Document Type : Original Article
Department of Mathematics, Faculty of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran
In this paper, for every unital $*$-ring $R$, we define the notions of $R$-convexity, as a kind of noncommutative convexity, $R$-face and $R$-extreme point, the relative face and extreme point, for general bimodules over R. The relation between the $C^*$-convex subsets of $R$ and $R$-convex subsets of $M_n(R)$, the set of all $n$ by $n$ matrices with entries in $R$, as well as, the relation between the $C^*$-faces ($C^*$-extreme points) of these $C^*$-convex sets and $R$-faces ($R$-extreme points) of $R$-convex sets in $M_n(R)$ is given. Also, we prove the same results for diagonal matrices in $M_n(R)$. Finally, we show that, if the entries restricted to the positive elements in the unital $*$-ring $R$, then the set of all diagonal matrices is an $R$-face of the set of all lower (upper) triangular matrices, and all of these sets are $R$-faces of $M_n(R^+)$.