On \(C^*\)-algebras cut down by closed projections: Characterizing elements via the extreme boundary (Q5944189)

This paper presents a further investigation into the structure of the \(W^*\)-envelope \(A^{**}\) of a \(C^*\)-algebra \(A\). The authors show that, for a particular class of closed projections \(p\) in \(A^{**}\), the atomic part \(zpap\) of an element \(pap\) of the hereditary subalgebra \(pA^{**}p\) of \(A^{**}\) actually lies in \(zpAp\) if and only if \(pap\) is uniformly weak\(^*\)-continuous on the extreme boundary \(X_0\) of the weak\(^*\)-closed face \(F(p)\) of the quasi-state space \(Q(A)\) of \(A\) corresponding to \(p\). They go on to show that, when \(p\) is semi-atomic, the same result holds if and only if \(pap\) is weak\(^*\)-continuous on the closure \(\overline X\) of the set of elements in \(X_0\) of norm one.