Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T13:43:12.617Z Has data issue: false hasContentIssue false
  • English
  • Français

2-LOCAL DERIVATIONS ON SEMISIMPLE BANACH ALGEBRAS WITH MINIMAL LEFT IDEALS

Part of: Linear spaces and algebras of operators Rings and algebras with additional structure

Published online by Cambridge University Press:  24 July 2020

WENBO HUANG  and
JIANKUI LI
WENBO HUANG
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China e-mail: [email protected]
JIANKUI LI*
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let ${\mathcal{A}}$ be a semisimple Banach algebra with minimal left ideals and $\text{soc}({\mathcal{A}})$ be the socle of ${\mathcal{A}}$. We prove that if $\text{soc}({\mathcal{A}})$ is an essential ideal of ${\mathcal{A}}$, then every 2-local derivation on ${\mathcal{A}}$ is a derivation. As applications of this result, we can easily show that every 2-local derivation on some algebras, such as semisimple modular annihilator Banach algebras, strongly double triangle subspace lattice algebras and ${\mathcal{J}}$-subspace lattice algebras, is a derivation.

Keywords

derivation2-local derivationmodular annihilator algebrasocletrace

MSC classification

Primary: 47L35: Nest algebras, CSL algebras
Secondary: 16W25: Derivations, actions of Lie algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 3 , June 2021 , pp. 321 - 332
Check for updates
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by A. Sims

This paper was partially supported by the National Natural Science Foundation of China (Grant No. 11871021).

References

Aupetit, B. and Mouton, H. du T., ‘Trace and determinant in Banach algebras’, Stud. Math. 121 (1996), 115–136.Google Scholar
Ayupov, S. and Arzikulov, F., ‘2-local derivations on semi-finite von Neumann algebras’, Glagow Math. J. 56 (2014), 9–12.Google Scholar
Ayupov, S. and Kudaybergenov, K., ‘2-local derivations on von Neumann algebras’, Positivity 19 (2015), 445–455.Google Scholar
Barnes, B., ‘Examples of modular annihilator algebras’, Rocky Mountain J. Math. 1 (1971), 657–665.Google Scholar
Beidar, K., Martindale, W. and Mikhalev, A., Rings with Generalized Identities (Dekker, New York, 1996).Google Scholar
Bonsall, F. and Duncan, J., Complete Normed Algebras (Springer, Berlin, 1973).Google Scholar
Brešar, M. and Šemrl, P., ‘Finite rank elements in semisimple Banach algebras’, Stud. Math. 128 (1998), 287–298.Google Scholar
Cusack, J., ‘Jordan derivations on rings’, Proc. Amer. Math. Soc. 53 (1975), 321–324.Google Scholar
Dalla, L., Giotopoulos, S. and Katseli, N., ‘The socle and finite-dimensionality of a semiprime Banach algebras’, Stud. Math. 92 (1989), 201–204.Google Scholar
Herstein, I., ‘Jordan derivations of prime rings’, Proc. Amer. Math. Soc. 8 (1957), 1104–1110.Google Scholar
Johnson, B. and Sinclair, A., ‘Continuity of derivations and a problem of Kaplansky’, Amer. J. Math. 90 (1968), 1067–1073.Google Scholar
Kim, S. and Kim, J., ‘Local automorphisms and derivations on M n’, Proc. Amer. Math. Soc. 132 (2004), 1389–1392.Google Scholar
Lambrou, M. and Longstaff, W., ‘Finite rank operators leaving double triangles invariant’, J. Lond. Math. Soc. 45 (1992), 153–168.Google Scholar
Longstaff, W., ‘Remarks on semi-simple reflexive algebras’, in: Proceedings, Conference on Automatic Continuity and Banach Algebras, Canberra, Proceedings of the Centre for Mathematical Analysis, 21 (Centre for Mathematical Analysis, Australian National University, Canberra, 1989), 273–287.Google Scholar
Palmer, T., Banach Algebras and the General Theory of *-Algebras, Vol. I (Cambridge University Press, Cambridge, 1994).Google Scholar
Pang, Y. and Ji, G., ‘Algebraic isomorphisms and strongly double triangle subspace lattices’, Linear Algebra Appl. 422 (2007), 265–273.Google Scholar
Schulz, F., Brits, R. and Braatvedt, G., ‘Trace characterizations and socle identifications in Banach algebras’, Linear Algebra Appl. 472 (2015), 151–166.Google Scholar
Šemrl, P., ‘Local automorphisms and derivations on B (H)’, Proc. Amer. Math. Soc. 125 (1997), 2677–2680.Google Scholar
Yood, B., ‘Ideals in topological rings’, Canad. J. Math. 16 (1964), 28–45.Google Scholar
Zhang, J. and Li, H., ‘2-local derivations on digraph algebras’, Acta Math. Sin. (Chin. Ser.) 49 (2006), 1401–1406.Google Scholar