On the structure of contractively decomposable projections on noncommutative $L^p$-spaces and Schatten spaces

We show that the range of a contractively decomposable projection on a noncommutative Haagerup $L^p$-space, $L^p(\mathcal{M},\phi )$, for $1<p<\mathrm{\infty }$, is completely isometrically isomorphic to a corner of a noncommutative $L^p$-space, that is $e{L}^p(\mathcal{N},\psi )(1-e)$, with $e\in \mathcal{N}$ a projection. In the setting of Schatten spaces, we obtain a more precise description: the range of a contractively decomposable projection on $S^p(K,H)$ is isometric to an ${\mathrm{\ell }}^p$ direct sum of subspaces of the form $S^p(K^{\mathrm{\prime }},H^{\mathrm{\prime }})$. Furthermore, we show that contractively 1-pseudo decomposable projections on Schatten spaces are automatically contractively decomposable, establishing the equivalence between these two notions in this setting.