Grassmannians of a finite algebra in the strong operator topology

ESTEBAN ANDRUCHOW, LÁZARO RECHT

INTERNATIONAL JOURNAL OF MATHEMATICS

World Scientific

Lugar: Singapore; Año: 2006 vol. 17 p. 477 - 491

0129-167X

If $\m$ is a type II$_1$ von Neumann algebra with a faithfultrace $\tau$, we consider the set $\p$ of selfadjoint projections of $\m$ as asubset of the Hilbert space $\h=\l2$. We prove that though it is not adifferentiable submanifold, the geodesics of the natural Levi-Civita connectiongiven by the trace have minimal length. More precisely: the curves of the form$\gamma(t)=e^{itx}pe^{-itx}$ with $x^*=x$, $pxp=(1-p)x(1-p)=0$ have minimallength when measured in the Hilbert space norm of $\h$, provided that the {\itoperator}norm $\|x\|$ is less or equal than $\pi/2$. Moreover, any two projections whichare unitary equivalent are joined by at least one such minimal geodesic, and only unitary equivalent projections can be joined by a smooth curve.Finally, we prove that these geodesics have also minimal length if one measuresthem with the Schatten $k$-norms of $\tau$, $\|x\|_k=\tau((x^*x)^{k/2})^{1/k}$,for all $k\in\zR$, $k\ge 0$. We also characterize curves of unitaries which haveminimal length with these $k$-norms.