The separable case of Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866911690201759744 |
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| author | He, Yixin Tang, Quanyu Zhang, Teng |
| author_facet | He, Yixin Tang, Quanyu Zhang, Teng |
| contents | 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 $τ$, then $L^2(M,τ)$ admits an orthonormal basis consisting of self-adjoint unitaries in $M$. Consequently, we affirm the separable case of the Kadison problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15006 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The separable case of Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors He, Yixin Tang, Quanyu Zhang, Teng Operator Algebras 46L10, 46L51, 46L30 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 $τ$, then $L^2(M,τ)$ admits an orthonormal basis consisting of self-adjoint unitaries in $M$. Consequently, we affirm the separable case of the Kadison problem. |
| title | The separable case of Kadison's problem on orthonormal bases of unitaries for type $\mathrm{II}_1$ factors |
| topic | Operator Algebras 46L10, 46L51, 46L30 |
| url | https://arxiv.org/abs/2605.15006 |