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Maps on Positive Cones of C*-algebras

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Abstract

We prove that a surjective map (on the positive cones of unital C*-algebras) preserves the minimum spectrum values of harmonic means if and only if it has a Jordan *-isomorphism extension to the whole algebra. We represent weighted geometric mean preserving bijective maps on the positive cones of prime C*-algebras in terms of Jordan *-isomorphisms of the algebras. We also characterize order isomorphisms and orthoisomorphisms of the projection lattice of the von Neumann algebra of all bounded linear operators on a Hilbert space, answering an open question arisen by Dye. Finally, we give a description for Fuglede—Kadison determinant preserving maps on the positive cone of a finite von Neumann algebra and improve Gaal and Nayak’s work on this topic.

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Acknowledgements

The authors would like thank the referee(s) for the comments about the original manuscript to clarify the work in this paper.

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Authors and Affiliations

  1. Department of Mathematics, Louisiana Christian University, Pineville, LA, 71359, USA

    Ming Chu Gao

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, P. R. China

    Gui Mei An

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  1. Ming Chu Gao
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  2. Gui Mei An
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Correspondence to Gui Mei An.

Additional information

The first named author was supported by Louisiana Christian University Carolyn and Adams Dawson Professorship Fund (220 625151 5302); the second named author was supported by the NSFC (Grant No. 11101220) and the Fundamental Research Funds for the Central Universities (Grant No. 96172373)

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Gao, M.C., An, G.M. Maps on Positive Cones of C*-algebras. Acta. Math. Sin.-English Ser. 39, 387–398 (2023). https://doi.org/10.1007/s10114-023-1356-y

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  • DOI: https://doi.org/10.1007/s10114-023-1356-y

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