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General Preservers of Quasi-Commutativity

Published:2010-05-20
Printed: Aug 2010
• Gregor Dolinar,
Faculty of Electrical Engineering, University of Ljubljana, Ljubljana, Slovenia
• Bojan Kuzma,
Institute of Mathematics, Physics, and Mechanics, Ljubljana, Slovenia
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Abstract

Let ${ M}_n$ be the algebra of all $n \times n$ matrices over $\mathbb{C}$. We say that $A, B \in { M}_n$ quasi-commute if there exists a nonzero $\xi \in \mathbb{C}$ such that $AB = \xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi \colon M_n \to M_n$ which preserve quasi-commutativity in both directions.
 Keywords: general preservers, matrix algebra, quasi-commutativity
 MSC Classifications: 15A04 - Linear transformations, semilinear transformations 15A27 - Commutativity 06A99 - None of the above, but in this section