On certain properties of the relative entropy of states of operator algebras (Q1812947)

Several properties of the relative entropy S(\(\phi\),\(\omega\)) of the states \(\phi\) and \(\omega\) of a \(C^*\)-algebra are established. It is proven that \(S(\phi,\omega)=S(\phi | {\mathcal A}_ 1,\omega | {\mathcal A}_ 1)| S(\omega \circ E,\omega)\) if \({\mathcal A}_ 1\), \({\mathcal A}\) are \(C^*\)-algebras and E is a \(\phi\)-preserving conditional expectation onto \({\mathcal A}_ 1\). A fairly general version of the strong superadditivity is obtained and used to prove the existence of the mean relative entropy for an abstract noncommutative stationary Markov chain.