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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