\[\phi: M_{n}(\complex) \to M_{n+1}(\complex) \quad a \mapsto \begin{bmatrix}a & 0 \\ 0 & \frac{\tau(a)}{n}\end{bmatrix}\]
then $\phi$ is an embedding of operator systems, but $\phi(M_{n}(\complex))$ is not a $C^{*}$-subalgebra.
then $\phi$ is an embedding of operator systems, but $\phi(M_{n}(\complex))$ is not a $C^{*}$-subalgebra.
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