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Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define

$$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$ which is a C*-algebra. Is there an accessible topological description of the maximal ideal space of $c_{\mathcal{U}}$? At least for ultrafilters?

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1 Answer 1

(I will use $\omega$ for the ultrafilter since using a lowercase letter will improve readability)

The way I see it, your algebra $c_\omega$ is simply $$ c_\omega=\{f:\ f(\omega)=0\}\subset C(\beta \mathbb N). $$ So you can make the identification $c_\omega=C_0(\beta\mathbb N\setminus\{\omega\})$.

Note that $c_\omega$ is an ideal in $C(\beta\mathbb N)$, so the ideals in $c_\omega$ are ideals in $C(\beta\mathbb N)$. This is important because, with $\beta\mathbb N$ being compact, the ideals of $C(\beta\mathbb N)$ are precisely the sets of functions that annihilate a fixed closed subset.

Thus, the ideals of $c_\omega$ are the sets of the form $$ \{f\in C_0(\beta\mathbb N\setminus\{\omega\}):\ f=0\ \mbox{ on }\{\omega\}\cup K\} $$ for a fixed closed $K\subset\mathbb N\setminus\{\omega\}$.

We conclude that maximal ideals of $c_\omega$ are of the form $$ \{f\in C_0(\beta\mathbb N\setminus\{\omega\}):\ f(\omega)=f(\eta)=0\} $$ for some $\eta\in\beta\mathbb N\setminus\{\omega\}$.

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Slightly more general situation (arbitrary locally compact space instead of $\beta\mathbb N$ with one point omitted) is described in Conway's Functional analysis, p. 222, Exercise 7: If $X$ is locally compact, show that $x\mapsto \delta_x$ is a homeomorphism of $X$ onto the maximal ideal space of $C_0(X)$. Exercise 8 gives a description of all maximal ideals of $C_0(X)$. –  Martin Sleziak Jun 17 '13 at 7:00

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