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Let $G$ = $\mathbb{Z_n}$ and let $A$ = $\ell^1(G)$ with convolution, over $\mathbb{C}$. Let $A_{\mathbb{R}}$ denote the subring of real valued functions in $A$, so $A_{\mathbb{R}}$ is an algebra over $\mathbb{R}$.

I'm being asked to find the maximal ideals of $A_{\mathbb{R}}$, their dimensions, and the corresponding quotient fields.

I've found that $m$ = {$f \in A_{\mathbb{R}}$ : $\sum_{m=0}^{n-1} f([m])$ = 0} is a maximal ideal of dimension n-1 with corresponding quotient field $\mathbb{R}$. I also know that $m$ isn't the only the maximal ideal as the function which takes 1 on every element of G is not invertible and isn't in $m$. I don't really know where to go with this problem. I spent a fair bit of time trying to determine the invertibility of sums of delta functions, which seemed to depend on the parity of n, but that didn't lead anywhere. I'm way up the river on this, and any pointers in the right direction would be helpful.

For context, the previous two parts of the problem were to determine the maximal ideal space of $A$ and whether or not the Gelfand transform of $A$ was isometric.

Edit: With the help of the suggestions below, I've been able to identify a number of maximal ideals of $A_\mathbb{R}$. Namely, $\mathfrak{m} \cap A_\mathbb{R}$, where $\mathfrak{m}$ is a maximal ideal in $A$. I want to say that these are the only maximal ideals of $A_\mathbb{R}$ (or at least I think I do), but I am unsure as to how to approach this.

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    I see now that $\mathfrak{m} \cap A_\mathbb{R}$ is a maximal ideal. I would like to say that every maximal ideal of $A_\mathbb{R}$ is of this form, but I don't see how to approach something like that.2011-10-02

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Note that $f \in A_{\mathbb R}$ iff $\hat{f}(\overline{\alpha}) = \overline{\hat{f}(\alpha)}$ for all $\alpha$. If $M_\alpha$ is the maximal ideal $\{f: \hat{f}(\alpha) = 0\}$ of $A$, then $M_{\alpha} \cap A_{\mathbb R} = M_{\overline{\alpha}} \cap A_{\mathbb R}$. For any $\alpha$, there is $h_\alpha \in A_{\mathbb R}$ such that $\widehat{h_\alpha}(\alpha) = \widehat{h_\alpha}(\bar{\alpha}) = 1$ and $\widehat{h_\alpha}(g) = 0$ for all other $g$.
Use this to show that $M_\alpha \cap A_{\mathbb R}$ is a maximal ideal in $A_{\mathbb R}$.

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    Nevermind, I've figured it out. Thanks for the help everyone.2011-10-05