i) – h’_n(f(i))| \leq | h_n(i) – h_n(f(i)| + 2/n$.
• Since all but finitely many $n$ have $1 / n < \epsilon / 3$ we have $$\{ h_n: \sup_i |h’_n(i) – h’_n(f(i)| < \varepsilon \} \supseteq^* \{ h_n: \sup_i |h_n(i) – h_n(f(i)| < \varepsilon / 3 \}.$$
• In other words, both sets lie in $p$ — as desired.

The above proposition tells us that we can now think of flat ultrafilters as simply living on $[\omega]^{<\omega}$, extending a certain filter — I don’t know about you, but for me that’s much easier to picture. We can, therefore, re-phrase our definitions.

Definition revisited. An ultrafilter $p\in \beta( [\omega]^{<\omega}) \cong \beta \omega$ is flat if for all $f:\omega \rightarrow \omega$ with $f(0)>0$ and $\varepsilon>0$,
$$\{ s \in [\omega]^{<\omega}: \sup_i |h_s (i) – h_s(f(i))| < \varepsilon \} \in p .$$

We’re still sort of hiding the fact that we can vary the bijection between $[\omega]^{<\omega}$ and $\omega$. But I still think this phrasing simplifies things a little.

### Convergence in the colums

The next important observation is that in each coordinate $i$, the values of the flatness scale at $i$ for a flat ultrafilter must converge to $1$ (along the ultrafilter).

Lemma If $p$ is flat, then $\lim_{s\rightarrow p} h_s(k) = 1$ for all $k \in \omega$.

Proof

• Suppose $\lim_{s\to p} h_s(k) < 1 – \varepsilon$ for some $k \in \omega$ and $1> \varepsilon > 0$.

• In other words, $X := \{ s: h_s(k) < 1 – \varepsilon \} \in p$.

• Pick some $f:\omega\to\omega$, increasing with $f(0) = k$ (note $k > 0$ since $h_s(0) = 1$).

• Then
$$\sup_i|h_s(i) – h_s (f(i)) | \geq |h_s(0) – h_s(k)| = 1 \geq \varepsilon$$
for every $n \in X$.

• In other words, $X \subseteq \{ s \in [\omega]^{<\omega}: \sup_i |h_s (i) – h_s(f(i))| \geq \varepsilon \}$

• But this contradics $\{ s \in [\omega]^{<\omega}: \sup_i |h_s (i) – h_s(f(i))| < \varepsilon \} \in p$.

## A proof that P-points are not flat.

In their paper Farah, Philips and Steprans announced that P-points are not flat. Francois, Andreas and I overlooked that statement and re-proved the result as follows. (See also the comments at the beginning as well as later.)

Suppose for this section that $p$ is a flat $P$-point as witnessed by $H_S$.

As the first step, we will improve the p-convergence from earlier to actual convergence, i.e., $p$ contains $H\subseteq H_S$ that satisfies $\lim_{s\in H} h_s(k) = 1$ for every $k$.

In the second step, we will show that no ultrafilter can have such a flatness scale.

Lemma If $p$ is a flat P-point, then $p$ contains $H\subseteq H_S$ that satisfies $\lim_{s\in H} h_s(k) = 1$ for every $k$.

Proof

• Since $\lim_{s\to p} h_s(k) = 1$ for all $k$ we know that $$X_{k,l} := \{ s: h_s(k) \geq 1 – 2^{-l} \} \in p,$$ for all $k,l \in \omega$.
• Since $p$ is a P-point, we find $H \in p$ such that $H \subseteq^* X_{k,l}$ for all $k, l \in \omega$.
• But that means precisely that $H$ has $\lim_{s\in H} h_s(k) = 1$ for every $k$.
• Of course, restricting to $H$ does not change the fact that $$\lim_{n\to p} \sup_i|h_s(i) – h_s(f(i))| = 0$$ holds for every increasing function $f:\omega\to\omega$ with $f(0) > 0$ — so we’re done.

We may thus suppose that our original flatness scale satisfies $\lim_{n\to\infty} h_n(k) = 1$ for every $k$. But this is impossible for any ultrafilter!

Lemma No ultrafilter $p$ has a flatness scale $H$ with $\lim_{h\in H} h(k) = 1$.

Proof.

• Else, for any fixed $\color{red}k$, there are only finitely many $\color{blue}h$ such that $h(k) < 3/4$.

