Acta mathematica scientia, Series B >
THE BANACH-LIE GROUP OF LIE TRIPLE AUTOMORPHISMS OF AN |H*-ALGEBRA
Received date: 2007-11-10
Revised date: 2008-10-07
Online published: 2010-07-20
Supported by
Supported by the PCI of the UCA `Teorí a de Lie y Teor\'\i a de Espacios de Banach', the PAI with project numbers FQM-298 and FQM-336, and the project of the Spanish Ministerio de Educaci\'on y Ciencia MTM2004-06580-C02-02 and with fondos FEDER
We study the Banach-Lie group Ltaut}(A) of Lie triple automorphisms of a complex associative H*-algebra A. Some consequences about its Lie algebra, the algebra of Lie triple derivations of A, Ltder(A), are obtained. For a topologically simple A, in the infinite-dimensional case we have Ltaut(A)0=Aut(A) implying Ltder(A)= Der(A). In the finite-dimensional case Ltaut(A)0 is a direct product of aut(A) and a certain subgroup of Lie derivations δ from A to its center, annihilating commutators.
Key words： Banach-Lie group; Lie triple automorphism; Lie triple derivation
A. J.Calderon Martí , C.Martín González . THE BANACH-LIE GROUP OF LIE TRIPLE AUTOMORPHISMS OF AN |H*-ALGEBRA[J]. Acta mathematica scientia, Series B, 2010 , 30(4) : 1219 -1226 . DOI: 10.1016/S0252-9602(10)60118-X
[1] Cuenca J A, Rodríguez A. Structure theory for noncommutative Jordan H*-algebras. J Algebra, 1987, 106: 1--14
[2] Ambrose W. Structure theorems for a special class of Banach algebras. Trans Amer Math Soc, 1945, 57: 364--386
[3] Cabrera M, Rodríguez A. Extended centroid and central closure of semiprime normed algebras: a first approach. Comm Algebra, 1990, 18: 2293--2326
[4] Calder\'on A J, Martín C. Lie isomorphisms on H*-algebras. Comm Algebra, 2003, 31(1): 333--343
[5] Cuenca J A, García A, Martín C. Structure Theory for L*-algebras. Math Proc Cambridge Philos Soc, 1990, 107(2): 361--365
[6] Schue J R. Hilbert space methods in the theory of Lie algebras. Trans Amer Math Soc, 1960, 95: 69--80
[7] Cuenca J A, Rodríguez A. Isomorphisms of H*-algebras. Math Proc Cambridge Philos Soc, 1985, 97: 93--99
[8] Villena A R. Continuity of Derivations on H*-algebras. Proc Amer Math Soc, 1994, 122: 821--826
[9] Bre\v{s}ar M. Commuting traces of biadditive mappings, commutativity preserving mappings and Lie mappings. Trans Amer Math Soc, 1993, 335: 525--546
[10] Jacobson N. Lie algebras. Interscience, 1962
[11] Jacobson N. Structure of Rings. Colloq Publ Vol 37. Second Edition. Amer Math Soc, 1956
[12] Upmeier H. Symmetric Banach Manifolds and Jordan C*-algebras. North-Holland Math Studies, 1985, 104
[13] Schr\"oder H. On the topology of the group of invertible elements. arXiv: Math.KT/9810069v1,1998
[14] Sagle A A, Walde R E. Introduction to Lie groups and Lie algebras//Pure and Applied Mathematics. New York-London: Academic Press, 1973, 51
[15] Calder\'{o}n A J, Martín C. Hilbert space methods in the theory of Lie triple systems//Recent Progress in Functional Analysis: North-Holland Math Studies, 2001: 309--319
/
| 〈 |
|
〉 |