Topological Dynamics

I’m working through the first section on Topological Dynamics.

1) My first questions come towards the end of the proof of Theorem 1.3 on page 119.  We are given that for a unital C^*-algebra monomorphism \pi:C(K)\rightarrow C(L), there is a continuous surjection \Pi:K\rightarrow L such that \pi(f)=f\circ \Pi.  It says a little later that if \pi is a G-map, then \Pi is as well.  In trying to show that, I get so far as g\cdot \pi(f)(x)=g\cdot(f\circ\Pi)(x)=f(\Pi(g^{-1}x)) and \pi(g\cdot f)=(g\cdot f)\circ\Pi(x)=f(g^{-1}\cdot\Pi(x)).  So these are equal since \pi is a G-map, but to get that \Pi is a G-map from here it looks like we need that f is injective.  It also looks like we need this later when they conclude \Pi(1_G)=x_0 from f(x_0)=f(\Pi(1_G)), but it doesn’t seem that we can assume that.

2) My next questions come at the top of page 120 concerning the inverse limit of G-flows.  I’ve been getting used to them as I’m reading through this part of the paper, but I am still unsure about how to prove \lim_{\leftarrow}X_i is nonempty and compact.  So I would like some direction on those.  A smaller question later in the paragraph is in the statement of the universal property: if \phi_i:X\rightarrow X_i are homomorphisms and i\leq j\Rightarrow \phi_i=\phi_{ij}\circ\phi_j\ldots.  We aren’t given the \phi_{ij} mentioned here, so I wasn’t sure if those were supposed to be the \pi_{ij} or if they were supposed to be introduced along with the \phi_i.

3) On page 121, where they start looking at countable discrete G (part B), they say that S(G) is identical to \beta G, the space of ultrafilters on G.  Earlier they defined S(G) to be the space of all continuous homomorphisms \phi:RUC^b(G)\rightarrow \mathbb{C} and in this case RUC^b(G)=l^{\infty}(G).  So I’m not really sure how to get from that to the space of ultrafilters.

4) On page 122 (part C) they mention Veech’s theorem about locally compact G acting freely on S(G).  There is a proof in the appendix and I was wondering if this would be a worthwhile proof to look through.

I also had a general question about reading this (or I suppose any) paper.  There are many times in the paper, even the little bit I’ve read, where something is “clear”, or “easily shown”, etc.  These are usually showing something is continuous or is a homomorphism or something like that.  I was wondering if it is a good idea to stop and work through each of these, since most times they aren’t obvious to me, or if it would be better to try and just get through the paper even if some details aren’t completely worked out.

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2 comments on “Topological Dynamics

  1. Towards my third question, I found this pdf on Stone-Cech compactifications: http://www.math.cornell.edu/~riley/Teaching/Topology2009/essays/chitra.pdf

    I skimmed it over and it looks helpful, so I’ll take a look at it a little closer and see if I still have questions.

  2. Jindrich Zapletal says:

    My apologies, my travel program kept me busy last week, now it is going to be better. Regarding point 2, as a warm-up show that the inverse limit of compact spaces is compact. The topology on the inverse limit comes from preimages of open sets on the spaces forming the limit. Suppose that Oi: i in I forms a cover of the limit without an infinite subcover, and build a sequence of points in the spaces that falls out of the union, this will give a contradiction. Let Pn be the open subset of the space Xn which is the union of all open sets in the cover coming from the n-th coordinate or earlier. Note that the complement of Pn is compact. its image under the phi0n map is compact subset Kn of the space X0 as images of compact spaces under continuous maps are compact again. the Kns form a descending sequence of nonempty compact sets in X0, so there will be a point in the intersection. This will be the first element of your desired sequence. The points later on are obtained in a similar way.

    The generalization to G-flows should be easy once you have this.

    Point 3, The pdf file you found is not a particularly good source, even though everything in it seems to be correct. I will think about a better one. The connection between homomorphisms and ultrafilters is this: suppose for simplicity that the group is Z. if you have an ultrafilter U on Z and a bounded function f on Z, you can form the U-limit of f: just start with a finite interval such that all values of f belong to it, divide it in half, and choose the half such that U-majority of the values of f belong to that half. Then cut that half in half again according to the same rule, and keep doing it until you get a single point –it is the unique real r such that the f-preimage of any neighborhood of x is in U. the map f to U-lim f is a continuous homomorphism. On the other hand, if you see a continuous homomorphism h, just consider the collection of those sets such that h of their characteristic function gives 1. Check that this is an ultrafilter.

    I will get to the other points later

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