2020/04/02 by Esteban Andruchow, Andruchow, Esteban
Mathematics · #46L05 #53C22 #58B20 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA #msc:46L05 #msc:53C22 #msc:58B20
paper · pdf · doi:10.48550/arxiv.2004.01158
arxiv created 2020/04/16 · arxiv updated 2020/04/20
Let \cal C(\cal H)=\cal B(\cal H) / \cal K(\cal H) be the Calkin algebra (\cal B(\cal H) the algebra of bounded operators on the Hilbert space \cal H, \cal K(\cal H) the ideal of compact operators and π:\cal B(\cal H)→ \cal C(\cal H) the quotient map), and \cal P_\cal C(\cal H) the differentiable manifold of selfadjoint projections in \cal C(\cal H). A projection p in \cal C(\cal H) can be lifted to a projection P∈\cal B(\cal H): π(P)=p. We show that given p,q ∈ \cal P_\cal C(\cal H), there exists a minimal geodesic of \cal P_\cal C(\cal H) which joins p and q if and only there exist lifting projections P and Q such that either both N(P-Q± 1) are finite dimensional, or both infinite dimensional. The minimal geodesic is unique if p+q- 1 has trivial anhihilator. Here the assertion that a geodesic is minimal means that it is shorter than any other piecewise smooth curve γ(t) ∈ \cal P_\cal C(\cal H), t ∈ I, joining the same endpoints, where the length of γ is measured by ∫I ‖γ(t)‖ d t.