\[A = \begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}\]
then $A$ is a self-adjoint element of $M_{2}(\complex)$. By Proposition 35.1.5, the mapping $T \mapsto \dpn{Tv, v}{\complex^2}$ is a pure state on $M_{2}(\complex)$ for every unit vector $v \in \complex^{2}$. In particular, if $v = (\sqrt{2}, \sqrt{2})/2$, then $\dpn{Tv, v}{\complex^2}= 0 \not\in \sigma_{M_2(\complex)}(A)$. Therefore
\[\sigma_{M_2(\complex)}(A) \subsetneq \bracsn{\dpn{T, \phi}{M_2(\complex)}|\phi \in P(M_2(\complex))}\]
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