Positive extensions on \(C^*\)-algebras (Q2716841)

For \(k\geq 2\), let \(M_k\) be the \(C^*\)-algebra of all complex \(k\times k\) matrices. In this paper, we consider two problems related to positive extensions. One is to prove that for each unital selfadjoint subspace \({\mathcal S}\subset M_k\), and for each unital \(C^*\)-algebra \({\mathcal A}\), every \(n\)-positive map from \({\mathcal S}\) to \({\mathcal A}\) has an \(n\)-positive extension on \(M_k\) if and only if \(k=2\) and \(n=1\). The other problem is concerned about injective \(C^*\)-algebras. We prove that for unital Abelian \(C^*\)-algebras \({\mathcal B}_i\), \(1\leq i\leq k\), the \(C^*\)-algebra \(\bigoplus^k_{i=1}{\mathcal B}_i\otimes M_{n_i}\) is injective if and only if each \({\mathcal B}_i\) is isometrical \(*\)-isomorphic to \(C({\mathbf X}_i)\), where each \({\mathbf X}_i\) is a c compact Hausdorff and extremely disconnected topological space.