Banach Journal of Mathematical Analysis

Linear maps between C-algebras preserving extreme points and strongly linear preservers

María J. Burgos, Antonio C. Márquez-García, Antonio Morales-Campoy, and Antonio M. Peralta

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We study new classes of linear preservers between C-algebras and between JB-triples. Let E and F be JB-triples with e(E1). We prove that every linear map T:EF strongly preserving Brown–Pedersen quasi-invertible elements is a triple homomorphism. Among the consequences, we establish that, given two unital C-algebras A and B, for each linear map T strongly preserving Brown–Pedersen quasi-invertible elements, there exists a Jordan -homomorphism S:AB satisfying T(x)=T(1)S(x) for every xA. We also study the connections between linear maps strongly preserving Brown–Pedersen quasi-invertibility and other clases of linear preservers between C-algebras like Bergmann-zero pairs preservers, Brown–Pedersen quasi-invertibility preservers, and extreme points preservers.

Article information

Banach J. Math. Anal., Volume 10, Number 3 (2016), 547-565.

Received: 22 July 2015
Accepted: 11 November 2015
First available in Project Euclid: 22 July 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B49: Transformers, preservers (operators on spaces of operators)
Secondary: 15A09: Matrix inversion, generalized inverses 46L05: General theory of $C^*$-algebras 47B48: Operators on Banach algebras

$\mathrm{C}^{*}$-algebra $\mathrm{JB}^{*}$-triple triple homomorphism linear preservers extreme points preserver strongly Brown–Pedersen quasi-invertibility preserver


Burgos, María J.; Márquez-García, Antonio C.; Morales-Campoy, Antonio; Peralta, Antonio M. Linear maps between $\mathrm{C}^{*}$ -algebras preserving extreme points and strongly linear preservers. Banach J. Math. Anal. 10 (2016), no. 3, 547--565. doi:10.1215/17358787-3607288.

