Journal of Operator Theory

Volume 88, Issue 2, Fall 2022  pp. 365-406.

Transformations preserving the norm of means between positive cones of general and commutative $C^*$-algebras

Authors:  Yunbai Dong (1), Lei Li (2), Lajos Molnar (3), Ngai-Ching Wong (4)
Author institution:(1) Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China
(2) School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China
(3) Bolyai Institute, University of Szeged, H-6720 Aradi vertanuk tere 1, Szeged, Hungary, and Institute of Mathematics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary
(4) Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan, and School of Mathematical Sciences, Tiangong University, Tianjin 300387, China

Summary: We consider a (nonlinear) transformation $T$ on the set of invertible positive elements in a $C^*$-algebra or in an AW$^*$-algebra which preserves the norm of any of the three fundamental means (arithmetic mean, geometric mean, harmonic mean) of positive invertible elements. We show that $T$ extends to a Jordan $*$-isomorphism between the underlying algebras. In the commutative case, we can relax the surjectivity assumption and show that, under the condition that $T$ preserves the norm of any of those means for all finite collections of elements, $T$ is a generalized composition operator.

DOI: http://dx.doi.org/10.7900/jot.2021fev26.2318
Keywords: means in $C^*$-algebras, norm additive maps, Jordan $*$-isomorphisms, composition operators, order isomorphisms