The rectifiable distance in the unitary Fredholm group

Esteban Andruchow, and Gabriel Larotonda. "The rectifiable distance in the unitary Fredholm group." Studia Mathematica 196.2 (2010): 151-178. <http://eudml.org/doc/285458>.

@article{EstebanAndruchow2010,
abstract = {Let $U_\{c\}()$ = u: u unitary and u-1 compact stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_\{∞\}$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_\{c\}()$. If $u₀,u₁,u ∈ U_\{c\}()$ and the geodesic β joining u₀ and u₁ in $U_\{c\}()$ satisfy $d_\{∞\}(u,β) < π/2$, then the map $f(s) = d_\{∞\}(u,β(s))$ is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in $U_\{c\}()$ is π/4. The same convexity property holds in the p-Schatten unitary groups $U_\{p\}()$ = u: u unitary and u-1 in the p-Schatten class for p an even integer, p ≥ 4 (in this case, the distance is strictly convex). The same results hold in the unitary group of a C*-algebra with a faithful finite trace. We apply this convexity result to establish the existence of curves of minimal length with given initial conditions, in the unitary orbit of an operator, under the action of the Fredholm group. We characterize self-adjoint operators A such that this orbit is a submanifold (of the affine space A + (), where () = compact operators).},
author = {Esteban Andruchow, Gabriel Larotonda},
journal = {Studia Mathematica},
keywords = {Fredholm group; rectifiable distance; geodesics},
language = {eng},
number = {2},
pages = {151-178},
title = {The rectifiable distance in the unitary Fredholm group},
url = {http://eudml.org/doc/285458},
volume = {196},
year = {2010},
}

TY - JOUR
AU - Esteban Andruchow
AU - Gabriel Larotonda
TI - The rectifiable distance in the unitary Fredholm group
JO - Studia Mathematica
PY - 2010
VL - 196
IS - 2
SP - 151
EP - 178
AB - Let $U_{c}()$ = u: u unitary and u-1 compact stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_{∞}$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_{c}()$. If $u₀,u₁,u ∈ U_{c}()$ and the geodesic β joining u₀ and u₁ in $U_{c}()$ satisfy $d_{∞}(u,β) < π/2$, then the map $f(s) = d_{∞}(u,β(s))$ is convex for s ∈ [0,1]. In particular, the convexity radius of the geodesic balls in $U_{c}()$ is π/4. The same convexity property holds in the p-Schatten unitary groups $U_{p}()$ = u: u unitary and u-1 in the p-Schatten class for p an even integer, p ≥ 4 (in this case, the distance is strictly convex). The same results hold in the unitary group of a C*-algebra with a faithful finite trace. We apply this convexity result to establish the existence of curves of minimal length with given initial conditions, in the unitary orbit of an operator, under the action of the Fredholm group. We characterize self-adjoint operators A such that this orbit is a submanifold (of the affine space A + (), where () = compact operators).
LA - eng
KW - Fredholm group; rectifiable distance; geodesics
UR - http://eudml.org/doc/285458
ER -