Maps on matrix algebras preserving idempotents. (English) Zbl 1031.15003

Let \(M_n({\mathbb C})\) be the algebra of all \(n\times n\) matrices with entries from the field of complex numbers \({\mathbb C}\), \(P_n\) be the set of all idempotents in \(M_n({\mathbb C})\). P. Šemrl [ibid. 361, 161-179 (2003; Zbl 1035.15004)], characterized bijective continuous maps \(T : M_n({\mathbb C}) \to M_n({\mathbb C})\), \(n\geq 3\), satisfying the condition \[ (A-\lambda B)\in P_n \text{ if and only if } (T(A)-\lambda T(B))\in P_n \tag{1} \] for all \(A,B \in M_n({\mathbb C})\), \(\lambda \in {\mathbb C}\). Namely, he proved that such transformations are exhausted by the transposition and similarity.
In this paper the theorem by Šemrl is extended by removing the assumption of continuity and by relaxing the assumption of bijectivity to the assumption of surjectivity. It turns out that the characterization remains the same.
In addition, it is shown that the result also holds for \(n=1, 2\). Moreover, in this case all transformations satisfying the condition (1) are automatically surjective. The author states the question, if the assumption of surjectivity is really necessary in the case \(n>2\).


15A04 Linear transformations, semilinear transformations
15A30 Algebraic systems of matrices


Zbl 1035.15004
Full Text: DOI


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