Brown measure support and the free multiplicative Brownian motion

Abstract

The free multiplicative Brownian motion bt is the large-N limit of Brownian motion BtN on the general linear group GL(N;C). We prove that the Brown measure for bt—which is an analog of the empirical eigenvalue distribution for matrices—is supported on the closure of a certain domain Σt in the plane. The domain Σt was introduced by Biane in the context of the large-N limit of the Segal–Bargmann transform associated to GL(N;C).We also consider a two-parameter version, bs,t: the large-N limit of a related family of diffusion processes on GL(N;C) introduced by the second author. We show that the Brown measure of bs,t is supported on the closure of a certain planar domain Σs,t, generalizing Σt, introduced by Ho.In the process, we introduce a new family of spectral domains related to any operator in a tracial von Neumann algebra: the Lpn-spectrum for n∈N and p≥1, a subset of the ordinary spectrum defined relative to potentially-unbounded inverses. We show that, in general, the support of the Brown measure of an operator is contained in its L22-spectrum.