Hua’s fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of n × n hermitian matrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. We managed to make three natural improvements for the complex hermitian matrices, i.e. we removed the bijectivity assumption, we replaced the assumption of preserving adjacency in both directions by the assumption of preserving adjacency in one direction only, and we also characterized maps acting between the spaces of hermitian matrices of different sizes.
COBISS.SI-ID: 14901337
The problem of characterizing multiplicative maps on matrices over a principal ideal domain was solved by Jodeit and Lam. Pierce showed that their result does not hold true for matrices over an arbitrary integral domain. The motivation to study multiplicative maps on matrices over an arbitrary division ring comes from the Wedderburn-Artin theorem. We managed to describe the general form of endomorphisms of matrix semigroups over an arbitrary not necessarily commutative division ring.
COBISS.SI-ID: 14651737