Completely isometric subspaces of noncommutative ${\mathrm{L}}^p$-spaces and contractive projections

Abstract

We investigate the relation between the complete isometry class of subspaces of noncommutative ${\mathrm{L}}^p$-spaces and their contractive complementability, where $1<p<\infty$ with $p\ne 2$. We show that if $P:{\mathrm{L}}^p(\mathcal{M})\to {\mathrm{L}}^p(\mathcal{M})$ is a positive contractive projection whose range is completely isometric to another noncommutative ${\mathrm{L}}^p$-space, then $P$ is necessarily completely positive. This provides a converse to the main result of [ArR24] and the positivity assumption on $P$ is essential. We further establish a rectangular analogue of this result. More precisely, we prove that every closed subspace of a noncommutative ${\mathrm{L}}^p$-space which is completely isometric to a rectangular noncommutative ${\mathrm{L}}^p$-space of the form $e{\mathrm{L}}^p(\mathcal{N})(1-e)$ is the range of a contractively decomposable projection. Combined with the known converse implication, this yields a characterization of the ranges of contractively decomposable projections as precisely the subspaces completely isometric to rectangular ${\mathrm{L}}^p$-spaces associated with ${\mathrm{W}}^{\ast }$-ternary rings of operators.

22 pages