$C^*$ algebra, existence of particular state

Question

If we have $a\in A$ be arbitrary element of $C^*$ algebra $A$. Can we find a faithful state $\phi$ such that $\phi(a) = k$ for $k$ in $spec(a)$?

Answer 1

First, not every C $^*$-algebra has a separable state.

But even then, it may fail, and even when $a\geq0$. Take $A=C_0 (\mathbb R)+\mathbb C\,1$, and $a $ the function $a (t)=\tfrac1{1+t^2} $. The spectrum of $a $ is its range, $[0,1] $. So take $1\in\sigma (a) $ and suppose there is a state $\phi $ with $\phi (a)=1$. Then $$\phi(1-a)=\phi (1)-\phi (a)=0. $$ As $\phi $ is faithful, this would imply $1-a=0$, a contradiction. So no such $\phi $ exists.