Abstract. Let $\phi $ be a bijective continuous map on the algebra of all $n\times n$ matrices, $n\ge2$, preserving commutativity in both directions (no linearity is assumed). Then $\phi $ is a similarity transformation composed with a locally polynomial map, possibly composed with the transposition and the entrywise complex conjugation. The main tool in the proof is the characterization of bijective maps defined on rank one idempotents that preserve orthogonality in both directions. This result, related to some problems in quantum mechanics, is considered also in the infinite-dimensional setting.

AMS Subject Classification (1991): 15A27, 47B49, 51A05

Received May 13, 2005, and in revised form September 16, 2005. (Registered under 5900/2009.)