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.