Metric geometry in infinite dimensional Stiefel manifolds

Abstract

Let 3 be a separable Banach ideal in the space of bounded operators acting in a Hilbert space 7T and T the set of partial isometries in 7T. Fix v g T. In this paper we study metric properties of the 3-Stiefel manifold associated to v, namely Stj(v) = {v0 eZ: v-v0 e 3, j(vqV0, v*v) =0), MSC: primary 22E65 secondary 47B10, 58B20 where j(.) is the Fredholm index of a pair of projections. Let ¿Yj(7T) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in 3. Then St-j(v) coincides with the orbit of v under the action of ¿Y-jCH) z ¿Y-jCH) on T given by Keywords: Partial isometry Banach ideal Finsler metric Minimal curves (u, w) • Vo = uvow*. u, w 6 Uj(7i) and Vo 6 Stj(v). We endow 5tj(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra ofi/jCH) zT/jCH) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of ¿Yj(7T) z i/jfH) by the isotropy group. Hence this metric defines the quotient topology in Stj(v). The other results concern with minimal curves in 3-Stiefel manifolds when the ideal 3 is fixed as the compact operators in 7T. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length.

Geometry of unitary orbits of pinching operators

“…We begin by recalling that (U I , d U I ) is a complete metric space and G is d U I -closed in U I (see e.g. [12,Lemma 2.4]). Thus the quotient distanceḋ U I is well defined.…”

“…The Section 5 is concerned with the metric structure of U I (P ). Motivated by similar results on other homogeneous spaces [2,12] we study the rectifiable distance induced by quotient Finsler metric on U I (P ). Under the assumption that the quotient topology on U I (P ) coincide with the inherited topology from B(I), we prove that the rectifiable distance defines these topologies.…”

The set of partial isometries as a quotient Finsler space

“…Starting with Halmos and Mc Laughlin [14], who characterized the connected components. Later on, other papers appeared studying geometric or topological aspects of the set of partial isometries, for instance: [16], [18], [19], [1], [2], [4], [8], [9].…”

“…This problem is also related with non-commutative C * -metrics and Leibniz seminorms [22]. An important background to the present work are the papers by E. Chiumiento [8], [9]. In these papers, quotient metrics and minimal liftings are studied in the orbits of partial isometries under the action of the so called restricted groups of unitaries (i.e., unitaries which are of the form 1 + K, for K in an operator ideal).…”

Geometría en espacios homogéneos de grupos unitarios

“…Los resultados originales expuestos en este trabajo están basados en los artículos [ACL09], [Ch08], [Ch09a] y [Ch09b]. Los primeros dos artículos son acerca de geometría métrica en espacios homogéneos del grupo unitario de unálgebra de von Neumann finita.…”