The phase diagram of injective-to-projective tensor distortion
Abstract
For finite-dimensional Banach spaces E and F, set ρ(E,F):=sup0≠ z∈ E⊗ Fπ(z)/ε(z). The square growth of ρ(ℓpd,ℓqd) was determined by Bonet, Defant, Peris and Ramanujan. We study the rectangular problem with the two dimensions varying independently. If r=min{n,m} and η(t)=min{1/t,1/t'}, then, whenever p and q lie on the same side of $2$, ρ(ℓpn,ℓqm)asympp,q r1/2min{nη(p),mη(q)}. Thus the classical square exponent splits into two dimensional scales. In the mixed range 1≤ p≤2≤ q≤∞, writing a=1/p and b=1/q, we obtain the lower estimate ρp,q(n,m)≳p,q max {rmin{a+b,2−a−b}, mb−1/2r1/2min{n1−a,m1/2}}, and the upper estimate ρp,q(n,m)≲p,q min {n1−ar1−b,mb ra,r}. Moreover, if (n−m)(a+b−1)≤0, then ρp,q(n,m)asympp,q rmin{a+b, 2−a−b}. In the interior mixed range 1<p<2<q<p', if αp,q:= ((a+b−1)(1/2−b))/(a−1/2), then ρp,q(n,m)≲p,qm1−αp,q, supn≥1ρp,q(n,m)asympp,qm1−αp,q. The corresponding statements in the reversed mixed range follow by duality and symmetry.
Related
- Finitistic injective dimension exceeding finitistic projective dimension for a commutative ring
- Reciprocity for projective and injective direct summands
- On strongly FP-injective accessibility and weakly FP-projective periodicity
- Cyclic group tensors and the volume ratio of projective tensor products
- Resultant multiplicity via projective degrees and applications to tensor eigenvalues
BibTeX
@misc{nunezalarcon2026phase,
title = {{The phase diagram of injective-to-projective tensor distortion}},
author = {Daniel Nunez-Alarcon and Daniel Pellegrino and Joedson Santos},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04469},
eprint = {2609.04469},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04469},
}
RIS
TY - UNPB
TI - The phase diagram of injective-to-projective tensor distortion
AU - Daniel Nunez-Alarcon
AU - Daniel Pellegrino
AU - Joedson Santos
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.04469
UR - https://arxiv.org/abs/2609.04469
AB - For finite-dimensional Banach spaces E and F, set ρ(E,F):=sup_0≠ z∈ E⊗ Fπ(z)/ε(z). The square growth of ρ(ℓ_p^d,ℓ_q^d) was determined by Bonet, Defant, Peris and Ramanujan. We study the rectangular problem with the two dimensions varying independently. If r=min{n,m} and η(t)=min{1/t,1/t'}, then, whenever p and q lie on the same side of $2$, ρ(ℓ_p^n,ℓ_q^m)asymp_p,q r^1/2min{n^η(p),m^η(q)}. Thus the classical square exponent splits into two dimensional scales. In the mixed range 1≤ p≤2≤ q≤∞, writing a=1/p and b=1/q, we obtain the lower estimate ρ_p,q(n,m)≳_p,q max {r^min{a+b,2−a−b}, m^b−1/2r^1/2min{n^1−a,m^1/2}}, and the upper estimate ρ_p,q(n,m)≲_p,q min {n^1−ar^1−b,m^b r^a,r}. Moreover, if (n−m)(a+b−1)≤0, then ρ_p,q(n,m)asymp_p,q r^min{a+b, 2−a−b}. In the interior mixed range 1<p<2<q<p', if α_p,q:= ((a+b−1)(1/2−b))/(a−1/2), then ρ_p,q(n,m)≲_p,qm^1−α_p,q, sup_n≥1ρ_p,q(n,m)asymp_p,qm^1−α_p,q. The corresponding statements in the reversed mixed range follow by duality and symmetry.
ER -