union of Stone-Cech remainders

2012-03-26T23:02:16Z

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.

Answer by KP Hart for union of Stone-Cech remainders

2012-03-29T08:46:38Z

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 $C(Y_\infty)$.

union of Stone-Cech remainders - MathOverflow

union of Stone-Cech remainders

Answer by KP Hart for union of Stone-Cech remainders