Metrics on the set of states of a \(W^ *\)-algebra (Q1087798)

Let M be a \(W^*\)-algebra, \(\psi_ 1\), \(\psi_ 2\) two states on M and \(q\in [1,\infty)\). We define \[ d_ q(\psi_ 1,\psi_ 2)=\sup \{\sum^{n}_{k=1}| \psi_ 1(p_ k)^{1/q}-\psi_ 2(p_ k)^{1/q}|^ q\}^{1/q} \] where the supremum is taken over all finite families of mutually orthogonal projections of M. We show that \(d_ q\) is a metric on the set of states and all these metrics are uniformly equivalent. We investigate the limit of \(d_ q\) as \(q\to \infty\) and we give alternative expressions. Metrics defined by other authors (Bures, Cantoni, Jauch-Misra and Gibson, Wooters, Gudder) are shown to be monotone functions of some \(d_ q\), hence they are all uniformly equivalent.