Non-Subhomogeneity of Minimal Operator Systems over Positive Semidefinite and Lorentz Cones

• URL: http://arxiv.org/abs/2608.24648v1
• Date: Tue, 25 Aug 2026 14:58:09 GMT
• Title: Non-Subhomogeneity of Minimal Operator Systems over Positive Semidefinite and Lorentz Cones
• Abstract summary: The minimal operator systems over $rm Mat_k(mathbb C)_+$, $kgeqslant2$, and the Lorentz cones $mathcal L_m$, $mgeqslant4$, are not subhomogeneous.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The minimal operator systems over ${\rm Mat}_k(\mathbb C)_+$, $k\geqslant2$, and the Lorentz cones $\mathcal L_m$, $m\geqslant4$, are not subhomogeneous. For the $2\times2$ cone we construct extreme positive maps of arbitrarily large output dimension. Positive retracts give the remaining cases. Equivalently, for fixed $k\geqslant2$ and $d$, there exist $s>d$ and an entangled positive operator on $\mathbb C^k\otimes\mathbb C^s$ such that every compression of the second factor to $\mathbb C^d$ is separable.

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