Ground state representations of topological groups
Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Neeb, Karl-Hermann [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2022 |
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| Anmerkung: |
© The Author(s) 2022 |
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| Übergeordnetes Werk: |
Enthalten in: Mathematische Annalen - Berlin : Springer, 1869, 388(2022), 1 vom: 03. Dez., Seite 615-674 |
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| Übergeordnetes Werk: |
volume:388 ; year:2022 ; number:1 ; day:03 ; month:12 ; pages:615-674 |
| Links: |
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| DOI / URN: |
10.1007/s00208-022-02531-4 |
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| Katalog-ID: |
SPR05449978X |
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| 100 | 1 | |a Neeb, Karl-Hermann |e verfasserin |0 (orcid)0000-0001-7654-5750 |4 aut | |
| 245 | 1 | 0 | |a Ground state representations of topological groups |
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| 520 | |a Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations. | ||
| 700 | 1 | |a Russo, Francesco G. |0 (orcid)0000-0002-5889-783X |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Mathematische Annalen |d Berlin : Springer, 1869 |g 388(2022), 1 vom: 03. Dez., Seite 615-674 |w (DE-627)254630715 |w (DE-600)1462120-4 |x 1432-1807 |7 nnns |
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10.1007/s00208-022-02531-4 doi (DE-627)SPR05449978X (SPR)s00208-022-02531-4-e DE-627 ger DE-627 rakwb eng Neeb, Karl-Hermann verfasserin (orcid)0000-0001-7654-5750 aut Ground state representations of topological groups 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations. Russo, Francesco G. (orcid)0000-0002-5889-783X aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 388(2022), 1 vom: 03. Dez., Seite 615-674 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:388 year:2022 number:1 day:03 month:12 pages:615-674 https://dx.doi.org/10.1007/s00208-022-02531-4 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 388 2022 1 03 12 615-674 |
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10.1007/s00208-022-02531-4 doi (DE-627)SPR05449978X (SPR)s00208-022-02531-4-e DE-627 ger DE-627 rakwb eng Neeb, Karl-Hermann verfasserin (orcid)0000-0001-7654-5750 aut Ground state representations of topological groups 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations. Russo, Francesco G. (orcid)0000-0002-5889-783X aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 388(2022), 1 vom: 03. Dez., Seite 615-674 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:388 year:2022 number:1 day:03 month:12 pages:615-674 https://dx.doi.org/10.1007/s00208-022-02531-4 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 388 2022 1 03 12 615-674 |
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10.1007/s00208-022-02531-4 doi (DE-627)SPR05449978X (SPR)s00208-022-02531-4-e DE-627 ger DE-627 rakwb eng Neeb, Karl-Hermann verfasserin (orcid)0000-0001-7654-5750 aut Ground state representations of topological groups 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations. Russo, Francesco G. (orcid)0000-0002-5889-783X aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 388(2022), 1 vom: 03. Dez., Seite 615-674 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:388 year:2022 number:1 day:03 month:12 pages:615-674 https://dx.doi.org/10.1007/s00208-022-02531-4 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 388 2022 1 03 12 615-674 |
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10.1007/s00208-022-02531-4 doi (DE-627)SPR05449978X (SPR)s00208-022-02531-4-e DE-627 ger DE-627 rakwb eng Neeb, Karl-Hermann verfasserin (orcid)0000-0001-7654-5750 aut Ground state representations of topological groups 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations. Russo, Francesco G. (orcid)0000-0002-5889-783X aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 388(2022), 1 vom: 03. Dez., Seite 615-674 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:388 year:2022 number:1 day:03 month:12 pages:615-674 https://dx.doi.org/10.1007/s00208-022-02531-4 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 388 2022 1 03 12 615-674 |
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10.1007/s00208-022-02531-4 doi (DE-627)SPR05449978X (SPR)s00208-022-02531-4-e DE-627 ger DE-627 rakwb eng Neeb, Karl-Hermann verfasserin (orcid)0000-0001-7654-5750 aut Ground state representations of topological groups 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s) 2022 Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations. Russo, Francesco G. (orcid)0000-0002-5889-783X aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 388(2022), 1 vom: 03. Dez., Seite 615-674 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:388 year:2022 number:1 day:03 month:12 pages:615-674 https://dx.doi.org/10.1007/s00208-022-02531-4 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_267 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 388 2022 1 03 12 615-674 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000naa a22002652 4500</leader><controlfield tag="001">SPR05449978X</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20240125064742.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">240125s2022 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00208-022-02531-4</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR05449978X</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s00208-022-02531-4-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Neeb, Karl-Hermann</subfield><subfield code="e">verfasserin</subfield><subfield code="0">(orcid)0000-0001-7654-5750</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Ground state representations of topological groups</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2022</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© The Author(s) 2022</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. 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ground state representations of topological groups |
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Ground state representations of topological groups |
| abstract |
Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations. © The Author(s) 2022 |
| abstractGer |
Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations. © The Author(s) 2022 |
| abstract_unstemmed |
Abstract Let %$\alpha : {{\mathbb {R}}}\rightarrow \mathop {\textrm{Aut}}\nolimits (G)%$ define a continuous %${{\mathbb {R}}}%$-action on the topological group G. A unitary representation %$(\pi ^\flat ,\mathcal {H})%$ of the extended group %$G^\flat := G \rtimes _\alpha {{\mathbb {R}}}%$ is called a ground state representation if the unitary one-parameter group %$\pi ^\flat (e,t) = e^{itH}%$ has a non-negative generator %$H \ge 0%$ and the subspace %$\mathcal {H}^0 := \ker H%$ of ground states generates %$\mathcal {H}%$ under G. In this paper, we introduce the class of strict ground state representations, where %$(\pi ^\flat ,\mathcal {H})%$ and the representation of the subgroup %$G^0 := \mathop {\textrm{Fix}}\nolimits (\alpha )%$ on %$\mathcal {H}^0%$ have the same commutant. The advantage of this concept is that it permits us to classify strict ground state representations in terms of the corresponding representations of %$G^0%$. This is particularly effective if the occurring representations of %$G^0%$ can be characterized intrinsically in terms of concrete positivity conditions. To find such conditions, it is natural to restrict to infinite dimensional Lie groups such as (1) Heisenberg groups (which exhibit examples of non-strict ground state representations); (2) Finite dimensional groups, where highest weight representations provide natural examples; (3) Compact groups, for which our approach provides a new perspective on the classification of unitary representations; (4) Direct limits of compact groups, as a class of examples for which strict ground state representations can be used to classify large classes of unitary representations. © The Author(s) 2022 |
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| title_short |
Ground state representations of topological groups |
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https://dx.doi.org/10.1007/s00208-022-02531-4 |
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| author2 |
Russo, Francesco G. |
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Russo, Francesco G. |
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| doi_str |
10.1007/s00208-022-02531-4 |
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2024-07-04T01:54:33.883Z |
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| score |
7.401063 |