An analog of Anderson's result in C* algebra setting [closed]

Question

Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.

For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$

It's known that $W(a)$ is a non-empty convex compact set in the complex plane $\mathbb{C}$.

Let $I$ is an essential ideal of $\mathcal{A}$.

Conjecture: If $a\in I$ and $W(a) \subset \overline{\mathbb{D}}$ and $W(a) \cap \mathbb{T}$ is infinite, then $W(a) =\overline{\mathbb{D}}$

Redefine $W(a) =\{\phi(a):\phi\in \mathcal{A}^{*} : \phi(1)=1=\|\phi\|\}$

Answer 1

No, this is not true. For example, let $\mathcal{A} = \ell^\infty \oplus \ell^\infty$ and $I = \ell^\infty \oplus c_0$. Let $\{\alpha_n\}_n$ be a dense sequence in the right half of the closed unit disk in the complex plane (i.e., the closure of $\{\alpha_n\}_n$ is $\{x \in \overline{\mathbb{D}}: \text{Re}(x) \geq 0\}$). Let $a = (\alpha_1, \alpha_2, \cdots) \oplus 0 \in I$. It is then not hard to check that $W(a)$ is $\{x \in \overline{\mathbb{D}}: \text{Re}(x) \geq 0\}$, contradicting your conjecture.