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Real Operator Algebras and Real Positive Maps

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Abstract

We present some foundations for a theory of real operator algebras and real Jordan operator algebras, and the various morphisms between these. A common theme is the ingredient of real positivity from papers of the first author with Read, Neal, Wang, and other coauthors, which we import to the real scalar setting here.

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Acknowledgements

Several results here (in particularly, many in Sects. 2–4) are from the Ph.D. thesis of the second author W. Tepsan [44]. Other complementary facts, alternative proofs, and additional theory may be found there, and there may also possibly be a forthcoming paper on that topic. We thank M. Kalantar for several helpful discussions, which we mention in more detail in and around Theorem 2.11 (see also 1.1). We also thank Ángel Rodríguez Palacios for some comments. In terms of future directions, the noncommutative topology of real Jordan operator algebras in the spirit of [9] looks like a fruitful topic that should be pursued elsewhere, as well as some other features of the real positive cone that have not been explored for the real case here or in [44]. There are no doubt some interesting ‘operator space aspects’ of the theory of real associative operator algebras that are worth developing. In addition, there are several open questions stated in this paper. Finally it would be very worthwhile to find many more interesting examples of real Jordan operator algebras, particularly if they might be important in quantum physics. It does not seem hard to find examples of real Jordan operator algebras beyond those already mentioned in this paper or [44]. For example if one looks inside the upper triangular (real or complex) matrices \({\mathcal U}_n\) one quickly spots several new examples. E.g. the matrices in \({{\mathcal {U}}}_2({{\,\mathrm{{{\mathbb {C}}}}\,}})\) with \(a_{22} = \overline{a_{11}}\) and \(a_{12}\) real. Or certain matrices in \({{\mathcal {U}}}_n\) that are ‘symmetric’ with respect to the diagonal connecting \(a_{n1}\) to \(a_{1n}\).

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, TX, 77204-3008, USA

    David P. Blecher

  2. International College of Digital Innovation, Chiang Mai University, 239 Nimmanhaemin Road, Suthep, Muang, Chiang Mai, 50200, Thailand

    Worawit Tepsan

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  1. David P. Blecher
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Correspondence to David P. Blecher.

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Blecher, D.P., Tepsan, W. Real Operator Algebras and Real Positive Maps. Integr. Equ. Oper. Theory 93, 49 (2021). https://doi.org/10.1007/s00020-021-02665-1

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