We introduce Poisson boundaries of II $_1$ factors with respect to density operators that give the traces. The boundary is a von Neumann algebra contains factor and particular example unital completely positive map as introduced by Izumi. Studying inclusion into its boundary, we develop number notions, such double ergodicity entropy, can be seen natural analogues results regarding Furstenberg. ...

background: diabetes mellitus is a serious public health problem worldwide and particularly in developing countries. in algeria, this metabolic disorder occurs with a wide variety or atypical forms that linked to multiple risk factors including local habits and traditions. this study aimed to determine the impact of risk factors (metabolic syndrome, social, cultural, physical activity, family h...