For \({\mathcal A}\) any subset of \({\mathcal B}({\mathcal H})\) (the bounded operators on a Hilbert space) containing the unit, and \(\sigma\) and \(\rho\) restrictions of states on \({\mathcal B}({\mathcal H})\) to \({\mathcal A}\), \(ent_{{\mathcal A}}(\sigma | \rho)\)- the entropy of \(\sigma\) relative to \(\rho\) given the information in \({\mathcal A}\)- is defined and given an axiomatic characterisation. It is compared with \(ent^ S_{{\mathcal A}}(\sigma | \rho)\)- the relative entropy introduced by Umegaki and generalised by various authors - which is defined only for \({\mathcal A}\) an algebra. It is proved that ent and \(ent^ S\) agree on pairs of normal states on an injective von Neumann algebra. It is also proved that ent always has all the most important properties known for \(ent^ S:\) monotonicity, concavity, \(w^*\) upper semicontinuity, etc. It is shown, as an extension of the property of semi-discreteness, that a von Neumann algebra (\({\mathcal A},{\mathcal H})\) is injective if and only if there exists a net \((\lambda_{\alpha})_{\alpha \in I}\) of normal completely positive finite rank maps \(\lambda_{\alpha}:{\mathcal B}({\mathcal H})\to {\mathcal A}\) such that \(\lambda_{\alpha}(1)=1\) and such that \(\lambda_{\alpha | {\mathcal A}}\) tends to the identity map on \({\mathcal A}\) in the topology of simple \(w^*\) convergence.