Jewell theorem for higher derivations on C*-algebras.
DOI:
https://doi.org/10.4067/S0716-09172010000200003Keywords:
Derivation, higher derivation, automatic continuity, Sakai theorem, derivación, derivación de grados superiores, continuidad automática, teorema de Sakai.Abstract
Let A be an algebra. A sequence {dn} of linear mappings on A is called a higher derivation if
for each a, b ∈ A and each nonnegative integer n. Jewell [Pacific J. Math. 68 (1977), 91-98], showed that a higher derivation from a Banach algebra onto a semisimple Banach algebra is continuous provided that ker(d0) ⊆ ker(dm), for all m = 1. In this paper, under a different approach using C*-algebraic tools, we prove that each higher derivation {dn} on a C*-algebra A is automatically continuous, provided that it is normal, i. e. d0 is the identity mapping on A.
References
H. G. Dales, Banach Algebras and Automatic Continuity, London Mathematical Society Monographs, New Series, 24. The Clarendon Press, Oxford University Press, Oxford, (2000).
H. Hasse and F. K. Schmidt, Noch eine Begrüdung der theorie der höheren Differential quotienten in einem algebraischen Funtionenkörper einer Unbestimmeten, J. Reine Angew. Math. 177, pp. 215—237, (1937).
S. Hejazian, A.R. Janfada, M. Mirzavaziri and M.S. Moslehian, Achievement of continuity of (φ, ψ)-derivations without linearity, Bull. Belg. Math. Soc.-Simon Stevn., 14, No. 4, pp. 641—652, (2007).
S. Hejazian, T. L. Shatery, Automatic continuity of higher derivations on JB*-algebras, Bull. Iranian Math. Soc., 33, No. 1, pp. 11—23, (2007).
N. P. Jewell, Continuity of module and higher derivations, Pacific J. Math. 68, pp. 91—98, (1977).
I. Kaplansky, Functional analysis, Some aspects of analysis and probability, Surveys in Applied Mathematics. Vol. 4 John Wiley & Sons, Inc., New York; Chapman & Hall, London, (1958).
R. J. Loy, Continuity of higher derivations, Pros. Amer. Math. Soc. 5, pp. 505—510, (1973).
M. Mirzavaziri and M. S. Moslehian, Automatic continuity of σ-derivations in C*-algebras, Proc. Amer. Math. Soc., 134, No. 11, pp. 3319—3327, (2006).
J. G. Murphy, Operator Theory and C*-algebras, Academic Press, Inc., Boston, MA, (1990).
T. W. Palmer, Banach algebras and the general theory of *-algebras, Vol. I. Algebras and Banach algebras, Encyclopedia of Mathematics and its Applications 49, Cambridge University Press, Cambridge, (1994).
S. Sakai, On a conjecture of Kaplansky, Tohoku Math. J. (2) 12, pp. 31—33, (1960).
S. Sakai, Operator Algebra in Dynamical Systems. Cambridge Univ. press, (1991).
Y. Uchino and T. Satoh, Functional field modular forms and higher derivations, Math. Ann. 311, pp. 439—466, (1998).
A. R. Villena, Lie derivations on Banach algebras, J. Algebra 226, pp. 390-409, (2000).
H. Hasse and F. K. Schmidt, Noch eine Begrüdung der theorie der höheren Differential quotienten in einem algebraischen Funtionenkörper einer Unbestimmeten, J. Reine Angew. Math. 177, pp. 215—237, (1937).
S. Hejazian, A.R. Janfada, M. Mirzavaziri and M.S. Moslehian, Achievement of continuity of (φ, ψ)-derivations without linearity, Bull. Belg. Math. Soc.-Simon Stevn., 14, No. 4, pp. 641—652, (2007).
S. Hejazian, T. L. Shatery, Automatic continuity of higher derivations on JB*-algebras, Bull. Iranian Math. Soc., 33, No. 1, pp. 11—23, (2007).
N. P. Jewell, Continuity of module and higher derivations, Pacific J. Math. 68, pp. 91—98, (1977).
I. Kaplansky, Functional analysis, Some aspects of analysis and probability, Surveys in Applied Mathematics. Vol. 4 John Wiley & Sons, Inc., New York; Chapman & Hall, London, (1958).
R. J. Loy, Continuity of higher derivations, Pros. Amer. Math. Soc. 5, pp. 505—510, (1973).
M. Mirzavaziri and M. S. Moslehian, Automatic continuity of σ-derivations in C*-algebras, Proc. Amer. Math. Soc., 134, No. 11, pp. 3319—3327, (2006).
J. G. Murphy, Operator Theory and C*-algebras, Academic Press, Inc., Boston, MA, (1990).
T. W. Palmer, Banach algebras and the general theory of *-algebras, Vol. I. Algebras and Banach algebras, Encyclopedia of Mathematics and its Applications 49, Cambridge University Press, Cambridge, (1994).
S. Sakai, On a conjecture of Kaplansky, Tohoku Math. J. (2) 12, pp. 31—33, (1960).
S. Sakai, Operator Algebra in Dynamical Systems. Cambridge Univ. press, (1991).
Y. Uchino and T. Satoh, Functional field modular forms and higher derivations, Math. Ann. 311, pp. 439—466, (1998).
A. R. Villena, Lie derivations on Banach algebras, J. Algebra 226, pp. 390-409, (2000).
Published
2011-01-07
How to Cite
[1]
S. Hejazian, M. Mirzavaziri, and E. O. Tehrani, “Jewell theorem for higher derivations on C*-algebras.”, Proyecciones (Antofagasta, On line), vol. 29, no. 2, pp. 101-108, Jan. 2011.
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