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On the Shalit-Shamovich Spectral Radius
Abstract
We show that the spectral radius introduced by Shalit and Shamovich uniquely determines the underlying operator space structure: If E_1 and E_2 are operator space structures on C^d such that the corresponding spectral radii ρ_{E_1} and ρ_{E_2} agree on every d-tuple of operators, then the identity map is completely isometric. This answers a question of Scherer, Shalit and Shamovich. Moreover, we obtain an analogous result for the minimal spectral radii, and showcase an application to the completely bounded isomorphism problem for algebras of bounded non-commutative functions on operator space unit balls.
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BibTeX
@misc{tornes2026shalitshamovich,
title = {{On the Shalit-Shamovich Spectral Radius}},
author = {Maximilian Tornes},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00816},
eprint = {2610.00816},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00816},
}
RIS
TY - UNPB
TI - On the Shalit-Shamovich Spectral Radius
AU - Maximilian Tornes
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00816
UR - https://arxiv.org/abs/2610.00816
AB - We show that the spectral radius introduced by Shalit and Shamovich uniquely determines the underlying operator space structure: If E_1 and E_2 are operator space structures on C^d such that the corresponding spectral radii ρ_{E_1} and ρ_{E_2} agree on every d-tuple of operators, then the identity map is completely isometric. This answers a question of Scherer, Shalit and Shamovich. Moreover, we obtain an analogous result for the minimal spectral radii, and showcase an application to the completely bounded isomorphism problem for algebras of bounded non-commutative functions on operator space unit balls.
ER -