LINEAR MAPS ON \(M_{n} (\mathbb{C})\) PRESERVING INNER LOCAL SPECTRAL RADIUS ZERO

Abstract

Let M_n(\(\mathbb{C}\)) be the algebra of all complex n × n matrices.  Let x₀ be a nonzero vector in \(\mathbb{C}\)ⁿ.  We show that a linear unital map ϕ : M_n(\(\mathbb{C}\)) \(\longrightarrow\) M_n(\(\mathbb{C}\)) preserves the inner local spectral radius zero at x₀, if and only if there exists an invertible matrix A ∈ M_n(\(\mathbb{C}\)) such that ϕ(T ) = AT A⁻¹.