• For these finitely many $\color{blue}h$, there is a common $\color{yellow}\ell$ (depending only on $\color{red}k$) such that $\color{blue}h(\color{yellow}\ell) = 0$, and thus $h(m) =0$ for all $m \geq \color{yellow}\ell$.

• In other words, we can define $\color{yellow}f:\omega\to\omega$ be an increasing function with $\color{yellow}f(0) > 0$ such that if $h(k) < 3/4$ then $h(\color{yellow}f(k)) = 0$.

• Since $H$ is a flatness scale, we have that $$\{h : (\forall k)(h(k) – h(\color{yellow}f(k)) \leq 1/4)\} \in p.$$

• So pick one of those $\color{blue}h$, i.e., with $\color{blue}h(k)-\color{blue}h(\color{yellow}f(k)) \leq 1/4$ for all $k$.

• For this fixed $\color{blue}h$ we can, of course, find the first $\color{red}k$such that $\color{blue}h(\color{red}k) < 3/4$. (Note that $\color{red} k > 0$ since $\color{blue}h(0) = 1$.)

• Since $\color{blue}h(\color{red}k-1) \geq 3/4$ and $\color{yellow}f(\color{red}k-1) \geq \color{red}k$, we see that $$\color{blue}h(\color{blue}k-1) – \color{blue}h(\color{red}k) \leq \color{blue}h(\color{red}k-1) – \color{blue}h(\color{yellow}f(\color{red}k-1)) \leq 1/4$$ by our choice of $h$.

• Therefore, $\color{blue}h(\color{red}k) \geq 1/2$.

• By choice of $\color{yellow}f$, we have that $\color{blue}h(\color{yellow}f(\color{red}k)) < 1/4$ and hence $\color{blue}h(\color{red}k) – \color{blue}h(\color{yellow}f(\color{red}k)) > 1/2 – 1/4 = 1/4$ — which contradicts our choice of $h$!

## Flat ultrafilter and rapidity

Since selective ultrafilters are not flat, I was for the longest time under the impression that Q-points and, more generally, rapid ultrafilters might be connected.

(Un)fortunately, this is not the case. An easy argument yields flat but rapid ultrafilters.

Proposition If $p$ is flat and $p \leq_{RK} q$, then $q$ is flat. By contraposition, if $q$ is not flat, then $p$ is not flat.

Proof

• If $H_S$ is a flatness scale for $p$ and $g:[\omega]^{<\omega} \rightarrow [\omega]^{<\omega}$ with $g(q)=p$, then $H_{g[S]}:=\{ h_{g(s)} :s \in \omega^{<\omega} \}$ is a flatness scale for $q$.
• Given $f$ and $\varepsilon$, simply calculate $$\{ s : \sup_i |h_{g(s)}(i) – h_{g(s)}(f(i))| < \varepsilon \} = g^{-1} [ \{ s : \sup_i |h_s(i) - h_s(f(i))| < \varepsilon \}] \in q.$$

This is all we need.

Proposition If there exists a rapid ultrafilter, there exists an ultrafilter both flat and rapid.

• Take $p$ rapid, and $q$ flat.
• Then $q \otimes p$ is rapid and flat
• Flat, since $q\otimes p \geq_{RK} q$.
• By a folklore exercise, if $p$ is rapid, then any $q\otimes p$ is rapid.

That’s about all I actually covered in my talk. There might be a continuation at a later point.

# Supervisor Precognition

I am currently working on a nice little proof. I was initially very confident that I could find the proof — mostly because I already gave a different proof for the same observation, albeit with completely different tools.

This time around the proof will probably end up as a series of stronger and stronger lemmas. Already 3 weeks ago I thought I had shown the strongest intermediary lemma I have formulated so far. Unfortunately, Andreas shot it down, then I found a different proof, and Andreas shot it down again, then I found another proof, and Francois shot it down. As painful as this sounds (and really, really is), I am so lucky to have such colleagues.

Finally, I think I have a found a proof for the intermediary lemma that will live (well, maybe I should wait until Francois sees it later today…). The creepiest thing about this proof, however, is Andreas’s precognition.

When I showed him the failed second attempt (which failed pretty much at line 1…), we discussed the phenomena involved and Andreas made two comments about the problem itself. The first was that due to the setting, an indirect proof seemed to him to be the way to go; second, he gave a very simple example, a special case of the problem that should turn up in some general form. And as you might expect, these two predictions came true — in almost every respect.

