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

In 1967, Kadison asked whether every type $\mathrm{II}_1$ 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 symm