# A compactification of the non-negative rationals with the discrete topology

Let $S$ be the set of non-negative rational numbers. (If it makes any difference, feel free to take the non-negative dyadic rationals instead.) Let $B=\ell_\infty(S)$; as a ${\rm C}^*$-algebra this is isomorphic to $\ell_\infty({\bf N})$ and so the Gelfand spectrum of $B$ is homeomorphic to $\beta{\bf N}$.

Now let $$A=\{ f \in B \colon f(t) \to 0 \hbox{ as t\to+\infty} \}.$$ (That is, for all $\varepsilon>0$ there exists $K\in [0,\infty)$ such that $|f(t)| < \varepsilon$ whenever $t\in S\cap (K,\infty)$.) This is a closed ideal in $B$, in particular it's an abelian ${\rm C}^*$-algebra so we can consider its Gelfand spectrum $\Phi_A$.

Let $\Omega_S$ be the one-point compactification of $\Phi_A$, which is homeomorphic to a quotient of $\beta {\bf N}$.

Question. Does this compactification of the discrete space $S$ have a name in the literature, and does it admit any description more concrete than saying "it consists of equivalence classes of certain ultrafilters"?

I guess that one could make a more abstract version of the question by looking at a total order on a countable discrete space $T$ and considering the corresponding compactification of $T$. Note that I'm not considering $T$ with the order topology, but the discrete topology.

• The ideal of $A$ of functions supported in $[N,N+1)$ is isomorphic to $\ell_\infty(\mathbb N)$, and $A$ seems to be a direct sum of all these (in the category of C*-algebras). So $\Omega_S$ should be the one point compactification of a countable union of copies of $\beta N$. – Leonel Robert Aug 31 '14 at 16:23
• @LeonelRobert that seems to be correct - thanks. The description feels somehow "non-canonical", since there are lots of ways to chop up the rationals to run the same argument, but maybe that just illustrates the large automorphism group of this space – Yemon Choi Aug 31 '14 at 17:38