The Banach-Lie Group of Lie Automorphisms of an $H^*$-Algebra

Calderón Martín, Antonio J. ; Martín González, Candido

Bollettino dell'Unione Matematica Italiana, Tome 10-A (2007), p. 623-631 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

We study the Banach-Lie group $\operatorname{Aut}(A^-)$ of Lie automorphisms of a complex associative $H^*$-algebra. Also some consequences about its Lie algebra, the algebra of Lie derivations of $A$, are obtained. For a topologically simple $A$, in the infinite-dimensional case we have $\operatorname{Aut}(A^-)_0 = \operatorname{Aut}(A)$ implying $\operatorname{Der}(A) = \operatorname{Der}(A^-)$. In the finite dimensional case $\operatorname{Aut}(A^{-})_{0}$ is a direct product of $\operatorname{Aut}(A)$ and a certain subgroup of Lie derivations $\delta$ from $A$ to its center, annihilating commutators.

Résumé

Studiamo il gruppo di Banach-Lie $\operatorname{Aut}(A^-)$ degli automorfismi di Lie di una $H^*$-algebra associativa complessa. Vengono anche ottenute alcune conseguenze riguardanti la sua algebra di Lie, cioè l'algebra delle derivazioni di Lie di $A$. Per una $A$ topologicamente semplice, nel caso di dimensione infinita si ha $\operatorname{Aut}(A^-)_0 = \operatorname{Aut}(A)$, il che implica che $\operatorname{Der}(A) = \operatorname{Der}(A^-)$. Nel caso di dimensione finita, $\operatorname{Aut}(A^-)_{0}$ è il prodotto diretto di $\operatorname{Aut}(A)$ e di un certo sottogruppo di derivazioni di Lie $\delta$ da $A$ al suo centro, che annullano i commutatori.

Détail

Publié le : 2007-10-01

MR 2351534 | Zbl 1145.46033

Bibliographie

[1] Ambrose, W., Structure theorems for a special class of Banach Algebras, Trans. Amer. Math. Soc.57 (1945), 364-386. | MR 13235 | Zbl 0060.26906

[2] Brešar, M., Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings, Trans. Amer. Math. Soc.335 (1993), 525-546. | MR 1069746 | Zbl 0791.16028

[3] Cabrera, M. - Martínez, J. - Rodríguez, A., Structurable $H^*$-algebras, J. Algebra 147 (1992) 19-62. | MR 1154673

[4] Calderón, A. J. - Martín, C., Lie isomorphisms on $H^*$-algebras, Comm. Algebra, 31 No. 1 (2003), 333-343. | MR 1969226

[5] Cuenca, J. A., Sobre $H^*$-álgebras no asociativas. Teoría de estructura de las $H^*$-álgebras de Jordan no conmutativas semisimples, Tesis Doctoral, Universidad de a Granada, 1982. | MR 853913

[6] Cuenca, J. A. - García, A. - Martín, C., Structure Theory for $L^*$-álgebras, Math. Proc. Cambridge Philos. Soc. 107, No. 2 (1990), 361-365. | MR 1027788

[7] Cuenca, J. A. - Rodríguez, A., Structure Theory for noncommutative Jordan $H^*$-algebras, J. Algebra, 106 (1987), 1-14. | MR 878465

[8] Jacobson, N., Lie algebras, Interscience. 1962. | MR 143793

[9] Jacobson, N., Structure of Rings, Colloq. Publ. Vol. 37, Amer. Math. Soc., second edition, 1956. | MR 81264 | Zbl 0073.02002

[10] Kaplansky, I., Lie algebras and locally compact groups, The University of Chicago Press. 1971. | MR 276398 | Zbl 0223.17001

[11] Mathieu, M. - Villena, A. R., Lie and Jordan derivations from Von Neumann Algebras, Preprint.

[12] Schue, J. R., Hilbert Space methods in the theory of Lie algebras, Trans. Amer. Math. Soc., 95 (1960), 69-80. | MR 117575 | Zbl 0093.30601

[13] Schröder, H., On the topology of the group of invertible elements, arXiv:math.KT/9810069v1, 1998.

[14] Upmeier, H., Symmetric Banach Manifolds and Jordan $C^*$-algebras, North-Holland Math. Studies, 104 (1985). | MR 776786

[15] Villena, A. R., Continuity of Derivations on $H^*$-algebras, Proc. Amer. Math. Soc., 122 (1994), 821-826. | MR 1207543 | Zbl 0822.46061