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

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 $\mathrm{C}^2$ plurisubharmonic function $Φ$, then, at any point of its range, the Levi form of $Φ$ defines