= Characters of the bilateral absolutely summable convolution algebra
{title2=$\chi_t(\alpha)=\sum_{j\in\mathbb Z}\alpha_je^{ijt}$}
Every <character of an algebra> on $\ell^1(\mathbb Z)$ with <convolution> multiplication is uniquely of this form for $0\leq t<2\pi$. If $e_j$ is the point mass at $j$, then $e_je_k=e_{j+k}$. A character has <norm> one and $e_1e_{-1}=1$, so $|\chi(e_1)|\leq1$ and $|\chi(e_1)^{-1}|\leq1$. Hence $\chi(e_1)$ lies on the <unit circle>; multiplication determines all $\chi(e_j)$ and continuity determines the displayed series. Conversely <absolute convergence> permits rearrangement of the double series, proving multiplicativity. The <maximal ideals> are exactly their kernels: each consists of sequences whose <Fourier series> vanishes at one fixed point of the <unit circle>.
Back to article page
