$R$-CONVEX SUBSETS OF BIMODULES OVER $*$-RINGS

Document Type : Original Manuscript

Authors

1 Department of Mathematics, Payame Noor University, Tehran, Iran.

2 Department of Mathematics, Faculty of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran.

Abstract

Let $M$ and $N$ be bimodules over the unital $*$-rings $R$ and $B$, respectively.
We investigate the notion of $R$-convexity and the corresponding notion of $R$-extreme points.
We discuss the effect of an $f$-homomorphism on
$R$-convex subsets and its\linebreak $R$-extreme points.
Namely, we declare how an $f$-homomorphism
from $M$ to $N$ carries $R$-convex subsets and its $R$-extreme points to $B$-convex subsets and its $B$-extreme points
and vice versa.\linebreak Moreover, we confirm that the $R$-convex hull of invariant subsets under $f$-homomorphisms
remains invariant.

Keywords

References

 1. S. K. Berberian, Baer ∗-rings, Springer Verlag, Berlin, 1972.
2. A. Ebrahimi and G. H. Esslamzadeh, C∗-convexity and C∗-faces in ∗-rings, Turk. J. Math., 36 (2012), 131–145.
3. E. G. Effros and S. Winkler, Matrix Convexity: Operator Analogues of the Bipolar and Hahn-Banach Theorems, J. Funct. Anal., 144(1) (1997), 117–152.
4. D. R. Farenick, C∗-convexity and matricial ranges, Canad. J. Math., 44 (1992), 280–297.
5. D. R. Farenick, Krein-Milman type problems for compact matricially convex sets, Linear Algebra Appl., 162–164 (1992), 325–334. 
 6. D. R. Farenick and P. B. Morenz, C∗-extreme points of some compact C∗- convex sets, Proc. Amer. Math. Soc., 118 (1993), 765–775.
7. A. Hopenwasser, R. L. Moore and V. I. Pualsen, C∗-extreme points, Trans. Amer. Math. Soc. 266(1) (1981), 291–307.
8. T. W. Hungerford, Algebra, Springer-Verlag, New York, Inc. 1974.
9. A. Jencova, On the convex structure of process POVMs, J. Math. Phys., 57(1) (2016), Article ID: 015207.
10. R. Loebl, V. I. Paulsen, Some remarks on C∗-convexity, Linear Alg. Appl., 35 (1981), 63–78.
11. B. Magajna, C∗-convex sets and completely bounded bimodule homomorphisms, Proc. Roy. Soc. Edinburgh Section A., 130(2) (2000), 375–387.
12. B. Magajna, C∗-convexity and the Numerical Range, Canad. Math. Bull., 43(2) (2000), 193–207.
13. B. Magajna, On C∗-extreme points, Proc. Amer. Math. Soc., 129(3) (2000), 771–780.
14. P. B. Morenz, The structure of C∗-convex sets, Canad. J. Math., 46 (1994), 1007–1026.
15. I. Nikoufar, A note on non-unital homomorphisms on C∗-convex sets in ∗-rings, Acta Univ. M. Belii Ser. Math., 24, (2016), 21–24. 
Journal of Algebraic Systems
Volume 12, Issue 1
September 2024
Pages 91-103

Files

Share

How to cite

Statistics