A variant of Šemrl's preserver theorem for singular matrices

For positive integers 1≤k≤n let M_n be the algebra of all n×n complex matrices and M_n^≤k its subset consisting of all matrices of rank at most k. We first show that whenever k>[Formula presented], any continuous spectrum-shrinking map ϕ:M_n^≤k→M_n (i.e. sp(ϕ(X))⊆sp(X) for all X∈M_n^≤k) either preserves characteristic polynomials or takes only nilpotent values. Moreover, for any k there exists a real analytic embedding of M_n^≤k into the space of n×n nilpotent matrices for all sufficiently large n. This phenomenon cannot occur when ϕ is injective and either k>n−n or the image of ϕ is contained in M_n^≤k. We then establish a main result of the paper – a variant of Šemrl's preserver theorem for M_n^≤k: if n≥3, any injective continuous map ϕ:M_n^≤k→M_n^≤k that preserves commutativity and shrinks spectrum is of the form ϕ(⋅)=T(⋅)T⁻¹ or ϕ(⋅)=T(⋅)^tT⁻¹, for some invertible matrix T∈M_n. Moreover, when k=n−1, which corresponds to the set of singular n×n matrices, this result extends to maps ϕ which take values in M_n. Finally, we discuss the indispensability of assumptions in our main result.