Estelle Boffy | Search arXiv

arXiv2026

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, $L^p(\mathcal{M},φ)$, for $1<p<\infty$, is completely isometrically isomorphic to a corner of a noncommutative $L^p$-space, that is $eL^p(\mathcal{N},ψ)(1-e)$, 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 $\ell^p$ direct sum of subspaces of the form $S^p(K',H')$. 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.

math.FA↗

arXiv2026

Positive contractive projections in Schatten Spaces

We characterize the positively 1-complemented subspaces of $S^p$, for $1\leq p<\infty$, where $S^p$ denotes the Schatten spaces. Building on the work of Arazy and Friedman, who described the 1-complemented subspaces of $S^p$, for $1\leq p\neq 2 <\infty$, we establish that there are five mutually distinct types of indecomposable positively 1-complemented subspaces in $S^p$. Moreover, every positively 1-complemented subspace of $S^p$ can be expressed as a direct sum of some of these indecomposable subspaces.

math.FA↗