The subgroup of unitaries in $C^*$-algebra is not open

Question

I could show that the subgroup of invertibles are open in a $C^*$-algebra, I know the fact that the subgroup of unitaries in a $C^*$-algebra is not open. I am trying to find an example. Looking for some hints.

Answer 1

Hint: if $U$ is a unitary, $\alpha U$ is invertible for any $\alpha\ne0$, but not unitary if $|\alpha|\ne1$.

Answer 2

The group of unitary elements in a unital $C^*$-algebra $A$ is closed.

Namely, let $(u_n)_{n=1}^\infty$ be a sequence of unitaries which converge to $u \in A$. We have $u_n^*u_n = u_nu_n^* = 1$ so letting $n\to\infty$ gives $u^*u = uu^* = 1$. Hence $u$ is also unitary so group of unitary elements is closed.

Since $A$ is connected, the group of unitary elements cannot also be open.