Property (T), finite-dimensional representations, and generic representations

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F19%3A00498908" target="_blank" >RIV/67985840:_____/19:00498908 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1515/jgth-2018-0030" target="_blank" >http://dx.doi.org/10.1515/jgth-2018-0030</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1515/jgth-2018-0030" target="_blank" >10.1515/jgth-2018-0030</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Property (T), finite-dimensional representations, and generic representations

Popis výsledku v původním jazyce

Let G be a discrete group with Property (T). It is a standard fact that, in a unitary representation of G on a Hilbert space H {mathcal{H}}, almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation σ, then the vector is close to a sub-representation isomorphic to σ: this makes quantitative a result of P. S. Wang. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot, that a group G with Property (T) and such that C ∗(G) {C^{∗}(G)} is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in Rep(G, H) {Rep(G,mathcal{H})} under the unitary group U(H) {U(mathcal{H})} is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in Rep(G, H) {mathrm{Rep}(G,mathcal{H})}.

Název v anglickém jazyce

Property (T), finite-dimensional representations, and generic representations

Popis výsledku anglicky

Let G be a discrete group with Property (T). It is a standard fact that, in a unitary representation of G on a Hilbert space H {mathcal{H}}, almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation σ, then the vector is close to a sub-representation isomorphic to σ: this makes quantitative a result of P. S. Wang. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot, that a group G with Property (T) and such that C ∗(G) {C^{∗}(G)} is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in Rep(G, H) {Rep(G,mathcal{H})} under the unitary group U(H) {U(mathcal{H})} is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in Rep(G, H) {mathrm{Rep}(G,mathcal{H})}.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10102 - Applied mathematics

Projekt

<a href="/cs/project/GF16-34860L" target="_blank" >GF16-34860L: Logika a topologie v Banachových prostorech</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Group Theory

ISSN

1433-5883

e-ISSN

—

Svazek periodika

22

Číslo periodika v rámci svazku

1

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

13

Strana od-do

1-13

Kód UT WoS článku

000454602000001

EID výsledku v databázi Scopus

2-s2.0-85052713220