Banach Journal of Mathematical Analysis

Jordan weak amenability and orthogonal forms on JB$^*$-algebras

Fatmah B. Jamjoom, Antonio M. Peralta, and Akhlaq A. Siddiqui

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We prove the existence of a linear isometric correspondence between the Banach space of all symmetric orthogonal forms on a JB$^*$-algebra $\mathcal{J}$ and the Banach space of all purely Jordan generalized Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$. We also establish the existence of a similar linear isometric correspondence between the Banach spaces of all anti-symmetric orthogonal forms on $\mathcal{J}$, and of all Lie Jordan derivations from $\mathcal{J}$ into $\mathcal{J}^*$.

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Banach J. Math. Anal., Volume 9, Number 4 (2015), 126-145.

First available in Project Euclid: 17 April 2015

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Primary: 46L57: Derivations, dissipations and positive semigroups in C-algebras
Secondary: 47B47: Commutators, derivations, elementary operators, etc. 17B40: Automorphisms, derivations, other operators 46L70: Nonassociative selfadjoint operator algebras [See also 46H70, 46K70] 46L05: General theory of $C^*$-algebras 46L89: Other "noncommutative" mathematics based on C-algebra theory [See also 58B32, 58B34, 58J22] 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups

(Jordan) weak amenability orthogonal form generalized derivation purely Jordan generalized derivation Lie Jordan derivation


Jamjoom, Fatmah B.; Peralta, Antonio M.; Siddiqui, Akhlaq A. Jordan weak amenability and orthogonal forms on JB$^*$-algebras. Banach J. Math. Anal. 9 (2015), no. 4, 126--145. doi:10.15352/bjma/09-4-8.

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  • J. Alaminos, M. Bresar, J. Extremera, and A. Villena, Maps preserving zero products, Studia Math. 193 (2009), no. 2, 131–159.
  • R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848.
  • T.J. Barton and R.M. Timoney, Weak$^*$-continuity of Jordan triple products and its applications, Math. Scand. 59 (1986), 177–191.
  • F. Bombal and I. Villanueva, Multilinear operators on spaces of continuous functions, Funct. Approx. Comment. Math. 26 (1998), 117–126.
  • M. Burgos, F.J. Fernández-Polo, J. Garcés, J. Martínez, and A.M. Peralta, Orthogonality preservers in C$^*$-algebras, JB*-algebras and JB*-triples, J. Math. Anal. Appl. 348 (2008), 220–233.
  • M. Burgos, F.J. Fernández-Polo, J.J. Garcés, and A.M. Peralta, Local triple derivations on C$^*$-algebras, Comm. Algebra 42 (2014), no. 3, 1276–1286.
  • M. Burgos, F.J. Fernández-Polo, and A.M. Peralta, Local triple derivations on C$^*$-algebras and JB$^*$-triples, Bull. London Math. Soc., (4) 46 (2014), 709–724.
  • M. Cabrera Garc\' ia and A. Rodr\' iguez Palacios, Non-Associative Normed Algebras, Volume 1. The Vidav-Palmer and Gelfand-Naimark Theorems, Part of Encyclopedia of Mathematics and its Applications, Cambridge University Press 2014.
  • Ch.-H. Chu, Jordan Structures in Geometry and Analysis, Cambridge Tracts in Math. 190, Cambridge. Univ. Press, Cambridge, 2012.
  • Ch.-H. Chu, B. Iochum, and G. Loupias, Grothendieck's theorem and factorization of operators in Jordan triples, Math. Ann. 284 (1989), 41–53.
  • A. Connes, On the cohomology of operator algebras, J. Funct. Anal. 28 (1978), 248–253.
  • H.G. Dales, Banach algebras and automatic continuity. London Mathematical Society Monographs. New Series, 24. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 2000.
  • C. M. Edwards and G. T. Rüttimann, Compact tripotents in bi-dual ${\rm JB}\sp *$-triples, Math. Proc. Cambridge Philos. Soc 120 (1996), no. 1, 155–173.
  • J. Garcés and A. M. Peralta, Orthogonal forms and orthogonality preservers on real function algebras, Linear Multilinear Algebra 62 (2014), no. 3, 275–296.
  • S. Goldstein, Stationarity of operator algebras, J. Funct. Anal. 118 (1993), no. 2, 275–308.
  • U. Haagerup, All nuclear C$^*$-algebras are amenable, Invent. Math. 74 (1983), no. 2, 305–319.
  • U. Haagerup and N.J. Laustsen, Weak amenability of C$^*$-algebras and a theorem of Goldstein, in Banach algebras '97 (Blaubeuren), 223–243, de Gruyter, Berlin, 1998.
  • H. Hanche-Olsen and E. Størmer, Jordan operator algebras, Monographs and Studies in Mathematics 21, Pitman, London-Boston-Melbourne 1984.
  • S. Hejazian and A. Niknam, Modules Annihilators and module derivations of JB$^*$-algebras, Indian J. pure appl. Math. 27 (1996), 129–140.
  • T. Ho, A.M. Peralta, and B. Russo, Ternary weakly amenable C$^*$-algebras and JB$^*$-triples,Q. J. Math. 64 (2013), 1109–1139.
  • G. Horn, Characterization of the predual and ideal structure of a ${\rm JBW}^*$-triple, Math. Scand. 61 (1987), no. 1, 117–133.
  • B.E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 3, 455–473.
  • W. Kaup, A Riemann Mapping Theorem for bounded symmentric domains in complex Banach spaces, Math. Z. 183 (1983), 503–529.
  • J. Li and Zh. Pan, Annihilator-preserving maps, multipliers, and derivations, Linear Algebra Appl. 423 (2010), 5–13.
  • C. Palazuelos, A.M. Peralta, and I. Villanueva, Orthogonally Additive Polynomials on C$^*$-algebras, Quart. J. Math. Oxford. (3) 59 (2008), 363–374.
  • A. M. Peralta and B. Russo, Automatic continuity of triple derivations on C$^*$-algebras and JB$^*$-triples, Journal of Algebra 399 (2014), 960–977.
  • R. Pluta, B. Russo, Triple derivations on von Neumann algebras, to appear in Studia Math. arXiv:1309.3526v2.
  • J.R. Ringrose, Automatic continuity of derivations of operator algebras, J. London Math. Soc. (2) 5 (1972), 432–438.
  • S. Sakai, $C\sp*$-algebras and $W\sp*$-algebras, in: Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60. Springer-Verlag, New York-Heidelberg. 1971.
  • H. Upmeier, Derivations of Jordan C$^*$-algebras, Math. Scand. 46 (1980), 251–264.