On classical adjoint-commuting mappings between matrix algebras
This paper extends the classification of additive inversion-preserving maps (see, e.g., p. 40 of \textit{M. Brešar} [Derivations, homomorphisms, and related mappings of rings and Banach algebras, Ph.D. Thesis, Univ. Ljubljana (in Slovene) (1990)]) in the spirit of \textit{G. Dolinar} and \textit{P. Šemrl} [Linear Algebra Appl. 348, No. 1--3, 189--192 (2002; Zbl 0998.15011)] to achieve that an additivity assumption is no longer required but rather built into the preserving property. Let \(M_n(F)\) denote the algebra of \(n\)-by-\(n\) matrices over a commutative field \(F\). One of the anti-multiplicative functions on \(M_n(F)\) is the adjoint operation \(A\mapsto\text{adj}\,A\), also known as the transposed matrix of cofactors. It maps invertible \(A\) into \((\det A)A^{-1}\), and annihilates every matrix with rank smaller than \(n-1\). The authors classify surjective maps \(\Phi\), between \(M_n(F)\) onto \(M_m(F)\), \(m,n\geq 3\), with the property that \[ \Phi(\text{adj}\,(A-B))=\text{ adj}\,(\Phi(A)-\Phi(B)). \] As usual, it turns out that such maps are automatically additive and standard. It is also shown that the surjectivity assumption is redundant under the stronger hypotheses that \(\Phi(\text{adj}\,(A-\alpha B))=\text{adj}\,(\Phi(A)-\alpha \Phi(B))\) holds for every \(\alpha\in F\), the field \(F\) has sufficiently many elements, and the identity is not annihilated (it turns out that such maps are automatically linear). If the identity is annihilated then \(\Phi\) must annihilate every invertible matrix and every matrix of rank-one. Examples of such nonstandard maps are also given.
scientific article; zbMATH DE number 5689424