Contractive linear preservers of absolutely compatible pairs between \(\hbox {C}^*\)-algebras

  • Nabin K. Jana
  • Anil K. Karn
  • Antonio M. PeraltaEmail author
Original Paper


Let a and b be elements in the closed ball of a unital C\(^*\)-algebra A (if A is not unital we consider its natural unitization). We shall say that a and b are domain (respectively, range) absolutely compatible (\(a\triangle _d b\), respectively, \(a\triangle _r b\), in short) if \(\Big | |a| -|b| \Big | + \Big | 1-|a|-|b| \Big | =1\) (respectively, \(\Big | |a^*| -|b^*| \Big | + \Big | 1-|a^*|-|b^*| \Big | =1\)), where \(|a|^2= a^* a\). We shall say that a and b are absolutely compatible (\(a\triangle b\) in short) if they are both range and domain absolutely compatible. In general, \(a\triangle _d b\) (respectively, \(a\triangle _r b\) and \(a\triangle b\)) is strictly weaker than \(ab^*=0 \) (respectively, \(a^* b =0\) and \(a\perp b\)). Let \(T: A\rightarrow B\) be a non-expansive bounded linear mapping between C\(^*\)-algebras. We prove that if T preserves domain absolutely compatible elements (i.e., \(a\triangle _d b\Rightarrow T(a)\triangle _d T(b)\)) then T is a triple homomorphism. A similar statement is proved when T preserves range absolutely compatible elements. It is finally shown that T is a triple homomorphism if, and only if, T preserves absolutely compatible elements.


Absolute compatibility Commutativity \(\hbox {C}^{*}\)-algebra von Neumann algebra Projection Partial isometry Linear absolutely compatible preservers 

Mathematics Subject Classification

Primary 46L10 Secondary 46B40 46L05 



We would like to thank the Referee for her/his valuable suggestions in a detailed report, and for sharing with us the result in Lemma 3.6. Third author partially supported by Junta de Andalucía Grant FQM375.


  1. 1.
    Akemann, C.A., Pedersen, G.K.: Ideal perturbations of elements in C\(^*\)-algebras. Math. Scand. 41(1), 117–139 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barton, T.J., Dang, T., Horn, G.: Normal representations of Banach Jordan triple systems. Proc. Am. Math. Soc. 102(3), 551–555 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brešar, M.: Jordan mappings of semiprime rings. J. Algebra 127, 218–228 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Burgos, M., Fernández-Polo, F.J., Garcés, J.J., Martínez Moreno, J., Peralta, A.M.: Orthogonality preservers in C\(^*\)-algebras, JB\(^*\)-algebras and JB\(^*\)-triples. J. Math. Anal. Appl. 348, 220–233 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Burgos, M., Fernández-Polo, F.J., Garcés, J.J., Peralta, A.M.: Orthogonality preservers revisited. Asian Eur. J. Math. 2(3), 387–405 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Burgos, M., Garcés, J.J., Peralta, A.M.: Automatic continuity of biorthogonality preservers between compact C\(^*\)-algebras and von Neumann algebras. J. Math. Anal. Appl. 376, 221–230 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Fernández-Polo, F.J., Martínez, J., Peralta, A.M.: Geometric characterization of tripotents in real and complex JB\(^*\)-triples. J. Math. Anal. Appl. 295, 435–443 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Fernández-Polo, F.J., Martínez, J., Peralta, A.M.: Contractive perturbations in JB\(^*\)-triples. J. Lond. Math. Soc. 85(2), 349–364 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Fernández-Polo, F.J., Peralta, A.M.: Partial isometries: a survey. Adv. Oper. Theory 3(1), 87–128 (2018)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Garcés, J.J., Peralta, A.M.: Orthogonal forms and orthogonality preservers on real function algebras. Linear Multilinear Algebra 62(3), 275–296 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Gardner, L.T.: Linear maps of C\(^*\)-algebras preserving the absolute value. Proc. Am. Math. Soc. 76(2), 271–278 (1979)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Gardner, L.T.: A dilation theorem for \(|.|\)-preserving maps of C\(^*\)-algebras. Proc. Am. Math. Soc. 73(3), 341–345 (1979)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Guan, Y., Wang, C., Hou, J.: Additive maps on C\(^*\)-algebras commuting with \(|\cdot |^k\) on normal elements. Bull. Iran. Math. Soc. 41, 85–98 (2015)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Horn, G.: Characterization of the predual and ideal structure of a JBW\(^*\)-triple. Math. Scand. 61(1), 117–133 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Kadison, R.V.: Isometries of operator algebras. Ann. Math. 54, 325–338 (1951)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Karn, A.K.: Algebraic orthogonality and commuting projections in operator algebras. Acta Sci. Math. (Szeged) 84, 323–353 (2018)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Leung, C.-W., Tsai, C.-W., Wong, N.-C.: Linear disjointness preservers of W\(^*\)-algebras. Math. Z. 270, 699–708 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Liu, J.-H., Chou, C.-Y., Liao, C.-J., Wong, N.-C.: Disjointness preservers of AW\(^*\)-algebras. Linear Algebra Appl. 552, 71–84 (2018)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Liu, J.-H., Chou, C.-Y., Liao, C.-J., Wong, N.-C.: Linear disjointness preservers of operator algebras and related structures. Acta Sci. Math. (Szeged) 84(1–2), 277–307 (2018)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Molnár, L.: Two characterisations of additive \(^*\)-automorphisms of \(B(H)\). Bull. Austral. Math. Soc. 53(3), 391–400 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Oikhberg, T., Peralta, A.M.: Automatic continuity of orthogonality preservers on a non-commutative \(L^p(\tau )\) space. J. Funct. Anal. 264(8), 1848–1872 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Oikhberg, T., Peralta, A.M., Puglisi, D.: Automatic continuity of orthogonality or disjointness preserving bijections. Rev. Mat. Complut. 26(1), 57–88 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Pedersen, G.K.: \(\text{ C }^*\)-Algebras and Their Automorphism Groups. London Mathematical Society Monographs, vol. 14. Academic Press, London (1979)zbMATHGoogle Scholar
  24. 24.
    Peralta, A.M.: Orthogonal forms and orthogonality preservers on real function algebras revisited. Linear Multilinear Algebra 65(2), 361–374 (2017)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Radjabalipour, M.: Additive mappings on von Neumann algebras preserving absolute values. Linear Algebra Appl. 368, 229–241 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Radjabalipour, M., Seddighui, K., Taghavi, Y.: Additive mappings on operator algebras preserving absolute values. Linear Algebra Appl. 327, 197–206 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Sakai, S.: C\(^*\)-Algebras and \(W^*\)-Algebras. Springer, Berlin (1971)zbMATHGoogle Scholar
  28. 28.
    Schweizer, J.: Interplay between noncommutative topology and operators on C\(^*\)-algebras, Ph.D. Dissertation, Eberhard-Karls-Universität, Tübingen, Germany (1997)Google Scholar
  29. 29.
    Taghavi, A.: Additive mappings on C\(^*\)-algebras preserving absolute values. Linear Multilinear Algebra 60(1), 33–38 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Wolff, M.: Disjointness preserving operators in C\(^*\)-algebras. Arch. Math. 62, 248–253 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Wong, N.-C.: Triple homomorphisms of C\(^*\)-algebras. Southeast Asian Bull. Math. 29, 401–407 (2005)zbMATHMathSciNetGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.School of Mathematical ScienceNational Institute of Science Education and Research, HBNIBhubaneswarIndia
  2. 2.Departamento de Análisis Matemático, Facultad de CienciasUniversidad de GranadaGranadaSpain

Personalised recommendations