Wavelet and Linear Algebra

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.

Abstract Abstract. In this paper, we define the notion of C*-affine maps in the
unital *-rings and we investigate the C*-extreme points of the graph
and epigraph of such maps. We show that for a C*-convex map f on a
unital *-ring R satisfying the positive square root axiom with an additional
condition, the graph of f is a C*-face of the epigraph of f. Moreover,
we prove some results about the C*-faces of C*-convex sets in *-rings.
Keywords: C*-affine map, C*-convexity, C*-extreme point, C*-face.
MSC(2010): Primary: 52A01; Secondary: 16W10, 46L89.