The rectifiable distance in the unitary Fredholm group

Esteban Andruchow¹ ; Gabriel Larotonda¹

¹Instituto de Ciencias Universidad Nacional de General Sarmiento J. M. Gutierrez 1150 (B1613GSX) Los Polvorines Buenos Aires, Argentina and Instituto Argentino de Matemática “Alberto Calderón”, CONICET

Studia Mathematica, Tome 196 (2010) no. 2, pp. 151-178

Résumé

Let $U_{\rm c}(\mathcal{H})=\{u: u \hbox{ unitary and } u-1 \hbox{ compact}\}$ stand for the unitary Fredholm group. We prove the following convexity result. Denote by $d_\infty$ the rectifiable distance induced by the Finsler metric given by the operator norm in $U_{\rm c}(\mathcal{H})$. If $u_0,u_1,u\in U_{\rm c}(\mathcal{H})$ and the geodesic $\beta$ joining $u_0$ and $u_1$ in $U_{\rm c}(\mathcal{H})$ satisfy $d_\infty(u,\beta)\pi/2$, then the map $f(s)=d_\infty(u,\beta(s))$ is convex for $s\in[0,1]$. In particular, the convexity radius of the geodesic balls in $U_{\rm c}({\mathcal H})$ is $\pi/4$. The same convexity property holds in the $p$-Schatten unitary groups $U_p(\mathcal{H})=\{u: u$ unitary and $u-1$ in the $p$-Schatten class$\}$ for $p$ an even integer, $p\ge 4$ (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 $A+\mathcal{K}(\mathcal{H})$, where $\mathcal{K}(\mathcal{H})=\hbox{compact operators}$).

Détail

DOI : 10.4064/sm196-2-4

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