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## Studia Mathematica

1998 | 130 | 3 | 275-291
Tytuł artykułu

### Derivations with a hereditary domain, II

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EN
Abstrakty
EN
The nilpotency of the separating subspace of an everywhere defined derivation on a Banach algebra is an intriguing question which remains still unsolved, even for commutative Banach algebras. On the other hand, closability of partially defined derivations on Banach algebras is a fundamental problem motivated by the study of time evolution of quantum systems. We show that the separating subspace S(D) of a Jordan derivation defined on a subalgebra B of a complex Banach algebra A satisfies $B[B ∩ S(D)]B ⊂ Rad_B(A)$ provided that BAB ⊂ A and $dim(Rad_J(A) ∩ ⋂_{n=1}^∞ B^n) < ∞$, where $Rad_J(A)$ and $Rad_B(A)$ denote the Jacobson and the Baer radicals of A respectively. From this we deduce the closability of partially defined derivations on complex semiprime Banach algebras with appropriate domains. The result applies to several relevant classes of algebras.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
275-291
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-10-10
poprawiono
1997-12-22
Twórcy
autor
Bibliografia
• [1] O. Bratteli, Derivations, Dissipations and Group Actions on C*-algebras, Lecture Notes in Math. 1229, Springer, Berlin, 1986.
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• [3] J. Cusack, Automatic continuity and topologically simple radical Banach algebras, J. London Math. Soc. 16 (1977), 493-500.
• [4] H. G. Dales, Automatic continuity: a survey, Bull. London Math. Soc. 10 (1978), 129-183.
• [5] H. G. Dales, Convolution algebras on the real line, in: Radical Banach Algebras and Automatic Continuity, Lecture Notes in Math. 975, Springer, 1983, 180-209.
• [6] R. S. Doran and J. M. G. Fell, Representations of *-Algebras, Locally Compact Groups, and Banach *-Algebraic Bundles, Vol. 1, Basic Representation Theory of Groups and Algebras, Academic Press, 1988.
• [7] R. V. Garimella, Continuity of derivations on some semiprime Banach algebra, Proc. Amer. Math. Soc. 99 (1987), 289-292.
• [8] R. V. Garimella, On nilpotency of the separating ideal of a derivation, ibid. 117 (1993), 167-174.
• [9] I. M. Gelfand, D. A. Raikov and G. E. Shilov, Commutative Normed Rings, Chelsea, New York, 1964.
• [10] F. Ghahramani, Homomorphisms and derivations on weighted convolution algebras, J. London Math. Soc. 21 (1980), 149-161.
• [11] S. Grabiner, Derivations and automorphisms of Banach algebras of power series, Mem. Amer. Math. Soc. 146 (1974).
• [12] I. N. Herstein, Jordan derivations on prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110.
• [13] N. P. Jewell and A. M. Sinclair, Epimorphisms and derivations on $L^1(0,1)$ are continuous, Bull. London Math. Soc. 8 (1976), 135-139.
• [14] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073.
• [15] R. J. Loy, Continuity of derivations on topological algebras of power series, Bull. Austral. Math. Soc. 1 (1969), 419-424.
• [16] M. Mathieu and V. Runde, Derivations mapping into the radical, II, Bull. London Math. Soc. 24 (1992), 485-487.
• [17] T. W. Palmer, Banach Algebras and the General Theory of *-Algebras, Vol. I, Algebras and Banach Algebras, Cambridge Univ. Press, 1994.
• [18] G. K. Pedersen, C*-Algebras and Their Automorphism Groups, Academic Press, London, 1979.
• [19] V. Runde, Automatic continuity of derivations and epimorphisms, Pacific J. Math. 147 (1991), 365-374.
• [20] S. Sakai, Operator Algebras in Dynamical Systems, Cambridge Univ. Cambridge, 1991.
• [21] A. M. Sinclair, Automatic Continuity of Linear Operators, London Math. Soc. Lecture Note Ser. 21, Cambridge Univ. Press, 1976.
• [22] M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. Math. 159 (1993), 139-152.
• [23] A. R. Villena, Continuity of derivations on H*-algebras, Proc. Amer. Math. Soc. 122 (1994), 821-826.
• [24] A. R. Villena, Derivations on Jordan-Banach algebras, Studia Math. 118 (1996), 205-229.
• [25] A. R. Villena, Essentially defined derivations on semisimple Banach algebras, Proc. Edinburgh Math. Soc. 40 (1997), 175-179.
• [26] A. R. Villena, Derivations with a hereditary domain, J. London Math. Soc., to appear.
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Bibliografia
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