Trace Inequalities for von Neumann Algebras

Traces and weights on \(C^*\) -algebras are fundamental tools in operator theory and its applications. For a \(C^*\) -algebra \(\mathscr {A}\) , we denote its subset of projections by \(\mathscr {A}_\textrm{pr}\) . A projection \(P\in \mathscr {A}_\textrm{pr}\) is said to be a minimal projectionminimal projection (or atomatom) if \(P\mathscr {A}P=\mathbb {C}P\) indicating that P majorizes no other nonzero projection in \(\mathscr {A}\) .