Extreme points of intervals in \(C^*\)-algebras
Let A be a \(C^*\)-algebra with identity and let S denote its unit sphere, P its positive cone and \(A^ s\) its self-adjoint part. Given two distinct self-adjoint elements a and b with \(a\leq b\) and \(ab=ba\), we define the following order interval: \[ P_{a,b}=\{x\in A^ s: xa=ax,\quad xb=bx\quad and\quad a\leq x\leq b\}. \] The main theorem characterizes the extreme points \(ext(P_{a,b})\) of the convex set \(P_{a,b}\) as those members which satisfy the functional equation \((x- a)(b-x)=0.\) By taking special values for the vectors a and b we obtain two classical results of S. Sakai as consequences; \(viz.,\) ext(P\(\cap S)=class\) of \(projections,\) ext(A\({}^ s\cap S)=class\) of self-adjoint unitary elements.
scientific article; zbMATH DE number 3899583
scientific article