Strong k-commutativity preservers on complex standard operator algebras

Abstract

Let X be a complex Banach space with and be a standard operator algebra on X. Let k be a positive integer. For , the k-commutator of A and B is defined by with . Let be a map. We say that is strong k-commutativity preserving if for any . In this paper, it is shown that, if the range of contains all operators of rank one, then is strong k-commutativity preserving if and only if there exist a functional and a scalar with such that for all .

Keywords:

• k-commutators
• standard operator algebras
• complex Banach spaces
• preservers

AMS Subject Classifications:

• 47B49
• 47B47

Acknowledgements

The authors would like to give their thanks to the referees and the editor B. Kuzma for the helpful comments to improve the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is partially supported by Natural Science Foundation of China [grant number 11671294] [grant number 11671006] and Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi.