TY - JOUR ID - 46731 TI - The structure of the set of all $C^*$-convex maps in $*$-rings JO - Wavelet and Linear Algebra JA - WALA LA - en SN - 2383-1936 AU - Ebrahimi Meymand, Ali AD - Department of Mathematics, Faculty of Mathematical Sciences, Vali-e-Asr University of Rafsanjan,\\ Rafsanjan, Islamic Republic of Iran. Y1 - 2020 PY - 2020 VL - 7 IS - 2 SP - 43 EP - 51 KW - $C^*$-affine map KW - $C^*$-convex map KW - $C^*$-face KW - $*$-ring DO - 10.22072/wala.2020.125309.1282 N2 - In this paper, for every unital $*$-ring $\mathcal{S}$, we investigate the Jensen's inequality preserving maps on $C^*$-convex subsets of $\mathcal{S}$, which we call them $C^*$-convex maps on $\mathcal{S}$. We consider an involution for maps on $*$-rings, and we show that for every $C^*$-convex map $f$ on the $C^*$-convex set $B$ in $\mathcal{S}$, $f^*$ is also a $C^*$-convex map on $B$. We prove that  in the unital commutative $*$-rings, the set of all $C^*$-convex maps ($C^*$-affine maps) on a $C^*$-convex set $B$, is also a $C^*$-convex set. In addition, we prove some results for increasing $C^*$-convex maps. Moreover, it is proved that the set of all $C^*$-affine maps on $B$, is a $C^*$-face of the set of all $C^*$-convex maps on $B$ in the unital commutative $*$-rings. Finally, some examples of $C^*$-convex maps and $C^*$-affine maps in $*$-rings are given. UR - https://wala.vru.ac.ir/article_46731.html L1 - https://wala.vru.ac.ir/article_46731_27950884887f822551d8a043e7881345.pdf ER -