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For number field $F$, we consider two case 1) $E$=quadratic extension of $F$ , 2) $E = F \times F$.

Let V be a 2-dim hermition space over E.

In 1) case, by Cartan decompostion $U(2)$ can be decomposed as $KMK$. (here, $K$ is a compact subgroup of $U(2)$ and M={$x \in E^\times | \left\vert x \right\vert \le 1 $}

In 2) case, $U(2)=GL_2 (F)$ the cartan decomposition of $U(2)$ is $KMK$ where $K$ is compact subgroup and $M$={ $\begin{pmatrix} x & 0 \\ 0 & y \end{pmatrix}$ $\in GL_(2)$ | $|x|\le|y|$ }.

My question arises here.

If we consider $U(1)$, then, how can we think the Cartan decomposition of it in the above two cases? I don't know what the above $M$ should be in these case.

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