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Contractive projections and real positive maps on operator algebras

Volume 256 / 2021

David P. Blecher, Matthew Neal Studia Mathematica 256 (2021), 21-60 MSC: Primary 17C65, 46L05, 46L70, 47L05, 47L07, 47L30, 47L70; Secondary 46H10, 46B40, 46L07, 46L30, 47L75. DOI: 10.4064/sm190614-7-11 Published online: 20 May 2020


We study contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space. For example we find generalizations and variants of certain classical results on contractive projections on $C^*$-algebras and JB-algebras due to Choi, Effros, Størmer, Friedman and Russo, and others. In fact most of our arguments generalize to contractive ‘real positive’ projections on Jordan operator algebras, that is, on a norm closed space $A$ of operators on a Hilbert space with $a^2 \in A$ for all $a \in A$. We also prove many new general results on real positive maps which are foundational to the study of such maps, and of interest in their own right. Moreover, we prove a new Banach–Stone type theorem for isometries between operator algebras or Jordan operator algebras. An application of this is given to the characterization of symmetric real positive projections.


  • David P. BlecherDepartment of Mathematics
    University of Houston
    Houston, TX 77204-3008, U.S.A.
  • Matthew NealMath and Computer Science Department
    Denison University
    Granville, OH 43023, U.S.A.

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