Extreme points of the unit balls and isometries of noncommutative quasi-Banach Marcinkiewicz spaces

Abstract

We characterize extreme points of the (positive) unit balls of Marcinkiewicz spaces with respect to their natural quasi-norms. Having these results at hand, we show that every positive surjective isometry on a noncommutative Marcinkiewicz space is of elementary form. In particular, a linear mapping on a noncommutative Marcinkiewicz space over an atomic von Neumann algebra with all atoms having the same trace is a positive surjective isometry if and only if it is the restriction of a Jordan $\ast$-isomorphism which preserves the singular value functions. We also study (not necessarily positive) surjective isometries on weak $L_p$-spaces affiliated a $\sigma$-finite non-atomic von Neumann algebra for $1<p<\infty$, Marcinkiewicz sequence spaces, and weak ${\ell }_p$ operator ideals, $p>0$, as well as one-parameter groups of isometries on the latter two.