The state on C*-algebra

Question

Let $C$ be a C*-algebra, $A\subset C$ be a C*-subalgebra of $C$ and $B=A'\cap C$ (here, $A'$ denotes the commutant of $A$). If $\xi$ is a state on $C$ and we take an positive element $b\in B$, then the functionals $a\rightarrow \frac{1}{\xi(b)}\xi(ab)$ is also a state?

Answer 1

Not necessarily, you would need to assume e.g. that $b$ is positive.

Let $C = M_2(\mathbb{C})$, and let $A$ be the subalgebra of diagonal matrices. Take \begin{align*} \xi \left( \begin{pmatrix} x & y \\ z & w \\ \end{pmatrix} \right) = \frac{x+w}{2} && b = \begin{pmatrix} 3 & 0 \\ 0 & -1 \end{pmatrix}. \end{align*} or something to that effect. Then taking $a = \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ \end{pmatrix}$, which is a positive element of $A$, you get $\frac{1}{\xi(b)} \xi(ab) < 0$, so the "state" is not positive.