The rectifiable distance in the unitary Fredholm group
- Andruchow, Esteban
- Larotonda, Gabriel Andrés
DOIAbstract
Let Uc(H)={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 Uc(H). If u0,u1,u∈Uc(H) and the geodesic β joining u0 and u1 in Uc(H) 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 Uc(H) is π/4. The same convexity property holds in the p-Schatten unitary groups Up(H)={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 ...
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