Examples of homogeneous manifolds with uniformly bounded metric projection

CHIUMIENTO EDUARDO

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UNION MATEMATICA ARGENTINA

Año: 2012 vol. 53 p. 13 - 23

0041-6932

Let M be a finite von Neumann algebra with a faithful normal trace tr. Denote by L_a ^p  the skew-hermitian part of the non-commutative L^p space associated with (M,tr). Let  1< p <infty,  z in L_a^p and S be a real closed subspace of  L_a ^p . The metric projection Q:L_a ^p -->S is defined for every z in L_a ^p as the unique operator Q(z) in S such that |z- Q(z) |_p = min_{y in S} | z -y|_p.  We show the relation between  metric projection and  metric geometry of homogeneous spaces of the unitary group U of M, endowed with a  Finsler quotient metric induced by the p-norms of tr, |x|_p= r(|x|^p)^1/p, p an even integer. The problem of finding minimal curves in such homogeneous spaces leads to the notion of uniformly bounded metric projection. Then we show examples of metric projections of this type.