For local fields $F$, 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 $M$={ $\begin{pmatrix}
 x & 0 \\\
 0 & y
\end{pmatrix} \in GL\_{2}(F)$
 | $|x|\le|y|$ }.


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.



Cartan decomposition of a unitary group?
Let me formulate the Cartan decomposition for a reductive group over a local field (for more information see e.g. <http://www.math.tau.ac.il/~bernstei/Unpublished_texts/unpublished_texts/Bernstein93new-harv.lect.from-chic.pdf> section 2.1): 


Let $G$ be a reductive group defined over over a local field $F$ of characteristic $0$.
Let $A$ be the maximal split torus. That is, the maximal algebraic subgroup defined over $F$ of $G$ which is isomorphic to $(F^{\times})^k$ for some $k$. Note that it is unique up to a conjugation. Let $K$ be the maximal compact subgroup of $G$. Note that in the Archemedian case it is an algebraic subgroup and it is unique up to conjugation. In the non-Archemedian it is not algebraic and often it is not unique up to a conjugation. If your group is defined over the ring of integers $O$ of $F$, you can take $K=G(O)$.


The week version of Cartan decomposition says: $$G=KAK.$$


For the stronger version, let $\Lambda=Mor(F^\times, A)$ be the group of co-characters of $A$. Let $\Lambda^{++}$ be the Wile chamber of $\Lambda$. ruffly spicing it is the fundamental domain for $W:=N\_G(A)/Z\_G(A)$. We have an obvious map $\Lambda \times F^\times \to A$. Let $A^+$ be:\
1. In the Archemedian case the image of $\Lambda^{++} \times \mathbb{R}\_{>0}$.
2. In the non-Archemedian case the image of $\Lambda^{++} \times \pi$, where $\pi$ is the uniformizer of $F$.


Than $$G=KA^+K.$$ Moreover, any $K\times K$ double co-set have a unique element from $A^+$. 


The case of $U(1)$, this is not that interesting. First of all the group is commutative, so we do not need to write $K$ twice. In the non-split case, the group is compact, then $A$ is trivial and the Cartan decomposition says $G=K$. In the split case, $A$ is one dimensional and $A^+$ is either $\mathbb{R}\_{>0}$ or $\pi^\mathbb{Z}$.


