Multiplicative preservers of higher-dimensional numerical ranges

Let H be a complex separable Hilbert space and k be a positive integer smaller than the dimension of H  . In this paper, we characterize multiplicative maps ϕ:B(H)→B(H)ϕ:B(H)→B(H) that preserve the k-dimensional numerical ranges (no linearity, surjectivity or continuity of ϕ   is assumed). It is shown that such a map is a C⁎C⁎-isomorphism.