We know that if $A$ is a separable $C^{\*}$-algebra then $K\_1(A)$ is countable.


Can anybody give an example of a C\*-algebra for which $K\_1(A)$ is uncountable?



Example of a C*-algebra whose $K_1$ is uncountable
There must be tons of ways to do this, but a simple one is to start with an uncountable set $X$, equipped with the discrete topology, and consider $c\_0(X)$. There are uncountably many pairwise inequivalent minimal projections in this algebra, so its $K\_0$ group is uncountable. Now use $K\_0(c\_0(X)) \cong K\_1(Sc\_0(X))$ where $SA$ is the suspension of $A$.


