Nonsurjective Numerical Radius Isometries Between Matrix Algebras

Let ℳ_n(ℂ) be the algebra of all n× n complex matrices, and denote by V ( T ) and v ( T ) the numerical range and numerical radius of any matrix T∈ℳ_n(ℂ) , respectively. Let ϕ : ℳ_m(ℂ)⟶ℳ_n(ℂ) be a nonsurjective linear maps that satisfy v(ϕ (A))=v(A) for all A∈ℳ_m(ℂ) . In this note, we show that if m≥ 2 then ϕ (V) is spectraloid for any unitary matrix V∈ℳ_m(ℂ) . Also, under the mild condition that ‖ϕ (1_m)‖≤ 1 , we prove that then there are unimodular scalars λ _1, … , λ _p in the spectrum of ϕ (1_m) such that V(A)⊂ V( ⊕ _k=1^pλ_kϕ (A)) for all A∈ℳ_m(ℂ) .