Completely positive maps and *-isomorphism of \(C^*\)-algebras (Q1109300)

Let \(A\), \(B\) be unital \(C^*\)-algebras. \(\mathcal K_ A = \{\phi|\phi\) are all completely positive linear maps from \(M_ n(C)\) to \(A\) with \(\| a(\phi)\|\leq1\}\). \[ a(\varphi) = \begin{pmatrix} \varphi(e_{11}) & \ldots & \varphi(e_{1n}) \\ \vdots && \vdots \\ \varphi(e_{n1}) & \ldots & \varphi(e_ {nm}) \end{pmatrix}, \text{ where \(\{e_{ij}\}\) is the matrix unit of \(M_ n(C)\).} \] Let \(\alpha\) be the natural action of \(SU(n)\) on \(M_ n(C)\). For \(n\geq 3\), if \(\Phi\) is an \(\alpha\)-invariant affine isomorphism between \({\mathcal K}_ A\) and \(\mathcal K_ B\), \(\Phi(0)=0\), then \(A\) and \(B\) are \(*\)-isomorphic. In this pap er a counterexample is given for the case \(n=2\).