Haar states and L\'evy processes on the unitary dual group

TL;DR

This paper investigates states on the algebra generated by unitary matrix coefficients, establishing the non-existence of Haar states but identifying weaker Haar traces, and connecting these to limits of random matrix blocks.

Contribution

It introduces the concept of Haar traces as a weaker alternative to Haar states on the unitary dual group and links them to limits of Haar unitary matrices and free Lévy processes.

Findings

No Haar state exists for the convolutions considered.

Haar trace exists as a limit of blocks of Haar unitary matrices.

Free Lévy processes on the dual group are limits of classical random matrix blocks.

Abstract

We study states on the universal noncommutative *-algebra generated by the coefficients of a unitary matrix, or equivalently states on the unitary dual group. Its structure of dual group in the sense of Voiculescu allows to define five natural convolutions. We prove that there exists no Haar state for those convolutions. However, we prove that there exists a weaker form of absorbing state, that we call Haar trace, for the free and the tensor convolutions. We show that the free Haar trace is the limit in distribution of the blocks of a Haar unitary matrix when the dimension tends to infinity. Finally, we study a particular class of free L\'evy processes on the unitary dual group which are also the limit of the blocks of random matrices on the classical unitary group when the dimension tends to infinity.