Aspects of the theory of derivations

• Autorzy: Gerard J. Murphy
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Banach Center Publications
• Rocznik: 1994
• Tom: 30
• Numer: 1
• Strony: 267-275

Daty

wydano

1994

Twórcy

autor

Gerard J. Murphy

• Department of Mathematics, University College, Cork, Ireland

Bibliografia

• [1] W. Bade and P. Curtis, Homomorphisms of commutative Banach algebras, Amer. J. Math. 82 (1960), 589-608.
• [2] M. Brešar, Derivations decreasing the spectral radius, Arch. Math. (Basel) 61 (1993), 160-162.
• [3] M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc. 110 (1990), 7-16.
• [4] E. Christensen, Extensions of derivations II, Math. Scand. 50 (1882), 111-122.
• [5] E. Christensen and A. M. Sinclair, A survey of completely bounded operators, Bull. London Math. Soc. 21 (1989), 417-448.
• [6] B. E. Johnson and A. M. Sinclair, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968), 1067-1073.
• [7] I. Kaplansky, Functional analysis, in: Some Aspects of Analysis and Probability, Surveys Appl. Math. 4, New York, 1958, 1-34.
• [8] D. C. Kleinecke, On operator commutators, Proc. Amer. Math. Soc. 8 (1957), 535-536.
• [9] M. Mathieu, Derivations and completely bounded maps on C*-algebras. A survey, preprint, University of Tübingen, 1990.
• [10] M. Mathieu and G. J. Murphy, Derivations mapping into the radical, Arch. Math. (Basel) 57 (1991), 469-474.
• [11] M. Mathieu and V. Runde, Derivations mapping into the radical, II, Bull. London Math. Soc. 24 (1992), 485-487.
• [12] G. J. Murphy, C*-Algebras and Operator Theory, Academic Press, New York, 1990.
• [13] G. K. Pedersen, C*-Algebras and their Automorphism Groups, Academic Press, New York, 1979.
• [14] V. Pták, Commutators in Banach algebras, Proc. Edinburgh Math. Soc. 22 (1979), 207-211.
• [15] C. E. Rickart, General Theory of Banach Algebras, Van Nostrand, New York, 1960.
• [16] S. Sakai, Derivations of W*-algebras, Ann. of Math. 83 (1966), 273-279.
• [17] S. Sakai, Derivations of simple C*-algebras, J. Funct. Anal. 2 (1968), 202-206.
• [18] S. Sakai, Derivations of simple C*-algebras II, Bull. Soc. Math. France 99 (1971), 259-263.
• [19] S. Sakai, Operator Algebras in Dynamical Systems, Cambridge University Press, Cambridge, 1991.
• [20] F. V. Shirokov, Proof of a conjecture of Kaplansky, Uspekhi Mat. Nauk 11 (4) (1956), 167-168 (in Russian).
• [21] A. M. Sinclair, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969), 166-170.
• [22] I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129 (1955), 260-264.
• [23] M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128 (1988), 435-460.
• [24] M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras, Pacific J. Math. 159 (1993), 139-152.
• [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.
• [26] J. Zemánek, Properties of the spectral radius in Banach algebras, in: Spectral Theory, Banach Center Publ. 8, PWN, Warszawa, 1982, 579-595.