# union of Stone-Cech remainders

Can anyone point me to a reference or further information on the following construction? Let $X$ be a compact metric space such as $[0,1]$. Let $A$ be the commutative pre-C*-algebra consisting of [equivalence classes of] continuous bounded functions $f$ defined on cofinite subsets of $X$. Thus two such functions are equivalent if they agree on a cofinite subset of $X$; and addition and multiplication are performed by restricting to some cofinite subset on which both functions are defined and then adding or multiplying in the usual way. With the obvious norm, $A$ is a pre-C*-algebra with completion $B$. I am interested in the maximal ideal space of $B$.

Here is one way to describe it. Let $Y$ be the absolute of $X$ with the usual map $h: Y\to X$. Then for $x\in X$, $h$ factors through $\beta(Y\setminus\{x\})$ so there is a canonical map $h_x: Y\to \beta (Y\setminus \{x\})$. Define an equivalence relation $*$ on $Y$ by $u*v$ if $h(u)=h(v)$ and $h_x(u)=h_x(v)$ (where $x=h(u)$). Then I think that $Y/*$ is homeomorphic to the maximal ideal space of $B$.

One can think of this space as the union of the Stone-Cech remainders at all the points of $X$, hence the title of my question.

Another way of constructing the space is as follows: for every finite subset $F$ of $X$ let $Y_F=\beta (X\setminus F)$. If $F\subseteq G$ then there is a natural map $f^G_F:Y_G\to Y_F$; this gives us an inverse system (indexed by the finite subsets of $X$). We let $Y_\infty$ be the limit. The preimage $f^{-1}(x)$ of a point $x$ of $X$ under the natural map $F:Y_\infty\to X$ is indeed the remainder of $\beta(X\setminus\lbrace x\rbrace)$; this is so because $f^G_F$ is the identity on that remainder when $x\in F$. Every equivalence class of $A$ induces a continuous real-valued function on $Y_\infty$: fix one representative, $g$, with domain $X\setminus F$, say. For every $G\supseteq F$ we restrict $g$ to $X\setminus G$ and then let $g_G$ be the Čech-Stone extension of that restriction. The limit map of the $g_G$s is a real-valued continuous functionon $Y_\infty$. We obtain a subalgebra of $C(Y_\infty)$ that contains the constant functions and separates the points; by the Stone-Weierstrass theorem it is dense in $C(Y_\infty)$.
• Thanks for this. It looks a more natural way of describing the space from a topological viewpoint than my attempt. I take it from the general silence that people have not considered this space before, or at least have not found anything interesting to say about it. One friend that I consulted observed that if $X=[0,1]$ then $Y_{\infty}$ is not an F-space (in spite of having many subspaces which are F-sapces). This is because every open set of $Y_{\infty}$ has a dense cozero set and thus $Y_{\infty}$ would be extremally disconnected and hence would have to coincide with the absolute of $X$. – Douglas Somerset Mar 29 '12 at 17:23
• Since MathOverflow has been kind enough to re-promote this question, let me raise another issue. In the question I mentioned a dense subalgebra of the C*-algebra of continuous bounded function on this space (i.e continuous bounded functions on cofinite subsets of $X$ - see also KPH's answer). What do the rest of the functions in the C*-algebra look like? They must be continuous except on a co-countable subset $D$ of $X$, but what can $D$ look like? Specifically, if $X=[0,1]$ can $D$ be the set of rational numbers in the $X$? – Douglas Somerset Apr 14 '12 at 7:31