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Property (T), finite-dimensional representations, and generic representations
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SYSNO ASEP 0498908 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Property (T), finite-dimensional representations, and generic representations Author(s) Doucha, Michal (MU-W) RID, SAI, ORCID
Malicki, M. (PL)
Valette, A. (CH)Source Title Journal of Group Theory. - : Walter de Gruyter - ISSN 1433-5883
Roč. 22, č. 1 (2019), s. 1-13Number of pages 13 s. Language eng - English Country DE - Germany Keywords generic representations Subject RIV BA - General Mathematics OECD category Applied mathematics R&D Projects GF16-34860L GA ČR - Czech Science Foundation (CSF) Institutional support MU-W - RVO:67985840 UT WOS 000454602000001 EID SCOPUS 85052713220 DOI 10.1515/jgth-2018-0030 Annotation 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})}. Workplace Mathematical Institute Contact Jarmila Štruncová, struncova@math.cas.cz, library@math.cas.cz, Tel.: 222 090 757 Year of Publishing 2019
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