A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrm{L}^p$-spaces

Abstract: We develop a differential-geometric approach to contractive projections on complex Banach spaces based on Levi forms of plurisubharmonic functions. We show that if a projection $P$ contracts a $\mathrm{C}<sup>2$ plurisubharmonic function $Φ$, then, at any point of its range, the Levi form of $Φ$ defines a tangent Hilbertian geometry for which $P$ becomes an orthogonal projection. When the range is finite-dimensional and the relevant Levi forms are nondegenerate, comparing the Levi geometries associated with different points leads naturally to positive relative operators and associated one-parameter groups. We apply this framework to tracial noncommutative $\mathrm{L}<sup>4$-spaces. Exploiting the cubic form of the duality mapping, we characterize contractive projections by a Jordan triple orthogonality property and prove that a projection on $\mathrm{L}<sup>4(\mathcal{M})$ is $2$-contractive if and only if it is completely contractive. When the ambient von Neumann algebra is finite-dimensional, we identify the relative operator associated with the left and right tangent forms on the range with a restriction of an inverse square root of a modular operator on a linking von Neumann algebra. This entails that the range of any $2$-contractive projection is completely isometric to a rectangular noncommutative $\mathrm{L}<sup>4$-space associated with a $\mathrm{W}<sup>*$-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every $1 &lt; p &lt; \infty$ with $p \neq 2$, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative $\mathrm{L}<sup>p$-space is completely contractive. For $p &gt; 2$, we obtain a weighted ternary description of its range in terms of a finite-dimensional $\mathrm{W}<sup>*$-$\mathrm{TRO}$.