Liberation of Projections

TL;DR

This paper investigates the liberation process for projections in free probability, deriving a PDE for spectral measures, establishing subordination theory, and proving the Unification Conjecture for free entropy at trace 1/2.

Contribution

It introduces a PDE-based framework for spectral measures under liberation, develops subordination theory, and proves the Unification Conjecture for free entropy in the trace 1/2 case.

Findings

Spectral measure density is piecewise analytic for all positive times.

Established a holomorphic PDE for the Cauchy transform of spectral measures.

Proved the Unification Conjecture for free entropy in the trace 1/2 setting.

Abstract

We study the liberation process for projections: $(p,q)\mapsto (p_t,q)= (u_tpu_t^\ast,q)$ where $u_t$ is a free unitary Brownian motion freely independent from $\{p,q\}$. Its action on the operator-valued angle $qp_tq$ between the projections induces a flow on the corresponding spectral measures $\mu_t$; we prove that the Cauchy transform of the measure satisfies a holomorphic PDE. We develop a theory of subordination for the boundary values of this PDE, and use it to show that the spectral measure $\mu_t$ possesses a piecewise analytic density for any $t>0$ and any initial projections of trace $\frac12$. We us this to prove the Unification Conjecture for free entropy and information in this trace $\frac12$ setting.