C*-Extreme Points and C*-Faces oF the Epigraph iF C*-Affine Maps in *-Rings

Document Type: Research Paper

Author

Department of Mathematics, Faculty of mathematical sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.

10.22072/wala.2018.90202.1184

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.

Keywords


[1] S.K. Berberian, Baer $*$-Rings, New York:  Springer Verlag, 1972.

[2] A. Ebrahimi, On $C^*$-extreme points of the epigraph of $C^*$-affine maps, 9th National conf. on 

     mathematics of payame noor univ., Kerman, Iran, 2017.

[3] A. Ebrahimi, On $C^*$-extreme points of the graph of $C^*$-affine maps, 48th Annu. Iranian math. 

     conf., Hamedan, Iran, 2017.

[4] A. Ebrahimi Meymand and G.H. Esslamzadeh, $C^*$-convexity and $C^*$-faces in $*$-rings, Turk. J.  

     Math., 36 (2012), 131-145.

[5] D.R. Farenick and  P.B. Morenz, $C^*$-extreme points in the generalised state spaces of a 

     $C^*$-algebra, Trans. Am. Math. Soc., 349, (1997), 1725-1748.

[6] D.R. Farenick and  P.B. Morenz, $C^*$-extreme points of some compact $C^*$-convex sets, Proc. Am. 

     Math. Soc., 118 (1993),  765-775.

[7] A. Hopenwasser,  R.L. Moore, and V. I. Paulsen, $C^*$-extreme points, Trans. Am. Math. Soc., 163

     (1981), 291-307.

[8] M. Kian, Epigraph of operator functions, Quaest. Math., 39(5) (2016) 587-594.

[9] R. Loebl and V.I. Paulsen, Some remarks on $C^*$-convexity, Linear Algebra Appl., 35 (1981), 63-78.

[10] B. Magajna, $C^*$-convex sets and completely bounded bimodule homomorphisms, Proc. R. Soc. Edinb., 

       Sect. A, Math., 130(2) (2000), 375-387.

[11] B. Magajna, $C^*$-convexity and the numerical range, Canad. Math. Bull., 43(2), (2000), 193-207.

[12] B. Magajna, On $C^*$-extreme points, Proc. Am. Math. Soc., 129 (2000), 771-780.

[13] P.B. Morenz, The structure of $C^*$-convex sets, Canad. J. Math., 46 (1994), 1007-1026.

[14] I. Nikoufar, A note on non-unital homomorphisms on $C^*$-convex sets in $*$-rings, Acta Univ. M. Belii 

       Ser. Math., 39 (2016), 21-24.