Linear maps on matrices preserving commutativity up to a factor

Abstract

In this note we determine the general form of all bijective linear maps on the algebra of n × n complex matrices or on the space of n × n self-adjoint matrices which preserve the relation of commutativity up to a factor in both directions.

Keywords:

• Preservers
• Matrix spaces
• Commutativity up to a factor

1991 Mathematics Subject Classifications: :

• Primary
• 15A04
• 15A27

Acknowledgements

The author is very grateful to the referee for detecting an error in the proof of the original version of Lemma, for correcting it, and for presenting a proof of Theorem substantially shorter than the one given in the first version of the article. The author was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T043080, T046203 and by a joint Hungarian-Slovene grant (Reg. No. SLO-5/05).