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arXiv · 2605.15006

The separable case of Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors

Yixin He·Quanyu Tang·Teng Zhang
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Abstract

In 1967, Kadison asked ``does every type $\mathrm{II}_1$ factor have an orthonormal (with respect to the trace) basis consisting of unitaries?'' Using a noncommutative Lyapunov theorem of Akemann and Weaver, we prove that if $M$ is a separable diffuse finite von Neumann algebra with a normal faithful trace $\tau$, then $L^2(M,\tau)$ admits an orthonormal basis consisting of self-adjoint unitaries in $M$. Consequently, we affirm the separable case of the Kadison problem.

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Yixin He, Quanyu Tang, Teng Zhang. 2026-05-14. The separable case of Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors. https://arxiv.org/abs/2605.15006

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