About the image of strongly generalized derivations of order $n$

Amin Hosseini

doi:10.5269/bspm.64567

Amin Hosseini Kashmar Institute

Abstract

Let A and B be two algebras. A linear mapping 𝜟:A → B is called a strongly generalized
derivation of order n, if there exist the families {𝐸_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐹_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛}, {𝐺_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} 𝑎𝑛𝑑 {𝐻_𝑘: 𝐴 → 𝐵}_{𝑘 = 1}^{𝑛} of linear mappings which satisfy 𝛥(𝑎𝑏) = Σ [𝐸_𝑘(𝑎) 𝐹_𝑘(𝑏) + 𝐺_𝑘(𝑎)𝐻_𝑘(𝑏)] 𝑛𝑘 =1 for all 𝑎, 𝑏 ⋲ 𝐴.
The main purpose of this paper is to study the image of such derivations. Our main result on
the image of strongly generalized derivations of order one reads as follows: Let A be a unital,
commutative Banach algebra and let 𝛥: 𝐴 → 𝐴 be a continuous strongly generalized
derivation of order one; that is, there exist the linear mappings 𝐸, 𝐹, 𝐺, 𝐻: 𝐴 → 𝐴 satisfying
𝐷(𝑎𝑏) = 𝐸(𝑎) 𝐹(𝑏) + 𝐺(𝑎) 𝐻(𝑏) for all 𝑎, 𝑏 ⋲ 𝐴. Let 𝐸, 𝐹, 𝐺 and 𝐻 be continuous linear
mappings. We prove that, under certain conditions, 𝐻 (𝐴), 𝐸(𝐴) 𝑎𝑛𝑑 𝛥(𝐴) are contained in
the Jacobson radical of A. This result generalizes Singer-Wermer theorem about the image of
continuous derivations on commutative Banach algebras.

Downloads

Download data is not yet available.

References

M. Breˇsar, A. R. Villena, The noncommutative Singer-Wermer conjecture and -derivations, J. London Math. Soc. 66, 710-720, (2002).

O. Bratteli and D. R. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. I, Springer Verlag, New York, (1987).

O. Bratteli and D. R. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol. II, Springer Verlag, New York, (1997).

H. G. Dales, Banach Algebras and Automatic Continuity, Math. Soc. Monographs, New Series, 24, Oxford University Press, Oxford, (2000).

H. G. Dales, Pietro Aiena, Jorg Eschmeier, Kjeld Laursen and George A. Willis, Introduction to Banach Algebras, Operators and Harmonic Analysis, Cambridge University Press, (2003).

O. Elchinger, K. Lundengård, A. Makhlouf, S. Silvestrov, Brackets with (, )-derivations and (p, q)-deformations of Witt and Virasoro algebras, Forum Mathematicum, 28(4), 657-673, (2016).

A. Hosseini, M. Hassani and A. Niknam, Generalized -derivation on Banach algebras, Bull. Iranian. Math. Soc. 37, 81-94, (2011).

A. Hosseini, M. Hassani, A. Niknam and S. Hejazian, Some results on -derivations, Ann. Funct. Anal. 2(2), 75-84, (2011).

Amin Hosseini, Automatic continuity of (, )-double derivations on C-algebras, U.P.B. Sci. Bull. Series A. 79(3), 67-72, (2017).

Amin Hosseini, A new proof of Singer-Wermer theorem with some results on {g, h}-derivations, Int. J. Nonlinear Anal. Appl. 11(1), 453-471, (2020).

Amin Hosseini and Ajda Foˇsner, The Image of Jordan Left Derivations on Algebras, Bol. Soc. Paran. Mat. 38(6), 53-61, (2020).

J. T. Hartwig, D. Larsson, S. D. Silvestrov, Deformations of Lie algebras using -derivations, Journal of Algebra, 295, 314-361, (2006).

M. Heller, T. Miller, L. Pysiak and W. Sasin, Generalized derivations and general relativity, Canadian Journal of Physics, 91(10), 757-763, (2013).

Y. S. Jung, Results on the range of derivations, Bull. Korean Math. Soc. 37, 265-272, (2000).

Y. S. Jung, K. H. Park, Noncommutative Versions of the Singer-Wermer Conjecture with Linear Left -derivations, Acta Mathematica Sinica, English Series, 24, 1891-1900, (2008).

T. K. Lee and C.K. Liu, Spectrally bounded -derivations on Banach algebras, Proc. Amer. Math. Soc. 133, 1427-1435, (2004).

M. Mirzavaziri and E. Omidvar Tehrani, -double derivations on C-algebras, Bull. Iranian Math. Soc. 35(1), 147-154, (2009).

I. M. Singer and J. Wermer, Derivations on commutative normed algebras, Math. Ann. 129, 260-264, (1955).

M. P. Thomas, The image of a derivation is contained in the radical, Ann. of Math. 128(3), 435-460, (1988).

M. P. Thomas, Primitive ideals and derivations on non-commutative Banach algebras. Pacifc J. Math. 159, 139-152, (1993).

M. A. Toumi, When the image of a derivation on a uniformly complete f-algebra is contained in the radical, Forum Mathematicum, 32(6), 1561-1573, (2020).