This looks like an elementary exercise on group rings (I heard it somewhere), nonetheless it seems to be non-trivial to me. Any references much appreciated.
Suppose that we are given an (infinite) group $G$ and let us consider its group algebra $(\mathbb{C}[G], +, *)$ generated by Dirac deltas $\delta_g\;(g\in G)$. Let $V$ be a linear subspace of $\mathbb{C}[G]$, maximal with respect to the property:
if $f\in V$ then $f(g\,\cdot\, )\in V$ for each $g\in G$.
Furthermore suppose that there exist elements $g_1, \ldots, g_n\in V$ such that every $f\in V$ can be written as $f=a_1 * g_1 + \ldots + a_n * g_n$ for some $a_1, \ldots, a_n \in \mathbb{C}[G]$.
(Thus $V$ is really a maximal left ideal of $\mathbb{C}[G]$.)
I would like to known whether there must exist an idempotent $p\in V$ such that $V=\mathbb{C}[G]*p$.
Notation: By $f(g\cdot\,)$ I understand the function $h\colon G\to \mathbb{C}$ given by $h(k)=f(gk),\; k\in G$