Contractively complemented subspaces of pre-symmetric spaces

TL;DR

This paper extends classical results on contractive projections in $L^1$-spaces and preduals of von Neumann algebras to the broader setting of preduals of $JBW^*$-triples, identifying conditions under which such projections exist.

Contribution

It significantly broadens the scope of previous results by establishing the existence of contractive projections for subspaces of preduals of $JBW^*$-triples, with specific restrictions.

Findings

Existence of contractive projections for subspaces of preduals of $JBW^*$-triples.

The result holds under the condition that the target $JBW^*$-triple does not contain a certain type of summand.

The theorem generalizes earlier results from $L^1$-spaces and von Neumann algebra preduals.

Abstract

In 1965, Ron Douglas proved that if $X$ is a closed subspace of an $L^1$-space and $X$ is isometric to another $L^1$-space, then $X$ is the range of a contractive projection on the containing $L^1$-space. In 1977 Arazy-Friedman showed that if a subspace $X$ of $C_1$ is isometric to another $C_1$-space (possibly finite dimensional), then there is a contractive projection of $C_1$ onto $X$. In 1993 Kirchberg proved that if a subspace $X$ of the predual of a von Neumann algebra $M$ is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of $M$ onto $X$. We widen significantly the scope of these results by showing that if a subspace $X$ of the predual of a $JBW^*$-triple $A$ is isometric to the predual of another $JBW^*$-triple $B$, then there is a contractive projection on the predual of $A$ with range $X$, as long as $B$ does not have a direct summand which is isometric to a space of the form $L^\infty(\Omega,H)$, where $H$ is a Hilbert space of dimension at least two. The result is false without this restriction on $B$.