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Real Structure in Operator Spaces, Injective Envelopes and G-spaces

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Abstract

We present some more foundations for a theory of real structure in operator spaces and algebras, in particular concerning the real case of the theory of injectivity, and the injective, ternary, and \(C^*\)-envelope. We consider the interaction between these topics and the complexification. We also generalize many of these results to the setting of operator spaces and systems acted upon by a group.

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References

  1. Ayupov, S., Rakhimov, A., Usmanov, S.: Jordan, Real and Lie Structures in Operator Algebras, Mathematics and its Applications, 418. Kluwer, Dordrecht (1997)

  2. Blecher, D.P.: Real operator spaces and operator algebras, preprint (2023), to appear Studia Math

  3. Blecher, D.P., Goldstein, S., Labuschagne, L.E.: Abelian von Neumann algebras, measure algebras and \(L^\infty \)-spaces. Expo. Math. 40, 758–818 (2022)

    Article  MathSciNet  Google Scholar 

  4. Blecher, D.P., Kaad, J., Mesland, B.: Operator *-correspondences in analysis and geometry. Proc. Lond. Math Soc. 117, 303–344 (2018)

    Article  MathSciNet  Google Scholar 

  5. Blecher, D.P., Kalantar, M.: Operator space complexification transfigured. Int. Math. Res. Not. (2024). https://doi.org/10.1093/imrn/rnae026

    Article  Google Scholar 

  6. Blecher, D.P., Le Merdy, C.: Operator Algebras and their Modules: An Operator Space Approach. Oxford Univ. Press, Oxford (2004)

    Book  Google Scholar 

  7. Blecher, D.P., Neal, M.: Metric characterizations II. Illinois J. Math. 57, 25–41 (2013)

    Article  MathSciNet  Google Scholar 

  8. Blecher, D.P., Paulsen, V.: Multipliers of operator spaces and the injective envelope. Pacific J. Math. 200, 1–14 (2001)

    Article  MathSciNet  Google Scholar 

  9. Blecher, D.P., Tepsan, W.: Real operator algebras and real positive maps. J. Integral Equ. Oper. Theory 90(5), Paper No. 49, 33 pp (2021)

  10. Blecher, D.P., Wang, Z.: Involutive operator algebras. Positivity 24, 13–53 (2020)

    Article  MathSciNet  Google Scholar 

  11. Breuillard, E., Kalantar, M., Mehrdad, Kennedy, M., Ozawa, N.: \(C^*\)-simplicity and the unique trace property for discrete groups. Publ. Math. Inst. Hautes Études Sci. 126, 35–71 (2017)

  12. Bryder, R.S.: Injective envelopes and the intersection property. J. Oper. Theory 87, 3–23 (2022)

    Article  MathSciNet  Google Scholar 

  13. Cabrera Garcia, M., Rodriguez Palacios, A.: Non-associative Normed Algebras. Vol. 1, The Vidav–Palmer and Gelfand–Naimark theorems. Encyclopedia of Mathematics and its Applications, 154. Cambridge University Press, Cambridge (2014)

  14. Cecco, A.: A categorical approach to injective envelopes, preprint (2023)

  15. Cecco, A.: Ph.D. thesis, University of Houston (In progress)

  16. Chiribella, G., Davidson, K.R., Paulsen, V.I., Rahaman, M.: Counterexamples to the extendibility of positive unital norm-one maps. Linear Algebra Appl. 663, 102–115 (2023)

    Article  MathSciNet  Google Scholar 

  17. Chiribella, G., Davidson, K.R., Paulsen, V.I., Rahaman, M.: Positive maps and entanglement in real Hilbert spaces. Ann. Henri Poincaré 24, 4139–4168 (2023)

    Article  MathSciNet  Google Scholar 

  18. Connes, A.: A factor not anti-isomorphic to itself. Bull. Lond. Math. Soc. 7, 171–174 (1975)

    Article  MathSciNet  Google Scholar 

  19. Effros, E.G., Ruan, Z.-J.: Operator Spaces, London Mathematical Society Monographs. New Series, 23. The Clarendon Press, Oxford University Press, New York (2000)

  20. Ferenczi, V.: Uniqueness of complex structure and real hereditarily indecomposable Banach spaces. Adv. Math. 213, 462–488 (2007)

    Article  MathSciNet  Google Scholar 

  21. Goodearl, K.: Notes on Real and Complex \(C^*\)-algebras, Shiva Mathematics Series, vol. 5. Shiva Publishing Ltd., Nantwitch (1982)

