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Derivations of operator algebras, automatic closability

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 Abstract: In this note we review briefly the subject of bounded and unbounded derivations of operator algebras (C*-and W*-algebras). In particular, we mention some results on closability and continuity of such derivations. This is an invitation to work on three problems in this subject.
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  • Department of Mathematics, University of Mashhad, P. O. Box 1159-91775, Mashhad, Iran
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Bibliografia
[1] C. A. Akemann and G. K. Pedersen, Central sequence and inner derivations of separable C*-algebras, Amer. J. Math. 101 (1979), 1047-1061.
[2] B. Aupetit, Propriétés Spectrales des Algèbres de Banach, Lecture Notes in Math. 735, Springer, Berlin, 1979.
[3] B. Aupetit, Automatic continuity conditions for linear mapping from a Banach algebra onto a semi-simple Banach algebra, in: Radical Banach Algebras and Automatic Continuity, Proc. California State Univ., Long Beach, 1981, Lecture Notes in Math. 975, Berlin, 1983, 313-316.
[4] C. J. K. Batty, Unbounded derivations of commutative C*-algebras, Comm. Math. Phys. 61 (1978), 261-266.
[5] C. J. K. Batty, Connected spaces without derivations, Bull. London Math. Soc. 15 (1983), 349-352.
[6] O. Bratteli and D. W. Robinson, Unbounded derivations of C*-algebras, Comm. Math. Phys. 42 (1975), 253-263.
[7] O. Bratteli, Derivations, Dissipations and Group Actions on C*-algebras, Lecture Notes in Math. 1229, Springer, Berlin, 1986.
[8] D. P. Chi, Derivations in C*-algebra, Ph.D. thesis, Univ of Pensylvania, 1976.
[9] G. A. Elliott, Some C*-algebras with outer derivations, Ann. of Math. 105 (1977), 121-143.
[10] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073.
[11] R. V. Kadison, Derivation of operator algebras, Ann. of Math. 83 (1966), 280-293.
[12] I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839-859.
[13] H. Kurose, An example of a non-quasi-well behaved derivation in C(I), J. Funct. Anal. 43 (1981), 193-201.
[14] H. Kurose, Closed derivation in C(I), Tôhoku Math. J. 35 (1983), 341-347.
[15] R. Longo, Automatic relative boundedness of derivation in C*-algebras, J. Funct. Anal. 34 (1979), 21-28.
[16] R. J. Loy, Uniqueness of the complete norm topology and continuity of derivations on Banach algebras, Tôhoku Math. J. 22 (1970), 371-377.
[17] A. Niknam, Closability of linear mapping in normes algebras, Arch. Math. (Basel) 47 (1986), 131-134.
[18] A. Niknam, Closable derivations of simple C*-algebras, Glasgow Math. J. 24 (1983), 181-183.
[19] A. Niknam, On the domain of an unbounded derivation of a C*-algebra, Bull. Iranian Math. Soc. 9 (1981), 21-25.
[20] K. Nishio, A local kernel property of closed derivations on C(I×I), Proc. Amer. Math. Soc. 95 (1985), 537-576.
[21] S. Sakai, On a conjecture of Kaplansky, Tôhoku Math. J. 12 (1960), 31-33.
[22] S. Sakai, Derivations of W*-algebras, Ann. of Math. 33 (1966), 273-299.
[23] S. Sakai, The theory of unbounded derivations in C*-algebras, Copenhagen University Lecture Notes, 1977.
[24] S. Sakai, Recent development in the theory of unbounded derivations in C*-algebras, in: C*-Algebras and Applications to Physics, Proc. 2nd Japan-USA Seminar, Los Angeles, April 18-22, 1977, Lecture Notes in Math. 650, Springer, Berlin, 1973.
[25] A. M. Sinclair, Automatic Continuity of Linear Operators, London Math. Soc. Lecture Note Ser. 21, Cambridge Univ. Press, 1976.
[26] J. Tomiyama, The theory of closed derivations in the algebra of continuous functions on the unit interval, Lecture Notes, Tsing Hua Univ., 1983.
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