The first time I ever heard of such precognition was from a student of Stevo Todorcevic when I was just starting out on my PhD — and it scared the hell out of me to hear that he predicted a complication that the student only found after 3 months of work on that problem. I call this phenomenon ‘supervisor precognition’. It’s not like supervisors in mathematics always have an idea for an actual proof of a problem, usually not even for a strategy. However, supervisors often have a small but brilliant insight into the situation as such and might spot some critical properties far ahead, long before the student actually gets there.

I know that this strength has as much to do with experience as it does with mathematical talent, but it is both annoying and wonderful. Annoying, since I would like to pretend that I could be as productive without these amazing insights from other people. For Hikaru no Go fans, it feels like Akira playing Sai in the first game — it’s a move from far above, so to speak. To use a metaphor that I don’t really like, if we fight through a jungle to get on a mountain top, this precognition maybe is the equivalent of a greater height, allowing to see not the exact path ahead, but a few major obstacles ahead.

I think this precognition is perhaps the most important strength of a good supervisor. On some level, this precognition needs to be present to guide students to their own, independent research. Students should look out for signs of it (and ask around who’s got it) and researchers should try to develop it (if anyone can tell me how, please tell!). Now if only my proof was finished…

Maybe it was the summer heat, maybe the summer break at the UofM or something else. In any case, I did not feel like blogging the last couple of weeks. But this must change! So to get me back to writing I’ll start with something small tiny.

## Unforeseen

I have the great pleasure of spending my PostDoc at the University of Michigan. After spending a winter here 2/3 years ago, I knew a lot of things I could look forwards to — like the amazing grad students.

One of the unforeseen pleasures so far has been to meet Francois Dorais of MathOverflow-Admin fame. Last Friday we talked he told me about a proof by Michael Canjar (sorry for linking to a paywall) on Mathias forcing and there is this small observation that I think is really cool.

## (non) P-points

I mentioned them before, but repetition is never a bad thing.

An ultrafilter $p$ on $\omega$ is called a P-point if for every $f: \omega \rightarrow \omega$ there is $A\in p$ such that $f$ restricted to $A$ is either finite-to-one or constant.

P-points are truly classical ultrafilters having been studied since the dawn of time ultrafilters. They carry interesting properties and Shelah proved that they might not exist (though they do under reasonably weak assumptions like very weak versions of Martin’s Axiom).

The property of P-points somehow tells us that functions drastically ‘changes speed’ on a set in the ultrafilter. If you take a function which is ‘nowhere’ finite-to-one, i.e., every point has an infinite preimage, then a P-point either slows it down completely (by making it constant on a set) or speeds it up extremely (by making it finite-to-one).

But the cool thing Francois showed me (from Canjar’s proof) is what non P-points (so possibly all ultrafilters) can do. They can force any function to slow down in a weird fashion.

## Slowing to identity.

Even though the argument I want to mention holds for arbitrary functions, you should think of quickly growing functions, i.e., strictly increasing functions. So let us pick some $g: \omega \rightarrow \omega$.

Now if $p \in \beta \mathbb{N}$ is not a P-point, then there exists a function $f: \omega \rightarrow \omega$ which is not constant or finite-to-one on any set $A \in p$.

So what about $I_g := \{ n \in \mathbb{N} \ | \ g(f(n)) < n \}$?

On this set, $g \circ f$ is dominated by the identity. That’s slow!!! Just imagine $g$ was the Ackermann function or faster thatn all recursive functions! Suddenly, its only as fast as the identity? Wow…

And now the crazy part.

$I_g \in p$.

That’s right! On a set in $p$ $g\circ f$ slows down like that. That’s crazy!

Proof.

• $f$ is finite-to-one on $\omega \setminus I_g$.
• For $k\in \omega$, $f^{-1}(k) \cap (\omega \setminus I_k) = \{ i \in \omega \ | \ g(k)= g(f(i)) \geq i \}$
• But this is a finite set for any $k$.
• In other words, $f$ is finite-to-one.
• Therefore $\omega \setminus I_g \notin p$.
• Since $p$ is a maximal filter, $I_g \in p$.

That’s all.

### What you can do with this.

Michael Canjar used this fact to show that Mathias forcing with a non P-point adjoins a dominating real. This is not too difficult now since it is easy to see that a Mathias real will dominate all sets in the ultrafilter. But that’s all for today.