Aspects of the theory of derivations

Murphy, Gerard

Banach Center Publications, Tome 29 (1994), p. 267-275 / Harvested from The Polish Digital Mathematics Library

Résumé

We survey some old and new results in the theory of derivations on Banach algebras. Although our overview is broad ranging, our principal interest is in recent results concerning conditions on a derivation implying that its range is contained in the radical of the algebra.

Détail

Publié le : 1994-01-01

Zbl 0811.46045

EUDML-ID : urn:eudml:doc:262692

Bibliographie

[000] [1] W. Bade and P. Curtis, Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589-608. | Zbl 0093.12503

[001] [2] M. Brešar, Derivations decreasing the spectral radius, Arch. Math. (Basel) 61 (1993), 160-162. | Zbl 0818.46049

[002] [3] M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), 7-16. | Zbl 0703.16020

[003] [4] E. Christensen, Extensions of derivations II, Math. Scand. 50 (1882), 111-122. | Zbl 0503.47032

[004] [5] E. Christensen and A. M. Sinclair, A survey of completely bounded operators, Bull. London Math. Soc. 21 (1989), 417-448. | Zbl 0698.46044

[005] [6] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073. | Zbl 0179.18103

[006] [7] I. Kaplansky, Functional analysis, in: Some Aspects of Analysis and Probability, Surveys Appl. Math. 4, New York, 1958, 1-34.

[007] [8] D. C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc. 8 (1957), 535-536. | Zbl 0079.12904

[008] [9] M. Mathieu, Derivations and completely bounded maps on C*-algebras. A survey, preprint, University of Tübingen, 1990.

[009] [10] M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), 469-474. | Zbl 0714.46038

[010] [11] M. Mathieu and V. Runde, Derivations mapping into the radical, II, Bull. London Math. Soc. 24 (1992), 485-487. | Zbl 0733.46023

[011] [12] G. J. Murphy, C*-Algebras and Operator Theory, Academic Press, New York, 1990.

[012] [13] G. K. Pedersen, C*-Algebras and their Automorphism Groups, Academic Press, New York, 1979. | Zbl 0416.46043

[013] [14] V. Pták, Commutators in Banach algebras, Proc. Edinburgh Math. Soc. 22 (1979), 207-211. | Zbl 0407.46043

[014] [15] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, New York, 1960. | Zbl 0095.09702

[015] [16] S. Sakai, Derivations of W*-algebras, Ann. of Math. 83 (1966), 273-279. | Zbl 0139.30601

[016] [17] S. Sakai, Derivations of simple C*-algebras, J. Funct. Anal. 2 (1968), 202-206. | Zbl 0172.41104

[017] [18] S. Sakai, Derivations of simple C*-algebras II, Bull. Soc. Math. France 99 (1971), 259-263. | Zbl 0216.41401

[018] [19] S. Sakai, Operator Algebras in Dynamical Systems, Cambridge University Press, Cambridge, 1991. | Zbl 0743.46078

[019] [20] F. V. Shirokov, Proof of a conjecture of Kaplansky, Uspekhi Mat. Nauk 11 (4) (1956), 167-168 (in Russian). | Zbl 0070.34201

[020] [21] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166-170. | Zbl 0164.44603

[021] [22] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264. | Zbl 0067.35101

[022] [23] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460. | Zbl 0681.47016

[023] [24] M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. Math. 159 (1993), 139-152. | Zbl 0739.47014

[024] [25] Yu. V. Turovskiĭ and V. S. Shul'man, Conditions for massiveness of the range of the derivation of a Banach algebra and associated differential operators, Math. Notes 42 (1987), 669-674.

[025] [26] J. Zemánek, Properties of the spectral radius in Banach algebras, in: Spectral Theory, Banach Center Publ. 8, PWN, Warszawa, 1982, 579-595.