PreprintCC BY 4.0arXiv
A Separable Hilbertian Operator Space with CBAP and No Completely Bounded Basis
Abstract
Liu and Ruan showed that a separable operator space has the completely bounded approximation property if and only if it admits a completely bounded frame. We show that, even for Hilbertian operator spaces, such a frame need not give rise to a completely bounded basis. More precisely, we construct a separable Hilbertian operator space with the $1$-CBAP but with no completely bounded basis. Our construction relies on a theorem of Oikhberg that provides a structural description of the completely bounded maps on a suitable operator space.
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BibTeX
@misc{guillot2026separable,
title = {{A Separable Hilbertian Operator Space with CBAP and No Completely Bounded Basis}},
author = {Dominique Guillot and Nicolas Cutrona},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26515},
eprint = {2609.26515},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26515},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Separable Hilbertian Operator Space with CBAP and No Completely Bounded Basis
AU - Dominique Guillot
AU - Nicolas Cutrona
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26515
UR - https://arxiv.org/abs/2609.26515
AB - Liu and Ruan showed that a separable operator space has the completely bounded approximation property if and only if it admits a completely bounded frame. We show that, even for Hilbertian operator spaces, such a frame need not give rise to a completely bounded basis. More precisely, we construct a separable Hilbertian operator space with the $1$-CBAP but with no completely bounded basis. Our construction relies on a theorem of Oikhberg that provides a structural description of the completely bounded maps on a suitable operator space.
ER -