Multiplicative maps preserving the higher rank numerical ranges and radii

Let Mn be the semigroup of n×n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in Mn including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λk(A) is the rank-k numerical range and rk(A) is the rank-k numerical radius of A∈Mn. Multiplicative maps ϕ:S→Mn satisfying rk(ϕ(A))=rk(A) are characterized. From these results, one can deduce the structure of multiplicative preservers of Λk(A).