the spectrum of self-adjoint element

Question

If $x$ is a self-adjoint element in a $C^*$ algebra $A$,I know the fact $\sigma_A(x)\subset \mathbb{R}$,my question is :Is the following form possible for $\sigma_A(x)$?1.$\sigma_A(x)$ be unions of intervals in $\mathbb{R}$. 2.$\sigma_A(x)$ is the set of isolated points. Does there exist other possibilites?Can anyone show me some examples?Thanks

Answer 1

The spectrum can be any compact subset of $\mathbb R$. For an example where the spectrum is $K$, consider the $C^*$ algebra $C(K)$ of complex-valued continuous functions on $K$, with $x$ the function $x(t) = t$.