The geometry of the unit ball of some tensor product spaces (Q1001100)

Let \(A\) be a uniform algebra acting on a compact Hausdorff space \(\Omega\). Assume that \(F\) is an element of the unit ball of \(A \otimes \mathbb{M}_n\) such that the set \[ \Delta = \{x \in \Omega : F(x)F^*(x) = F^*(x)F(x) = 1\} \] is dense in \(\Omega\). The author shows that if there exists a diffuse measure \(\mu\) on \(\Omega\) such that \(\mu(\Omega) > 0\), then \(F\) is an exposed point. If no such measure exists, the unit ball has no exposed points. It is also shown that if, in addition, \(\Omega\) is metrizable, then every pure state of \(A \otimes \mathbb{M}_n\) is a weak*-exposed point.