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  • [1] T. Barton and R. M. Timoney, Weak$^{*}$-continuity of Jordan triple products and applications, Math. Scand. 59 (1986), no. 2, 177–191.
  • [2] L. G. Brown and G. K. Pedersen, On the geometry of the unit ball of a $\mathit{C}^{*}$-algebra, J. Reine Angew. Math. 469 (1995), 113–147.
  • [3] L. J. Bunce and C.-H. Chu, Compact operations, multipliers and Radon–Nikodym property in $\mathit{JB}^{*}$-triples, Pacific J. Math. 153 (1992), no. 2, 249–265.
  • [4] M. Burgos, F. J. Fernández-Polo, J. J. Garcés, J. Martínez Moreno, and A. M. Peralta. Orthogonality preservers in $\mathit{C}^{*}$-algebras, $\mathit{JB}^{*}$-algebras and $\mathit{JB}^{*}$-triples, J. Math. Anal. Appl. 348 (2008), no. 1, 220–233.
  • [5] M. Burgos, A. Kaidi, A. Morales, A. M. Peralta, and M. Ramírez, Von Neumann regularity and quadratic conorms in $\mathit{JB}^{*}$-triples and $\mathit{C}^{*}$-algebras, Acta Math. Sin. (Engl. Ser.) 24 (2008), no. 2, 185–200.
  • [6] M. Burgos, A. C. Márquez-García, and A. Morales-Campoy, Linear maps strongly preserving Moore–Penrose invertibility, Oper. Matrices 6 (2012), no. 4, 819–831.
  • [7] M. Burgos, A. C. Márquez-García, and A. Morales-Campoy, Strongly preserver problems in Banach algebras and $\mathit{C}^{*}$-algebras, Linear Algebra Appl. 437 (2012), no. 5, 1183–1193.
  • [8] C.-H. Chu, Jordan Structures in Geometry and Analysis, Cambridge Tracts in Math. 190, Cambridge Univ. Press, Cambridge, 2012.
  • [9] S. Dineen, “The second dual of a $\mathrm{JB}^{*}$-triple system" in Complex Analysis, Functional Analysis and Approximation Theory (ed. by J. Múgica), North-Holland Math. Stud. 125, North-Holland, Amsterdam, 1986, 67–69.
  • [10] C. M. Edwards and G. T. Rüttimann, Compact tripotents in bi-dual $\mathit{JB}^{*}$-triples, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 1, 155–174.
  • [11] A. Férnandez López, E. García Rus, E. Sánchez Campos, and M. Siles Molina, Strong regularity and generalized inverses in Jordan systems, Comm. Algebra 20 (1992), no. 7, 1917–1936.
  • [12] F. J. Fernández-Polo, J. Martínez Moreno, and A. M. Peralta, Contractive Perturbations in $\mathit{JB}^{*}$-triples, J. Lond. Math. Soc. (2) 85 (2012), no. 2, 349–364.
  • [13] Y. Friedman and B. Russo, Structure of the predual of a $\mathit{JBW}^{*}$-triple, J. Reine Angew. Math. 356 (1985), 67–89.
  • [14] R. Harte and M. Mbekhta, On generalized inverses in $\mathit{C}^{*}$-algebras, Studia Math. 103 (1992), no. 1, 71–77.
  • [15] L. K. Hua, On the automorphisms of a sfield, Proc. Natl. Acad. Sci. USA 35 (1949), 386–389.
  • [16] N. Jacobson, Structure and Representations of Jordan Algebras. Amer. Math. Soc. Colloq. Publ. 39, Amer. Math. Soc., Providence, 1968.
  • [17] F. Jamjoom, A. Peralta, A. Siddiqui, and H. Tahlawi, Approximation and convex decomposition by extremals and the $\lambda$-function in $\mathit{JBW}^{*}$-triples, Q. J. Math. 66 (2015), no. 2, 583–603.
  • [18] F. B. Jamjoom, A. A. Siddiqui, and H. M. Tahlawi, On the geometry of the unit ball of a $\mathit{JB}^{*}$-triple, Abstr. Appl. Anal. 2013, art. ID 891249.
  • [19] R. V. Kadison, A generalized Schwarz inequality and algebraic invariants for operator algebras, Ann. of Math. (2) 56 (1952), 494–503.
  • [20] W. Kaup, A Riemann mapping theorem for bounded symmentric domains in complex Banach spaces, Math. Z. 183 (1983), no. 4, 503–529.
  • [21] W. Kaup, On spectral and singular values in $\mathit{JB}^{*}$-triples, Proc. Roy. Irish Acad. Sect. A 96 (1996), no. 1, 95–103.
  • [22] L. E. Labuschagne and V. Mascioni, Linear maps between $\mathit{C}^{*}$-algebras whose adjoint preserve extreme points of the dual ball, Adv. Math. 138 (1998), no. 1, 15–45.
  • [23] O. Loos, Jordan pairs and bounded symmetric domains, math lectures at Univ. of California, Irvine, 1977,
  • [24] V. Mascioni and L. Molnár, Linear maps on factors which preserve the extreme points of the unit ball, Canad. Math. Bull. 41 (1998), no. 4, 434–441.
  • [25] M. Mbekhta, A Hua type theorem and linear preserver problems, Math. Proc. R. Ir. Acad. 109 (2009), no. 2, 109–121.
  • [26] K. Meyberg, Von Neumann regularity in Jordan triple systems. Arch Math. (Basel) 23 (1972), 589–593.
  • [27] B. Russo and H. A. Dye, A note on unitary operators in $\mathit{C}^{*}$-algebras, Duke Math. J. 33 (1966), 413–416.
  • [28] E. Størmer, On the Jordan structure of $\mathit{C}^{*}$-algebras, Trans. Amer. Math. Soc. 120 (1965), no. 3, 438–447.
  • [29] M. Takesaki, Theory of Operator Algebras, I, Springer, New York, 1979.