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

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Contributed by Cédric ARHANCET ↗

Version 1 / Nov 06, 2026/ arXiv: 2610.03375 / CC BY-SA 4.0

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 PP contracts a C2\mathrm{C}^2 plurisubharmonic function Φ\Phi, then, at any point of its range, the Levi form of Φ\Phi defines a tangent Hilbertian geometry for which PP 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 L4\mathrm{L}^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 L4(M)\mathrm{L}^4(\mathcal{M}) is 22-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 22-contractive projection is completely isometric to a rectangular noncommutative L4\mathrm{L}^4-space associated with a W∗\mathrm{W}^*-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every 1<p<∞1 < p < \infty with p≠2p \neq 2, we prove that every finite-rank 22-contractive projection on a tracial noncommutative Lp\mathrm{L}^p-space is completely contractive. For p>2p > 2, we obtain a weighted ternary description of its range in terms of a finite-dimensional W∗\mathrm{W}^*-TRO\mathrm{TRO}.

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