On some convex sets and their extreme points (Q1318083)

ft is shown that the extreme points of the set of Markov maps of \(M(2,K)\), \(K=\mathbb{R}\) or \(\mathbb{C}\), are precisely the set of automorphisms, and that this statement is false for \(M(n,\mathbb{C})\), \(n\geq 4\) and \(M(n,\mathbb{R})\), \(n\geq 3\). We also show that if \(N\subset M\) is a pair of finite factors with Jones index \([M:N]= r^{-1}<\infty\), the extreme points of the set \(C(M,N)= \{x\in M_ +\): \(E_ N x=r\}\) are precisely the projections in \(C(M,N)\) if \(r={1\over 2}\), and in general, for \(r<{1\over 2}\), there may exist other extreme points of \(C(M,N)\).