Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type

Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its...
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Autor*in:	Ando, Hiroshi [verfasserIn] Matsuzawa, Yasumichi Thom, Andreas Törnquist, Asger

Format:	Artikel

Erschienen:	2017

Anmerkung:	© 2019 Walter de Gruyter GmbH, Berlin/Boston

Übergeordnetes Werk:	Enthalten in: Journal für die reine und angewandte Mathematik - De Gruyter, 1826, 2020(2017), 758 vom: 17. Nov., Seite 223-251
Übergeordnetes Werk:	volume:2020 ; year:2017 ; number:758 ; day:17 ; month:11 ; pages:223-251

Links:	Volltext

DOI / URN:	10.1515/crelle-2017-0047

Katalog-ID:	OLC2141981530

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520			\|a Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))} ) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space.
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allfields	10.1515/crelle-2017-0047 doi (DE-627)OLC2141981530 (DE-B1597)crelle-2017-0047-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 17,1 ssgn Ando, Hiroshi verfasserin aut Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 2019 Walter de Gruyter GmbH, Berlin/Boston Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))} ) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space. Matsuzawa, Yasumichi aut Thom, Andreas aut Törnquist, Asger aut Enthalten in Journal für die reine und angewandte Mathematik De Gruyter, 1826 2020(2017), 758 vom: 17. Nov., Seite 223-251 (DE-627)129078824 (DE-600)3079-X (DE-576)014411393 0075-4102 nnns volume:2020 year:2017 number:758 day:17 month:11 pages:223-251 https://doi.org/10.1515/crelle-2017-0047 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_130 GBV_ILN_2002 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 2020 2017 758 17 11 223-251
spelling	10.1515/crelle-2017-0047 doi (DE-627)OLC2141981530 (DE-B1597)crelle-2017-0047-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 17,1 ssgn Ando, Hiroshi verfasserin aut Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 2019 Walter de Gruyter GmbH, Berlin/Boston Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))} ) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space. Matsuzawa, Yasumichi aut Thom, Andreas aut Törnquist, Asger aut Enthalten in Journal für die reine und angewandte Mathematik De Gruyter, 1826 2020(2017), 758 vom: 17. Nov., Seite 223-251 (DE-627)129078824 (DE-600)3079-X (DE-576)014411393 0075-4102 nnns volume:2020 year:2017 number:758 day:17 month:11 pages:223-251 https://doi.org/10.1515/crelle-2017-0047 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_130 GBV_ILN_2002 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 2020 2017 758 17 11 223-251
allfields_unstemmed	10.1515/crelle-2017-0047 doi (DE-627)OLC2141981530 (DE-B1597)crelle-2017-0047-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 17,1 ssgn Ando, Hiroshi verfasserin aut Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 2019 Walter de Gruyter GmbH, Berlin/Boston Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))} ) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space. Matsuzawa, Yasumichi aut Thom, Andreas aut Törnquist, Asger aut Enthalten in Journal für die reine und angewandte Mathematik De Gruyter, 1826 2020(2017), 758 vom: 17. Nov., Seite 223-251 (DE-627)129078824 (DE-600)3079-X (DE-576)014411393 0075-4102 nnns volume:2020 year:2017 number:758 day:17 month:11 pages:223-251 https://doi.org/10.1515/crelle-2017-0047 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_130 GBV_ILN_2002 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 2020 2017 758 17 11 223-251
allfieldsGer	10.1515/crelle-2017-0047 doi (DE-627)OLC2141981530 (DE-B1597)crelle-2017-0047-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 17,1 ssgn Ando, Hiroshi verfasserin aut Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 2019 Walter de Gruyter GmbH, Berlin/Boston Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))} ) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space. Matsuzawa, Yasumichi aut Thom, Andreas aut Törnquist, Asger aut Enthalten in Journal für die reine und angewandte Mathematik De Gruyter, 1826 2020(2017), 758 vom: 17. Nov., Seite 223-251 (DE-627)129078824 (DE-600)3079-X (DE-576)014411393 0075-4102 nnns volume:2020 year:2017 number:758 day:17 month:11 pages:223-251 https://doi.org/10.1515/crelle-2017-0047 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_130 GBV_ILN_2002 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 2020 2017 758 17 11 223-251
allfieldsSound	10.1515/crelle-2017-0047 doi (DE-627)OLC2141981530 (DE-B1597)crelle-2017-0047-p DE-627 ger DE-627 rakwb 510 VZ 510 VZ 17,1 ssgn Ando, Hiroshi verfasserin aut Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type 2017 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © 2019 Walter de Gruyter GmbH, Berlin/Boston Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))} ) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space. Matsuzawa, Yasumichi aut Thom, Andreas aut Törnquist, Asger aut Enthalten in Journal für die reine und angewandte Mathematik De Gruyter, 1826 2020(2017), 758 vom: 17. Nov., Seite 223-251 (DE-627)129078824 (DE-600)3079-X (DE-576)014411393 0075-4102 nnns volume:2020 year:2017 number:758 day:17 month:11 pages:223-251 https://doi.org/10.1515/crelle-2017-0047 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_120 GBV_ILN_130 GBV_ILN_2002 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2030 GBV_ILN_2088 GBV_ILN_4027 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4266 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4311 GBV_ILN_4318 GBV_ILN_4323 AR 2020 2017 758 17 11 223-251
source	Enthalten in Journal für die reine und angewandte Mathematik 2020(2017), 758 vom: 17. Nov., Seite 223-251 volume:2020 year:2017 number:758 day:17 month:11 pages:223-251
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author	Ando, Hiroshi
spellingShingle	Ando, Hiroshi ddc 510 ssgn 17,1 Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type
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title	Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type
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title_full	Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type
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title_sort	unitarizability, maurey–nikishin factorization, and polish groups of finite type
title_auth	Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type
abstract	Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))} ) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space. © 2019 Walter de Gruyter GmbH, Berlin/Boston
abstractGer	Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))} ) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space. © 2019 Walter de Gruyter GmbH, Berlin/Boston
abstract_unstemmed	Abstract Let Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))} ) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}} -factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space. © 2019 Walter de Gruyter GmbH, Berlin/Boston
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title_short	Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type
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up_date	2024-07-04T03:19:35.889Z
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