A LINEAR APPROACH TO LIE TRIPLE AUTOMORPHISMS OF H*-ALGEBRAS

Title & Authors
A LINEAR APPROACH TO LIE TRIPLE AUTOMORPHISMS OF H*-ALGEBRAS
Martin, A. J. Calderon; Gonzalez, C. Martin;

Abstract
By developing a linear algebra program involving many different structures associated to a three-graded H*-algebra, it is shown that if L is a Lie triple automorphism of an infinite-dimensional topologically simple associative H*-algebra A, then L is either an automorphism, an anti-automorphism, the negative of an automorphism or the negative of an anti-automorphism. If A is finite-dimensional, then there exists an automorphism, an anti-automorphism, the negative of an automorphism or the negative of an anti-automorphism F : A $\small{\rightarrow}$ A such that $\small{\delta}$:= F - L is a linear map from A onto its center sending commutators to zero. We also describe L in the case of having A zero annihilator.
Keywords
H*-algebra;graded algebra;Jordan pair;Lie triple automorphism;
Language
English
Cited by
1.
ADDITIVITY OF JORDAN TRIPLE PRODUCT HOMOMORPHISMS ON GENERALIZED MATRIX ALGEBRAS,;;

대한수학회보, 2013. vol.50. 6, pp.2027-2034
1.
Centralizing traces and Lie triple isomorphisms on generalized matrix algebras, Linear and Multilinear Algebra, 2015, 63, 9, 1786
2.
Centralizing traces and Lie triple isomorphisms on triangular algebras, Linear and Multilinear Algebra, 2015, 63, 7, 1309
3.
ADDITIVITY OF JORDAN TRIPLE PRODUCT HOMOMORPHISMS ON GENERALIZED MATRIX ALGEBRAS, Bulletin of the Korean Mathematical Society, 2013, 50, 6, 2027
References
1.
W. Ambrose, Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc. 57 (1945), 364-386.

2.
R. Banning and M. Mathieu, Commutativity preserving mappings on semiprime rings, Comm. Algebra 25 (1997), no. 1, 247-265.

3.
K. I. Beidar, M. Bresar, M. A. Chebotar, and W. S. Martindale, 3rd On Herstein's Lie map conjectures. III, J. Algebra 249 (2002), no. 1, 59-94.

4.
K. I. Beidar, W. S. Martindale, and A. V. Mikhalev, Lie isomorphisms in prime rings with involution, J. Algebra 169 (1994), no. 1, 304-327.

5.
M. I. Berenguer and A. R. Villena, Continuity of Lie isomorphisms of Banach algebras, Bull. London Math. Soc. 31 (1999), no. 1, 6-10.

6.
M. Bresar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (1993), no. 2, 525-546.

7.
M. Bresar, Functional identities: a survey, Algebra and its applications (Athens, OH, 1999), 93-109, Contemp. Math., 259, Amer. Math. Soc., Providence, RI, 2000.

8.
M. Cabrera, J. Martinez, and A. Rodriguez, Structurable $H^*$-algebras, J. Algebra 147 (1992), no. 1, 19-62.

9.
A. J. Calderon Martin and C. Martin Gonzalez, Dual pairs techniques in $H^*$-theories, J. Pure Appl. Algebra 133 (1998), no. 1-2, 59-63.

10.
A. J. Calderon Martin and C. Martin Gonzalez, Hilbert space methods in the theory of Lie triple systems, Recent progress in functional analysis (Valencia, 2000), 309-319, North-Holland Math. Stud., 189, North-Holland, Amsterdam, 2001.

11.
A. J. Calderon Martin and C. Martin Gonzalez, Lie isomorphisms on $H^*$-algebras, Comm. Algebra 31 (2003), no. 1, 333-343.

12.
A. J. Calderon Martin and C. Martin Gonzalez, A structure theory for Jordan $H^*$-pairs, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 7 (2004), no. 1, 61-77.

13.
A. J. Calderon Martin and C. Martin Gonzalez, On $L^*$-triples and Jordan $H^*$-pairs, Ring theory and algebraic geometry, Lecture Notes in Pure and Applied Math. Marcel Dekker Inc. (2004), 87-94.

14.
A. Castellon Serrano and J. A. Cuenca Mira, Isomorphisms of $H^*$-triple systems, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 4, 507-514.

15.
A. Castellon Serrano, J. A. Cuenca Mira, and C. Martin Gonzalez, Ternary $H^*$-algebras, Boll. Un. Mat. Ital. B (7) 6 (1992), no. 1, 217-228.

16.
M. A. Chebotar, On Lie automorphisms of simple rings of characteristic 2, Fundam. Prikl. Mat. 2 (1996), no. 4, 1257-1268.

17.
M. A. Chebotar, On Lie isomorphisms in prime rings with involution, Comm. Algebra 27 (1999), no. 6, 2767-2777.

18.
J. A. Cuenca Mira, A. Garcia, and C. Martin Gonzalez, Structure theory for $L^*$-algebras, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 361-365.

19.
J. A. Cuenca Mira and A. Rodriguez, Isomorphisms of $H^*$-algebras, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 1, 93-99.

20.
J. A. Cuenca Mira and A. Rodriguez, Structure theory for noncommutative Jordan $H^*$-algebras, J. Algebra 106 (1987), no. 1, 1-14.

21.
A. D'Amour, Jordan triple homomorphisms of associative structures, Comm. Algebra 19 (1991), no. 4, 1229-1247.

22.
N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10 Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962.

23.
N. Jacobson, Structure of Rings, American Mathematical Society, Colloquium Publications, vol. 37. American Mathematical Society, 190 Hope Street, Prov., R. I., 1956.

24.
P. Ji and L. Wang, Lie triple derivations of TUHF algebras, Linear Algebra Appl. 403 (2005), 399-408.

25.
K. McCrimmon and E. Zel'manov, The structure of strongly prime quadratic Jordan algebras, Adv. in Math. 69 (1988), no. 2, 133-222.

26.
M. Mathieu, Lie mappings of $C^*$-algebras, Nonassociative algebra and its applications (Sao Paulo, 1998), 229-234, Lecture Notes in Pure and Appl. Math., 211, Dekker, New York, 2000.

27.
C. R. Miers, Lie $^*-triple$ homomorphisms into von Neumann algebras, Proc. Amer. Math. Soc. 58 (1976), 169-172.

28.
E. Neher, On the classification of Lie and Jordan triple systems, Comm. Algebra 13 (1985), no. 12, 2615-2667.

29.
J. R. Schue, Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc. 95 (1960), 69-80.

30.
J.-H. Zhang, B.-W. Wu, and H.-X. Cao, Lie triple derivations of nest algebras, Linear Algebra Appl. 416 (2006), no. 2-3, 559-567.