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Amenability properties of unitary co-representations of locally compact quantum groups

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

E-mail Address: f_akhtari@math.iut.ac.ir

Corresponding author. The first author acknowledges that this research was in part supported by a grant from Iran National Science Foundation, INSF (No. 94027263).

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and

Rasoul Nasr-Isfahani

http://orcid.org/0000-0001-9233-3467

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

E-mail Address: isfahani@cc.iut.ac.ir

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https://doi.org/10.1142/S0129167X19500770Cited by:0 (Source: Crossref)

Abstract

For locally compact quantum groups $𝔾$ , we initiate an investigation of stable states with respect to unitary co-representations $U$ of $𝔾$ on Hilbert spaces $H_{U}$ ; in particular, we study the subject on the multiplicative unitary operator $W_{𝔾}$ of $𝔾$ with some examples on locally compact quantum groups arising from discrete groups and compact groups. As the main result, we consider the one co-dimensional Hilbert subspace of $H_{U}$ associated to a suitable vector $η$ , to present an operator theoretic characterization of stable states with respect to a related unitary co-representation $U_{η}$ . This provides a quantum version of an interesting result on unitary representations of locally compact groups given by Lau and Paterson in 1991.

Keywords:

AMSC: 22D10, 43A07, 43A65, 46L10, 46L89, 47C15

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