2011/05/09 by Chiumiento, Eduardo, Lucero, María E. Di Iorio y
#46T05 (Primary) #47B10 #47B49 #57N20 #58B20 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1105.1686
Let I be a symmetrically-normed ideal of the space of bounded operators acting on a Hilbert space H. Let pi1 w (1≤ w ≤ ∞) be a family of mutually orthogonal projections on H. The pinching operator associated with the former family of projections is given by P: I --> I, P(x)=∑i=1w pi x pi. Let UI denote the Banach-Lie group of the unitary operators whose difference with the identity belongs to I. We study several geometric properties of the orbit UI(P)=Lu P Lu^* : u ∈ UI, where Lu is the left representation of UI on the algebra B(I) of bounded operators acting on I. The results include necessary and sufficient conditions for UI(P) to be a submanifold of B(I). Special features arise in the case of the ideal K of compact operators. In general, UK(P) turns out to be a non complemented submanifold of B(K). We find a necessary and sufficient condition for UK(P) to have complemented tangent spaces in B(K). We also show that UI(P) is a covering space of another natural orbit of P. A quotient Finsler metric is introduced, and the induced rectifiable is studied. In addition, we give an application of the results on UI(P) to the topology of the UI-unitary orbit of a compact normal operator.