The Order of the Non-universal Tree of a Hilbert Space with Respect to the Haar Basis
Abstract
In his 1994 doctoral thesis, Bossard introduced the notion of the non-universal tree TNU(X) associated with each separable Banach space X which does not contain an isomorphic copy of C(2N). Together with the order operation defined on well-founded trees, we obtain a method of classifying the complexity of separable Banach spaces by the degree of isomorphism of finite-dimensional subspaces of C(2N). Despite further refinement of this concept in later years, we are unaware of any specific instances of direct exhibitions of the order of the non-universal tree for a concrete space and basis. We show that if H is any separable Hilbert space, then o(TNU(H)) = ω + 1 when taken with respect to the Haar basis for C(2N), demonstrating that the class of Hilbert spaces is the least complex class with respect to this measurement when considering this basis.
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BibTeX
@misc{whitmire2026order,
title = {{The Order of the Non-universal Tree of a Hilbert Space with Respect to the Haar Basis}},
author = {Sam Whitmire},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01885},
eprint = {2609.01885},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01885},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Order of the Non-universal Tree of a Hilbert Space with Respect to the Haar Basis
AU - Sam Whitmire
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01885
UR - https://arxiv.org/abs/2609.01885
AB - In his 1994 doctoral thesis, Bossard introduced the notion of the non-universal tree T_NU(X) associated with each separable Banach space X which does not contain an isomorphic copy of C(2^N). Together with the order operation defined on well-founded trees, we obtain a method of classifying the complexity of separable Banach spaces by the degree of isomorphism of finite-dimensional subspaces of C(2^N). Despite further refinement of this concept in later years, we are unaware of any specific instances of direct exhibitions of the order of the non-universal tree for a concrete space and basis. We show that if H is any separable Hilbert space, then o(T_NU(H)) = ω + 1 when taken with respect to the Haar basis for C(2^N), demonstrating that the class of Hilbert spaces is the least complex class with respect to this measurement when considering this basis.
ER -