A modular theory of contractive projections on Banach spaces with applications to noncommutative Lp-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 C2 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 L4-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) 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 L4-space associated with a W*-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every 1 < p < ∞ with p ≠ 2, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative Lp-space is completely contractive. For p > 2, we obtain a weighted ternary description of its range in terms of a finite-dimensional W*-TRO.
Related
- Functional Calculus on Noncommutative Tori, II. Complex Powers, Logarithms, and Sectorial Projections
- A thickness boundary and modular obstructions for two-set radial projections
- Higher Noncommutative Motives and Higher K-theory
- Free modules of vector-valued modular forms and applications to Jacobi forms
- Zeta elements for p-ordinary modular forms over imaginary quadratic fields and applications
BibTeX
@misc{arhancet2026modular,
title = {{A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrmL^p$-spaces}},
author = {Cédric Arhancet},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03375},
eprint = {2610.03375},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03375},
}
RIS
TY - UNPB
TI - A modular theory of contractive projections on Banach spaces with applications to noncommutative L^p-spaces
AU - Cédric Arhancet
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03375
UR - https://arxiv.org/abs/2610.03375
AB - 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 C^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 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 L^4(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 L^4-space associated with a W^*-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every 1 < p < ∞ with p ≠ 2, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative L^p-space is completely contractive. For p > 2, we obtain a weighted ternary description of its range in terms of a finite-dimensional W^*-TRO.
ER -