The structure of the set of all $C^*$-convex maps in $*$-rings

Abstract

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.