    Google Scholar 

  22. Hamana, M.: Injective envelopes of \(C^*\)-dynamical systems. Tohoku Math. J. 37, 463–487 (1985)

    Article  MathSciNet  Google Scholar 

  23. Hamana, M.: Injective envelopes of dynamical systems. Toyama Math. J. 34, 23–86 (2011)

    MathSciNet  Google Scholar 

  24. Hanche-Olsen, H., Størmer, E.: Jordan Operator Algebras. Pitman Publishing Inc., London (1984)

    Google Scholar 

  25. Isidro, J.M., Rodríguez-Palacios, A.: On the definition of real \(W^*\)-algebras. Proc. Am. Math. Soc. 124, 3407–3410 (1996)

    Article  MathSciNet  Google Scholar 

  26. Kadison, R.V.: Isometries of operator algebras. Ann. Math. 54, 325–338 (1951)

    Article  MathSciNet  Google Scholar 

  27. Kalantar, M., Kennedy, M.: Boundaries of reduced \(C^*\)-algebras of discrete groups. J. Reine Angew. Math. 727, 247–267 (2017)

    Article  MathSciNet  Google Scholar 

  28. Kalantar, M., Scarparo, E.: Boundary maps, germs and quasi-regular representations. Adv. Math. 394, Paper No. 108130, 31 pp (2022)

  29. Kallman, R.R.: A generalization of free action. Duke Math. J. 36, 781–789 (1969)

    Article  MathSciNet  Google Scholar 

  30. Kalton, N.J.: An elementary example of a Banach space not isomorphic to its complex conjugate. Canad. Math. Bull. 38, 218–222 (1995)

    Article  MathSciNet  Google Scholar 

  31. Kennedy, M., Kim, S.J., Li, X., Raum, S., Ursu, D.: The ideal intersection property for essential groupoid \(C^*\)-algebras, preprint (2021). arXiv:2107.03980

  32. Kennedy, M., Schafhauser, C.: Noncommutative boundaries and the ideal structure of reduced crossed products. Duke Math. J. 168, 3215–3260 (2019)

    Article  MathSciNet  Google Scholar 

  33. Li, B.: Real Operator Algebras. World Scientific, River Edge (2003)

    Book  Google Scholar 

  34. Moslehian, M.S., Peralta, A.M., et al.: Similarities and differences between real and complex Banach spaces: an overview and recent developments. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 116, Paper No. 88 (2022)

  35. Palmer, T.: Real \(C^*\)-algebras. Pacific J. Math 35, 195–204 (1970)

    Article  MathSciNet  Google Scholar 

  36. Paulsen, V.I.: Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics, vol. 78. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  37. Pedersen, G.K.: \(C^*\)-algebras and their Automorphism Groups. Academic Press, London (1979)

    Google Scholar 

  38. Pisier, G.: Introduction to Operator Space Theory, London Math. Soc. Lecture Note Series, vol. 294. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  39. Rosenberg, J.: Structure and application of real \(C^*\)-algebras. Contemp. Math. 671, 235–258 (2016)

    Article  MathSciNet  Google Scholar 

  40. Ruan, Z.-J.: On real operator spaces. Acta Math. Sinica 19, 485–496 (2003)

    Article  MathSciNet  Google Scholar 

  41. Ruan, Z.-J.: Complexifications of real operator spaces. Ill. J. Math. 47, 1047–1062 (2003)

    MathSciNet  Google Scholar 

  42. Sharma, S.: Real operator algebras and real completely isometric theory. Positivity 18, 95–118 (2014)

    Article  MathSciNet  Google Scholar 

  43. Tepsan, W.: Real Operator Spaces, Real Operator Algebras, and Real Jordan Operator Algebras, Ph.D. thesis, University of Houston (2020)

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Acknowledgements

We thank the referee for their helpful suggestions and insightful ideas. For example, Theorem 4.12 was inspired by their suggestion that perhaps \(C^*_{e,G}(X) = C^*_e(X)\) for all discrete groups (we had earlier proved this for finite groups). See also Remarks 2 and 3 after Theorem 3.1.

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

    David P. Blecher, Arianna Cecco & Mehrdad Kalantar

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  1. David P. Blecher
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  2. Arianna Cecco
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  3. Mehrdad Kalantar
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Correspondence to David P. Blecher.

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MK and AC are supported by the NSF Grant DMS-2155162. DB is supported by a Simons Foundation Collaboration Grant.

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Blecher, D.P., Cecco, A. & Kalantar, M. Real Structure in Operator Spaces, Injective Envelopes and G-spaces. Integr. Equ. Oper. Theory 96, 14 (2024). https://doi.org/10.1007/s00020-024-02766-7

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