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

TL;DR

This paper proves that for separable diffuse finite von Neumann algebras with a trace, there exists an orthonormal basis of self-adjoint unitaries, affirming the separable case of Kadison's problem.

Contribution

It establishes the existence of an orthonormal basis of self-adjoint unitaries in separable diffuse finite von Neumann algebras, solving a specific case of Kadison's problem.

Findings

Existence of orthonormal basis of self-adjoint unitaries in the specified algebras.

Use of noncommutative Lyapunov theorem to construct the basis.

Affirmation of the separable case of Kadison's problem.

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.