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

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