Metrics in the sphere of a C*-module

Abstract

Given a unital C * -algebra A and a right C * -module X over A, we consider the problem of finding short smooth curves in the sphere S X = {x ∈ X :< x, x >= 1}. Curves in S X are measured considering the Finsler metric which consists of the norm of X at each tangent space of S X . The initial value problem is solved, for the case when A is a von Neumann algebra and X is selfdual: for any element x 0 ∈ S X and any tangent vector v at x 0 , there exists a curve γ(t) = e tZ (x 0 ), Z ∈ L A (X ), Z * = −Z and Z ≤ π, such that γ(0) = x 0 andγ(0) = v, which is minimizing along its path for t ∈ [0, 1]. The existence of such Z is linked to the extension problem of selfadjoint operators. Such minimal curves need not be unique. Also we consider the boundary value problem: given x 0 , x 1 ∈ S X , find a curve of minimal length which joins them. We give several partial answers to this question. For instance, let us denoteb

Metric geometry of infinite-dimensional Lie groups and their homogeneous spaces

“…Remark 3. 10. Since the distance from a point to a closed set is positive in a metric space, this shows that if dist µ is in fact a metric in G, then d is a metric in M (whether it is reversible or not depends exclusively on the fact of the homogeneity of µ).…”

“…Actually, it will suffice to assume that exp | B R is a bijection with its image V R , and every piecewise C 1 path γ ⊂ V R starting at g = 1 has a unique piecewise Remark 4. 10. The straightforward example of our definition is given by a Banach-Lie group K and a bi-invariant norm µ = | · | that is equivalent to the original norm modeling the Banach space Lie(K).…”

“…34. Let U ⊂ Lie(K) be as in (10), with H ⊂ K a normal subgroup and K/H locally exponential. If e z = e v e w with v, w, z ∈ U and w ∈ Lie(H), then z − v ∈ Lie(H).…”

“…8. Let H ⊂ K be a normal Lie subgroup such that K 0 = K/H is locally exponential, assume B R ⊂ U where U ⊂ Lie(K) is as in (10). Let v ∈ Lie(K) with |v| < R/2.…”

“…Sphere of a C * -Hilbert module Quotient metrics for the sphere S X of Hilbert C * -module over a C * -algebra A were considered by Andruchow and Varela in [10], in connection with the ideas of the quotient metric for the quotients of unitary groups of C * algebras U A /U B developed by Recht et al in [32,33].…”