2025/01/12 by Chirvasitu, Alexandru, Gogić, Ilja, Tomašević, Mateo · 1 citation
#15A27 #47A10 #47B15 #47B49 #54D05 #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #Rings and Algebras (math.RA) #Spectral Theory (math.SP)
paper · doi:10.48550/arxiv.2501.06840
For a positive integer n let Xn be either the algebra Mn of n × n complex matrices, the set Nn of all n × n normal matrices, or any of the matrix Lie groups GL(n), SL(n) and U(n). We first give a short and elementary argument that for two positive integers m and n there exists a continuous spectrum-shrinking map ϕ: Xn → Mm (i.e. sp(ϕ(X))⊆ sp(X) for all X ∈ Xn) if and only if n divides m. Moreover, in that case we have the equality of characteristic polynomials kϕ(X)(⋅) = kX(⋅)^(m)/(n) for all X ∈ Xn, which in particular shows that ϕ preserves spectra. Using this we show that whenever n ≥ 3, any continuous commutativity preserving and spectrum-shrinking map ϕ: Xn → Mn is of the form ϕ(⋅)=T(⋅)T-1 or ϕ(⋅)=T(⋅)tT-1, for some T∈ GL(n). The analogous results fail for the special unitary group SU(n) but hold for the spaces of semisimple elements in either GL(n) or SL(n). As a consequence, we also recover (a strengthened version of) Šemrl's influential characterization of Jordan automorphisms of Mn via preserving properties.