On the angle operator between two projections in a von Neumann algebra (Q1203571)

In the paper under review the relative positions of projections in von Neumann algebras are investigated. Let \(M\) be a von Neumann algebra and let \(p\) and \(q\) be two projections in \(M\). Then the operators \(c(p,q) = (pqp - p \land q)^{1/2}\) and \(s(p,q) = (p_ 0 - c(p,q)^ 2)^{1/2}\), where \(p_ 0 = p - p \land q - p \land q^ \perp\), are called the cosine operator and the sine operator between \(p\) and \(q\), respectively. Furthermore, the angle operator \(\theta (p,q)\) is a positive operator uniquely defined by the equations: \(c(p,q) = \cos (\theta (p,q))\) and \(s(p,q) = \sin (\theta (p,q))\). The relation between the sets \(C(p) = \{c(p,q)\): \(q\) is a projection in \(M\}\) and \(E(p) = \{h \in M\): \(0 \leq h \leq p\) and 1 is not an eigenvalue of \(h\}\) is studied. It is shown that \(C(p) \subset E(p)\) for every projection \(p \in M\). The main result says that \(C(p) = E(p)\) if and only if \(p \prec 1 - p\); if and only if \(\{\theta (p,q)\); \(q\) is a projection in \(M\} = \pi/2\) \(E(P)\). The proofs are elegant and elementary.