The solution to Kadison's problem on orthonormal bases of unitaries for type II1 factors
Abstract
In 1967, Kadison asked whether every type II1 factor admits an orthonormal basis, with respect to its trace, consisting of unitaries. We resolve this problem in full generality and, more broadly, characterize the diffuse finite von Neumann algebras admitting such bases consisting of symmetries. Let M be a diffuse finite von Neumann algebra with a faithful normal tracial state τ, let κ be the density character of L2(M,τ), and identify M with its canonical image in L2(M,τ). We prove that there exists a family B⊂ s∈ M:s=s*=s−1, τ(s)=0 such that 1∪ B is an orthonormal basis of L2(M,τ) if and only if the density character of L2(zM,τ(z)−1τ|zM) equals κ for every nonzero central projection z∈ Z(M). In particular, every type II1 factor admits an orthonormal basis consisting of unitaries, thereby answering Kadison's question affirmatively. We also provide a Lean 4 formalization of the main results.
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BibTeX
@misc{he2026solution,
title = {{The solution to Kadison's problem on orthonormal bases of unitaries for type $\mathrmII_1$ factors}},
author = {Yixin He and Quanyu Tang and Zongben Xu and Teng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.18405},
eprint = {2608.18405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.18405},
}
RIS
TY - UNPB
TI - The solution to Kadison's problem on orthonormal bases of unitaries for type II_1 factors
AU - Yixin He
AU - Quanyu Tang
AU - Zongben Xu
AU - Teng Zhang
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.18405
UR - https://arxiv.org/abs/2608.18405
AB - In 1967, Kadison asked whether every type II_1 factor admits an orthonormal basis, with respect to its trace, consisting of unitaries. We resolve this problem in full generality and, more broadly, characterize the diffuse finite von Neumann algebras admitting such bases consisting of symmetries. Let M be a diffuse finite von Neumann algebra with a faithful normal tracial state τ, let κ be the density character of L^2(M,τ), and identify M with its canonical image in L^2(M,τ). We prove that there exists a family B⊂ s∈ M:s=s^*=s^−1, τ(s)=0 such that 1∪ B is an orthonormal basis of L^2(M,τ) if and only if the density character of L^2(zM,τ(z)^−1τ|_zM) equals κ for every nonzero central projection z∈ Z(M). In particular, every type II_1 factor admits an orthonormal basis consisting of unitaries, thereby answering Kadison's question affirmatively. We also provide a Lean 4 formalization of the main results.
ER -