Unique state extension and hereditary \(C^*\)-algebras (Q909202)

Let A be a \(C^*\)-algebra and let B be a \(C^*\)-subalgebra of A. This paper provides some necessary and sufficient conditions for every state of B to uniquely extend to a state of A. In particular, it is shown that every state of B uniquely extends to a state of A if and only if B is a hereditary \(C^*\)-subalgebra of A. For a \(C^*\)-dynamical system (A,G,\(\alpha)\), it is also shown that an \(\alpha\)-invariant \(C^*\)- subalgebra B of A is a hereditary \(C^*\)-subalgebra if and only if the \(C^*\)-crossed product \(B\times_{\alpha}G\) is a hereditary \(C^*\)- subalgebra of the \(C^*\)-crossed product \(A\times_{\alpha}G\).