Abstract
We consider the relationship between derivations d and g of a Banach algebra B that satisfy \({{\sigma}(g(x)) \subseteq {\sigma}(d(x))}\) for every \({x\in B}\) , where σ( . ) stands for the spectrum. It turns out that in some basic situations, say if B = B(X), the only possibilities are that g = d, g = 0, and, if d is an inner derivation implemented by an algebraic element of degree 2, also g = −d. The conclusions in more complex classes of algebras are not so simple, but are of a similar spirit. A rather definitive result is obtained for von Neumann algebras. In general C*-algebras we have to make some adjustments, in particular we restrict our attention to inner derivations implemented by selfadjoint elements. We also consider a related condition \({\|[b,x]\|\leq M\|[a,x]\|}\) for all selfadjoint elements x from a C*-algebra B, where \({a,b\in B}\) and a is normal.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alaminos J., Brešar M., Extremera J., Villena A.R.: Maps preserving zero products. Studia Math. 193, 131–159 (2009)
Aupetit B.: A primer on spectral theory. Springer, Berlin (1991)
Bonsall F.F., Duncan J.: Complete normed algebras. Springer, Berlin (1973)
Boudi N., Mathieu M.: Commutators with finite spectrum. Ill. J. Math. 48, 687–699 (2004)
Boudi N., Šemrl P.: Derivations mapping into the socle, III. Studia Math. 197, 141–155 (2010)
Brešar M., Mathieu M.: Derivations mapping into the radical, III. J. Funct. Anal. 133, 21–29 (1995)
Brešar M., Šemrl P.: Derivations mapping into the socle. Math. Proc. Camb. Philos. Soc. 120, 339–346 (1996)
Brešar M., Špenko Š.: Determining elements in Banach algebras through spectral properties. J. Math. Anal. Appl. 393, 144–150 (2012)
Chebotar M.A., Ke W.-F., Lee P.-H.: On a Brešar-Šemrl conjecture and derivations of Banach algebras. Q. J. Math. 57, 469–478 (2006)
Conway J.B.: A course in functional analysis. Springer, New York (1990)
Fong C.K.: Range inclusion for normal derivations. Glasgow Math. J. 25, 255–262 (1984)
Glimm J.: A Stone–Weierstrass theorem for C*-algebras. Ann. Math. 72, 216–244 (1960)
Halpern H.: Irreducible module homomorphisms of a von Neumann algebra into its centre. Trans. Am. Math. Soc. 140, 195–221 (1969)
Johnson E., Williams J.P.: The range of a normal derivation. Pac. J. Math. 58, 105–122 (1975)
Kadison R.V., Ringrose J.R.: Fundamentals of the theory of operator algebras, vol. II. Academic press, London (1986)
Kaplansky, I.: Algebraic and analytic aspects of operator algebras. Regional Conference Series in Mathematics 1. Amer. Math. Soc. (1970)
Kissin E., Shulman V.S.: On the range inclusion of normal derivations: variations on a theme by Johnson, Williams and Fong. Proc. Lond. Math. Soc. 83, 176–198 (2001)
Lee T.-K.: Derivations on noncommutative Banach algebras. Studia Math. 167, 153–160 (2005)
Magajna B.: On the distance to finite-dimensional subspaces in operator algebras. J. Lond. Math. Soc. 47, 516–532 (1993)
Pedersen G.K.: C*-algebras and their automorphism groups. Academic press Inc, London (1979)
Rickart C.E.: General theory of Banach algebras. D. Van Nostrand, Princeton (1960)
Turovskii, Yu.V., Shulman, V.S.: Conditions for the massiveness of the range of a derivation of a Banach algebra and of associated differential operators. Mat. Zametki 42, 305–314 (1987) [English transl., Math. Notes 42, 669–674 (1987)]
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by ARRS Grant P1-0288.
Rights and permissions
About this article
Cite this article
Brešar, M., Magajna, B. & Špenko, Š. Identifying Derivations Through the Spectra of Their Values. Integr. Equ. Oper. Theory 73, 395–411 (2012). https://doi.org/10.1007/s00020-012-1975-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00020-012-1975-7