The structure of the set of all C∗-convex maps in ∗-rings

In this paper, for every unital ∗-ring S, we investigate the Jensen’s inequality preserving maps on C ∗ -convex subsets of S, which we call them C ∗ -convex maps on 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 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.