Derivable maps and generalized derivations on nest and standard algebras

• Autorzy: Pan Z.
• Języki publikacji: EN

Abstrakty

EN

For an algebra A, an A-bimodule M, and m (…) M, define a relation on A by RA(m, 0) = {(a, b) (…) A x A : amb = 0}. We show that generalized derivations on unital standard algebras on Banach spaces can be characterized precisely as derivable maps on these relations. More precisely, if A is a unital standard algebra on a Banach space X then (…)L(A,B(X)) is a generalized derivation if and only if (..) is derivable on RA(M,0), for some M (…) B(X). We give an example to show this is not the case in general for nest algebras. On the other hand, for an idempotent P in a nest algebra A = algN on a Hilbert space H such that P is either left-faithful to N or right-faithful to N﬩, if δ (…)L(A, B)H)) is derivable on RA(P, 0) then δ is a generalized derivation.

Słowa kluczowe

EN

derivable map; derivation; nest algebra

PL

pochodne; algebra

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2016
• Tom: Vol. 49, nr 3
• Strony: 331--344
• Opis fizyczny: Bibliogr. 16 poz.

Twórcy

autor

Pan Z.

• Department of Mathematics Saginaw Valley State University University Center MI 48710, USA

Bibliografia

• [1] K. Davidson, Nest Algebras, Pitman Research Notes in Math Series 191, 1988.
• [2] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104–1110.
• [3] W. Jing, S. Lu, P. Li, Characterizations of derivations on some operator algebras, Bull. Austral. Math. Soc. 66 (2002), 227–232.
• [4] B. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Cambridge Philos. Soc. 120 (1996), 455–473.
• [5] J. Li, Z. Pan, Algebraic reflexivity of linear transformations, Proc. Amer. Math. Soc. 135 (2007), 1695–1699.
• [6] J. Li, Z. Pan, Annihilator-preserving maps, multipliers, and derivations, Linear Algebra Appl. 432 (2010), 5–13.
• [7] J. Li, Z. Pan, On derivable mappings, J. Math. Anal. Appl. 374 (2011), 311–322.
• [8] J. Li, Z. Pan, Q. Shen, Jordan and Jordan higher all-derivable points of some algebras, Linear and Multilinear Algebra 61 (2013), 831–845.
• [9] J. Li, Z. Pan, H. Xu, Characterizations of isomorphisms and derivations of some algebras, J. Math. Anal. Appl. 332 (2007), 1314–1322.
• [10] F. Lu, Characterizations of derivations and Jordan derivations on Banach algebras, Linear Algebra Appl. 430 (2009), 2233–2239.
• [11] Z. Pan, Derivable maps and derivational points, Linear Algebra Appl. 436 (2012), 4251–4260.
• [12] Z. Pan, Derivable maps and generalized derivations, Oper. Matrices 8 (2014), 1191–1199.
• [13] M. Rosenblum, On the operator equation BX – XA = Q, Duke Math. J. 23 (1956), 263–269.
• [14] X. Qi, J. Hou, Characterizations of derivations of Banach space nest algebras: all-derivable points, Linear Algebra Appl. 432 (2010), 3183–3200.
• [15] J. Zhou, Linear mappings derivable at some nontrivial elements, Linear Algebra Appl. 435 (2011), 1972–1986.
• [16] J. Zhu, C. Xiong, L. Zhang, All-derivable points in matrix algebras, Linear Algebra Appl. 430 (2009), 2070–2079.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę.