Linear maps preserving the higher numerical ranges of tensor products of matrices

For a positive integer n, let Mnbe the set of n × n complex matrices. Suppose m, n ≥ 2 are positive integers and k ∈ {1,..., mn - 1}. Denote by Wk(X) the k-numerical range of a matrix X ∈ Mmn. It is shown that a linear map Φ: Mmn→ Mmnsatisfies (Formula presented.) for all A ∈ Mmand B ∈ Mnif and only if there is a unitary U ∈ Mmnsuch that one of the following holds. (Formula presented.) where (1) φ is the identity map A ⊗ B {mapping} A ⊗ B or the transposition map A ⊗ B {mapping} (A⊗B)t, or (2) min{m, n} ≤ 2 and φ has the form A⊗B {mapping} A⊗Btor A⊗B {mapping} At⊗ B.