Generalized Triple Homomorphisms and Derivations

Garcés, Jorge J. ; Peralta, Antonio M.

Canadian journal of mathematics, Tome 65 (2013) no. 4, pp. 783-807

Résumé

We introduce generalized triple homomorphisms between Jordan–Banach triple systems as a concept that extends the notion of generalized homomorphisms between Banach algebras given by K. Jarosz and B. E. Johnson in 1985 and 1987, respectively. We prove that every generalized triple homomorphism between $\text{J}{{\text{B}}^{*}}$ -triples is automatically continuous. When particularized to ${{C}^{*}}$ -algebras, we rediscover one of the main theorems established by Johnson. We will also consider generalized triple derivations from a Jordan–Banach triple $E$ into a Jordan–Banach triple $E$ -module, proving that every generalized triple derivation from a $\text{J}{{\text{B}}^{*}}$ -triple $E$ into itself or into ${{E}^{*}}$ is automatically continuous.

Détail

DOI : 10.4153/CJM-2012-043-7

Mots-clés : 46L05, 46L70, 47B48, 17C65, 46K70, 46L40, 47B47, 47B49, generalized homomorphism, generalized triple homomorphism, generalized triple derivation, Banach algebra, Jordan-Banach triple, c*-algebra, JB*-triple

Bibliographie

[1] [1] Akkar, M. and Laayouni, M., Théorèmes de factorisation dans les algèbres normées complètes non associatives. Colloq. Math. 70(1996), 253–264. Google Scholar

[2] [2] Bade, W. G. and Curtis, P. C., Homomorphisms of commutative Banach algebras. Amer. J. Math. 82(1960), 589–608. Google Scholar | DOI

[3] [3] Barton, T. J., Dang, T., and Horn, G., Normal representations of Banach Jordan triple systems. Proc. Amer. Math. Soc. 102(1988), 551–555. Google Scholar | DOI

[4] [4] Becerra-Guerrero, J., Lopez, G., Peralta, A. M. and Rodríguez, A., Relatively weakly open sets in closed unit balls of Banach spaces and real JB*-triples of finite rank. Math. Ann. 330(2004), 45–58. Google Scholar

[5] [5] Bunce, L. J., Chu, C.-H., and Zalar, B., Structure spaces and decomposition in JB*-triples. Math. Scand. 86(2000), 17–35. Google Scholar

[6] [6] Burgos, M., Fernández-Polo, F. J., Garcés, J., Martínez, J., and Peralta, A. M., Orthogonality preservers in C*-algebras, JB*-algebras and JB*-triples. J. Math. Anal. Appl. 348(2008), 220–233. Google Scholar | DOI

[7] [7] Cleveland, S. B., Homomorphisms of non-commutative *-algebras. Pacific J. Math. 13(1963), 1097–1109. Google Scholar

[8] [8] Cuntz, J., On the continuity of semi-norms on operator algebras. Math. Ann. 220(1976), 171–183. Google Scholar | DOI

[9] [9] Dales, H. G., Automatic continuity: a survey. Bull. London Math. Soc. 10(1978), 129–183. Google Scholar | DOI

[10] [10] H. G. Dales, , Banach algebras and automatic continuity. London Math. Soc. Monogr. 24, Oxford University Press, New York, 2000. Google Scholar

[11] [11] Dang, T. and Friedman, Y., Classification of JBW*-triple factors and applications. Math. Scand. 61(1987), 292–330. Google Scholar

[12] [12] Edwards, C. M., Fernández-Polo, F. J., Hoskin, C. S., and Peralta, A. M., On the facial structure of the unit ball in a JB*-triple. J. Reine Angew. Math. 641(2010), 123–144. Google Scholar

[13] [13] Fernandez-Polo, F. J., Garces, Jorge J., and Peralta, A. M., A Kaplansky Theorem for JB*-triples. Proc. Amer. Math. Soc., to appear. Google Scholar | DOI

[14 [14] Fernández-Polo, F. J., Martánez Moreno, J., and Peralta, A. M., Surjective isometries between real JB*-triples. Math. Proc. Cambridge Philos. Soc. 137(2004), 709–723. Google Scholar

[15] [15] Friedman, Y. and Russo, B., Structure of the predual of a JBW*-triple. J. Reine Angew. Math. 356(1985), 67–89. Google Scholar

[16] [16] Hanche-Olsen, H. and Stormer, E., Jordan operator algebras. Monogr. Stud. Math. 21, Pitman, London–Boston–Melbourne, 1984. Google Scholar

[17] [17] Isidro, J. M.,Kaup, W., and RodrÍguez, A., On real forms of JB*-triples. Manuscripta Math. 86(1995), 311–335. Google Scholar | DOI

[18] [18] Jarosz, K., Perturbations of Banach algebras. Lecture Notes in Math. 1120, Springer-Verlag, Berlin, 1985. Google Scholar

[19] [19] Johnson, B. E., Continuity of homomorphisms of operator algebras II. J. London Math. Soc. (2) 1(1969), 81–84. Google Scholar | DOI

[20] [20] Johnson, B. E., Continuity of generalized homomorphisms. Bull. London Math. Soc. 19(1987), 67–71. Google Scholar | DOI

[21] [21] Johnson, B. E., Approximately multiplicative functional. J. London Math. Soc. (2) 37(1988), 294–316. Google Scholar | DOI

[22] [22] Kaup, W., A Riemann Mapping Theorem for bounded symmetric domains in complex Banach spaces. Math. Z. 183(1983), 503–529. Google Scholar | DOI

[23] [23] Kaup, W., On real Cartan factors. Manuscripta Math. 92(1997), 191–222. Google Scholar | DOI

[24] [24] Loos, O., Bounded symmetric domains and Jordan pairs. Math. Lectures, University of California, Irvine, 1977. Google Scholar

[25] [25] Peralta, A. M. and Russo, B., Automatic continuity of derivations on C*-algebras and JB*-triples. Preprint, 2010 Google Scholar

[26] [26] Rickart, C., The uniqueness of norm problem in Banach algebras. Ann. of Math. 51(1950), 615–628. Google Scholar | DOI

[27] [27] A. M. Sinclair, , Homomorphisms from C*-algebras. Proc. London Math. Soc. (3) 29(1975), 435–452. Google Scholar | DOI

[28] [28] A. M. Sinclair, , Automatic continuity of linear operators. London Math. Soc. Lecture Notes 21, Cambridge University Press, Cambridge–New York–Melbourne, 1976. Google Scholar

[29] [29] Yood, B., Homomorphisms on normed algebras.Pacific J. Math. 8(1958), 373–381. Google Scholar