TY - JOUR ID - 131349 TI - Noncommutative convexity in matricial *-rings JO - Khayyam Journal of Mathematics JA - KJM LA - en SN - AU - Ebrahimi Meymand, Ali AD - Department of Mathematics, Faculty of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran Y1 - 2022 PY - 2022 VL - 8 IS - 1 SP - 7 EP - 16 KW - R-convex set KW - R-face KW - R-extreme point KW - *-ring DO - 10.22034/kjm.2021.255330.2045 N2 - AbstractIn 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 $M_n(R^+)$. UR - https://www.kjm-math.org/article_131349.html L1 - https://www.kjm-math.org/article_131349_bf1a3cb7b7d42d710c6e23c22a8d0862.pdf ER -