Orthogonal Unitary Bases and a Subfactor Conjecture

TL;DR

This paper proves the existence of orthonormal unitary bases in finite-dimensional von Neumann algebras and subalgebras, and confirms a conjecture regarding bases in finite-index regular inclusions of $II_1$-factors.

Contribution

It establishes the existence of orthonormal unitary bases in finite-dimensional von Neumann algebras and verifies a conjecture on bases in finite-index regular inclusions of $II_1$-factors.

Findings

Finite-dimensional von Neumann algebras admit orthonormal unitary bases.

A subalgebra admits such a basis iff traces coincide.

Verification of the Bakshi-Gupta conjecture for $II_1$-factors.

Abstract

We show that any finite dimensional von Neumann algebra admits an orthonormal unitary basis with respect to its standard trace. We also show that a finite dimensional von Neumann subalgebra of $M_n(\mathbb{C})$ admits an orthonormal unitary basis under normalized matrix trace if and only if the normalized matrix trace and standard trace of the von Neumann subalgebra coincide. As an application, we verify a recent conjecture of Bakshi-Gupta, showing that any finite-index regular inclusion $N\subseteq M$ of $II_1$-factors admits an orthonormal unitary Pimsner-Popa basis.