Does every type II_1 factor M, equipped with its normalized trace tau, admit an orthonormal basis of L^2(M,tau) consisting of unitaries in M? Equivalently, does there exist a family u_i_i∈ I⊂ M such that each u_i is unitary, tau(u_i^u_j)=delta_ij for all i,j∈ I, and the linear span of u_i:i∈ I is dense in L^2(M,tau)?

Kadison’s orthonormal bases of unitaries problem

Kadison posed the question in 1967: whether every type $\mathrm{II}_1$ factor admits a complete orthonormal basis of trace vectors, equivalently an orthonormal basis of unitaries in the associated $L^2$ -space. A new preprint claims a full solution, extending an earlier claimed solution for the separable case.

Known results

Group von Neumann algebras of countable discrete groups and certain group-measure-space constructions were already known cases.
De and Mukherjee (2023) constructed uniformly bounded self-adjoint orthonormal bases for every separable diffuse finite von Neumann algebra, without proving that the basis elements are unitaries.
He, Tang, and Zhang (2026) claimed the unitary-basis result for diffuse finite von Neumann algebras with separable $L^2$ -space.

August 2026 full-resolution claim

The abstract of a new preprint claims to extend the separable result to arbitrary density character using a transfinite construction, thereby settling Kadison’s question for all type $\mathrm{II}_1$ factors. No peer-reviewed verification, independent exposition, identified gap, or retraction was found.

Current status (as of August 2026): The separable case and now the general case are claimed in preprints, but the all-factor resolution remains unverified.

Kadison’s orthonormal bases of unitaries problem

Progress summary

Primary source

Additional references

Solutions 0