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

Remember how our about page says that Booles' Rings is about best practices for an academic 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).

Flat Ultrafilters (2011/09/21 University of Michigan Logic Seminar) from Peter Krautzberger on Vimeo.

When I visited Toronto in June, Ilijas Farah introduced me to this somewhat strange new type of utrafilter on

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

Then

and

Finally, the Calkin Algebra is the quotient

Farah, Philips and Steprans were interested in the relative commutant of subalgebras

## The relative commutant in the ultrapower

If

(Similarly for

Kirchberg had shown that

As a response, Farah, Philips and Steprans showed that

Theorem(Farah, Philips, Steprans)

If

is a flatultrafilter (see below), then. If

is selective, then ; 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

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

Definition

Aflatness scaleH is a countable subset of

**Addendum** Nothing spectacular so far -- a flatness scale is just a bunch of sequences converging to

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

Definition[Farah, Philips, Steprans]

An ultrafilteris flatif there is a flatness scalesuch that for every increasing with In other words, for every

as above and every , we have We then say that

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

One of the things I found irritating was to think of them as ultrafilters on

Definition[reformulation] Given a flatness scaleand an ultrafilter on , we say that is flat, if for every increasingwith and every , In other words,

Here

You might wonder if we're not loosing too much information -- after all, the original definition did not forbid repetitions. 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**

are **infinite** for every increasing

Once we have this, we get the finite intersection property for free since

In other words, any ultrafilter

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

Proposition[Farah, Philips, Steprans] There is a flatness scalesuch that the set is infinite for every increasing

with and every .

**Proof**

- Consider the finite subsets of
, . - We can think of
as encoding a simple step function, dropping by at each of the elements of ; in other words, we define by

- Clearly,
is a flatness scale (starting at and converging to zero, in fact being eventually zero). - Now imagine we're given some increasing
with as well as some . - Then we can easily find infinitely many
with the following properties: - With the natural enumeration of
of ,

for each . - But this means that for all
,

- It follows that