Linear Maps Preserving the Higher Numerical Ranges of Tensor Products of Matrices

For a positive integer n, let Mn be 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 →Mmn satisfies Wk(φ(A⊗B)) = Wk(A⊗B) for all A ∈Mm and B ∈Mn if and only if there is a unitary U ∈Mmn such that one of the following holds. (i) For all A ∈Mm, B ∈Mn, φ(A⊗B) = U(φ(A⊗B))U. (ii) mn = 2k and for all A ∈Mm, B ∈Mn, φ(A⊗B) = (tr (A⊗B)/k)Imn − U(φ(A⊗B))U, where (1) φ is the identity map A ⊗ B 7→ A ⊗ B or the transposition map A ⊗ B 7→ (A ⊗ B), or (2) min{m,n} ≤ 2 and φ has the form A⊗B 7→ A⊗B or A⊗B 7→ A ⊗B. 2010 Math. Subj. Class.: 15A69, 15A86, 15A60, 47A12.