2022/12/25 by Qian, Wenhua, Xiao, Dandan, Tao, Tanghong +2
#47A25 #47B49 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2212.12895
Let \mathcal H be a finite dimensional complex Hilbert space with dimension n ≥ 3 and \mathcal P(\mathcal H) the set of projections on \mathcal H. Let φ: \mathcal P(\mathcal H) → \mathcal P(\mathcal H) be a surjective map. We show that φ shrinks the joint spectrum of any two projections if and only if it is joint spectrum preserving for any two projections and thus is induced by a ring automorphism on \mathbb C in a particular way. In addition, for an arbitrary k ≥ 3, φ shrinks the joint spectrum of any k projections if and only if it is induced by a unitary or an anti-unitary. Assume that ϕ is a surjective map on the Grassmann space of rank one projections. We show that ϕ is joint spectrum preserving for any n rank one projections if and only if it can be extended to a surjective map on \mathcal P(H) which is spectrum preserving for any two projections. Moreover, for any k >n, ϕ is joint spectrum shrinking for any k rank one projections if and only if it is induced by a unitary or an anti-unitary.