TITLE: Example of a C*-algebra whose $K_1$ is uncountable
QUESTION [8 upvotes]: 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?

REPLY [9 votes]: Nik's answer nails it but if you prefer something representable on a separable Hilbert space then you may consider the suspension $SM$ of any ${\rm II}_1$-factor $M$. Indeed, as $M$ is tracial, $K_0(M) \cong \mathbb R$ and the suspension simply reverses the $K$-groups.
Actually you may produce further commutative examples that are representable on separable Hilbert spaces: for example $S\ell_\infty$, since $K_0(\ell_\infty)$ comprises $\mathbb{Z}$-valued continuous functions on $\beta \mathbb N$.