Non-commutative central limits
For the non-commutative or quantum system \((M,\rho)\), where M is a von Neumann algebra of operators on a Hilbert space and \(\rho\) is a vector state on M, a CCR-\(C^*\)-algebra of normal fluctuations \(\overline{W(H,\sigma_{\rho})}\) of the system (M,\(\rho)\) and quasi-free state \(\omega_ s\) of \(\overline{W(H,\sigma_{\rho})}\) are defined and investigated (here \(H=M_{sa}\) and \(\sigma_{\rho}(x,y)=-i\rho ([x,y])\), \(x,y\in M_{sa})\). Using \(\overline{W(H,\sigma_{\rho})}\) and \(\omega_ s\) the central limit theorem for the operators from M is formulated and stability of this theorem under small perturbations is studied. In particular, two systems (M,\(\rho)\) and \((M,\rho '_ n)\), where \(\rho '_ n\) is a weakly perturbed state with respect to \(\rho\), are considered and it is proved that the central limit of the relative entropies of \(\rho\) and \(\rho '_ n\) yields the relative entropy of the limit quasi-free representations.
\(CCR\)-\(C^*\)-algebra of normal fluctuations
scientific article; zbMATH DE number 4138552
scientific article