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Seminario Teoría de grupos 17/02

Seminario Teoría de grupos 17/02

11:30 Aula Naranja ICMAT

Speaker: Jan Boschheidgen (UAM)

Title: Non convergence of eigenvalue measures associated to residual chains

Abstract: Let $G$ be a discrete residually finite group and let $A in text{Mat}_n(mathbb{C}[G])$. Let $G unrhd N_1 unrhd N_2 ldots$ be a chain of normal subgroups of finite index with trivial intersection. Set $G_i = G / N_i$. Then $A$ acts by right multiplication via reduction modulo $N_i$ on $mathbb{C}[G_i]^n$. Since $mathbb{C}[G_i] cong mathbb{C}^{vert G_i vert}$ as $mathbb{C}$-vector spaces, this action can be represented by a matrix $A_i in text{Mat}_{n cdot vert G_i vert}(mathbb{C})$. For every $i$ let now $lambda_1^{(i)}, ldots , lambda_{n cdot vert G_i vert }^{(i)}$ be the eigenvalues of $A_i$ and define

$mu_i = frac{1}{vert G_i vert}sumlimits_{k = 1}^{n cdot vert G_i vert } delta_{lambda_k^{(i)}}$,

Where $delta_c$ denotes the Dirac measure at $c in mathbb{C}$.

We now can ask the following questions:

(1) Does the limit $limlimits_{i to infty} mu_i ({0 })$ exist?
(2) If the answer to the first question is yes, does the limit depend on the chain ${N_i}$?
(3) Let $mathcal{N} (G)$ denote the group von Neumann algebra. We can consider $A$ as an element of the tracial von Neumann algebra $text{Mat}_n(mathcal{N} (G))$ acting on the Hilbert space $(ell^2 (G))^n$. Therefore we can define the Brown measure $mu_A$ of $A$ and ask: Does the sequence $mu_i$ converge weakly to $mu_A$?

All these questions have been already studied for the case when $A$ is normal. In this case the answer to all questions is positive. In this talk we will show that in general the answers to questions (2) and (3) are negative.

There will also be a coffee break before the talk, starting at 11:10.

As always, a list of the upcoming seminars can be found on the seminar webpage at the following address:
https://dominik.francoeur.xyz/seminar.html
Localización 11:30 Aula Naranja ICMAT