Banach space properties forcing a reflexive, amenable Banach algebra to be trivial
Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\f...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Runde, V. [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2001 |
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| Schlagwörter: |
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| Anmerkung: |
© Birkhäuser Verlag, Basel 2001 |
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| Übergeordnetes Werk: |
Enthalten in: Archiv der Mathematik - Berlin : Springer, 1948, 77(2001), 3 vom: 01. Sept., Seite 265-272 |
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| Übergeordnetes Werk: |
volume:77 ; year:2001 ; number:3 ; day:01 ; month:09 ; pages:265-272 |
| Links: |
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| DOI / URN: |
10.1007/PL00000490 |
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| Katalog-ID: |
SPR00004556X |
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| 024 | 7 | |a 10.1007/PL00000490 |2 doi | |
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| 041 | |a eng | ||
| 100 | 1 | |a Runde, V. |e verfasserin |4 aut | |
| 245 | 1 | 0 | |a Banach space properties forcing a reflexive, amenable Banach algebra to be trivial |
| 264 | 1 | |c 2001 | |
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| 500 | |a © Birkhäuser Verlag, Basel 2001 | ||
| 520 | |a Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional. | ||
| 650 | 4 | |a Banach Space |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Open Problem |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Space Property |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Banach Algebra |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Approximation Property |7 (dpeaa)DE-He213 | |
| 773 | 0 | 8 | |i Enthalten in |t Archiv der Mathematik |d Berlin : Springer, 1948 |g 77(2001), 3 vom: 01. Sept., Seite 265-272 |w (DE-627)253389976 |w (DE-600)1458441-4 |x 1420-8938 |7 nnns |
| 773 | 1 | 8 | |g volume:77 |g year:2001 |g number:3 |g day:01 |g month:09 |g pages:265-272 |
| 856 | 4 | 0 | |u https://dx.doi.org/10.1007/PL00000490 |z lizenzpflichtig |3 Volltext |
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10.1007/PL00000490 doi (DE-627)SPR00004556X (SPR)PL00000490-e DE-627 ger DE-627 rakwb eng Runde, V. verfasserin aut Banach space properties forcing a reflexive, amenable Banach algebra to be trivial 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Birkhäuser Verlag, Basel 2001 Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional. Banach Space (dpeaa)DE-He213 Open Problem (dpeaa)DE-He213 Space Property (dpeaa)DE-He213 Banach Algebra (dpeaa)DE-He213 Approximation Property (dpeaa)DE-He213 Enthalten in Archiv der Mathematik Berlin : Springer, 1948 77(2001), 3 vom: 01. Sept., Seite 265-272 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:77 year:2001 number:3 day:01 month:09 pages:265-272 https://dx.doi.org/10.1007/PL00000490 lizenzpflichtig 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_121 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_206 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_374 GBV_ILN_602 GBV_ILN_636 GBV_ILN_647 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_2043 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 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_2808 GBV_ILN_4012 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_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 77 2001 3 01 09 265-272 |
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10.1007/PL00000490 doi (DE-627)SPR00004556X (SPR)PL00000490-e DE-627 ger DE-627 rakwb eng Runde, V. verfasserin aut Banach space properties forcing a reflexive, amenable Banach algebra to be trivial 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Birkhäuser Verlag, Basel 2001 Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional. Banach Space (dpeaa)DE-He213 Open Problem (dpeaa)DE-He213 Space Property (dpeaa)DE-He213 Banach Algebra (dpeaa)DE-He213 Approximation Property (dpeaa)DE-He213 Enthalten in Archiv der Mathematik Berlin : Springer, 1948 77(2001), 3 vom: 01. Sept., Seite 265-272 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:77 year:2001 number:3 day:01 month:09 pages:265-272 https://dx.doi.org/10.1007/PL00000490 lizenzpflichtig 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_121 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_206 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_374 GBV_ILN_602 GBV_ILN_636 GBV_ILN_647 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_2043 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 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_2808 GBV_ILN_4012 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_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 77 2001 3 01 09 265-272 |
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10.1007/PL00000490 doi (DE-627)SPR00004556X (SPR)PL00000490-e DE-627 ger DE-627 rakwb eng Runde, V. verfasserin aut Banach space properties forcing a reflexive, amenable Banach algebra to be trivial 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Birkhäuser Verlag, Basel 2001 Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional. Banach Space (dpeaa)DE-He213 Open Problem (dpeaa)DE-He213 Space Property (dpeaa)DE-He213 Banach Algebra (dpeaa)DE-He213 Approximation Property (dpeaa)DE-He213 Enthalten in Archiv der Mathematik Berlin : Springer, 1948 77(2001), 3 vom: 01. Sept., Seite 265-272 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:77 year:2001 number:3 day:01 month:09 pages:265-272 https://dx.doi.org/10.1007/PL00000490 lizenzpflichtig 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_121 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_206 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_374 GBV_ILN_602 GBV_ILN_636 GBV_ILN_647 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_2043 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 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_2808 GBV_ILN_4012 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_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 77 2001 3 01 09 265-272 |
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10.1007/PL00000490 doi (DE-627)SPR00004556X (SPR)PL00000490-e DE-627 ger DE-627 rakwb eng Runde, V. verfasserin aut Banach space properties forcing a reflexive, amenable Banach algebra to be trivial 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Birkhäuser Verlag, Basel 2001 Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional. Banach Space (dpeaa)DE-He213 Open Problem (dpeaa)DE-He213 Space Property (dpeaa)DE-He213 Banach Algebra (dpeaa)DE-He213 Approximation Property (dpeaa)DE-He213 Enthalten in Archiv der Mathematik Berlin : Springer, 1948 77(2001), 3 vom: 01. Sept., Seite 265-272 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:77 year:2001 number:3 day:01 month:09 pages:265-272 https://dx.doi.org/10.1007/PL00000490 lizenzpflichtig 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_121 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_206 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_374 GBV_ILN_602 GBV_ILN_636 GBV_ILN_647 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_2043 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 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_2808 GBV_ILN_4012 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_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 77 2001 3 01 09 265-272 |
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10.1007/PL00000490 doi (DE-627)SPR00004556X (SPR)PL00000490-e DE-627 ger DE-627 rakwb eng Runde, V. verfasserin aut Banach space properties forcing a reflexive, amenable Banach algebra to be trivial 2001 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Birkhäuser Verlag, Basel 2001 Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional. Banach Space (dpeaa)DE-He213 Open Problem (dpeaa)DE-He213 Space Property (dpeaa)DE-He213 Banach Algebra (dpeaa)DE-He213 Approximation Property (dpeaa)DE-He213 Enthalten in Archiv der Mathematik Berlin : Springer, 1948 77(2001), 3 vom: 01. Sept., Seite 265-272 (DE-627)253389976 (DE-600)1458441-4 1420-8938 nnns volume:77 year:2001 number:3 day:01 month:09 pages:265-272 https://dx.doi.org/10.1007/PL00000490 lizenzpflichtig 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_121 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_206 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_374 GBV_ILN_602 GBV_ILN_636 GBV_ILN_647 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_2043 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_2116 GBV_ILN_2118 GBV_ILN_2119 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_2158 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2193 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_2808 GBV_ILN_4012 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_4346 GBV_ILN_4367 GBV_ILN_4393 GBV_ILN_4700 GBV_ILN_4753 AR 77 2001 3 01 09 265-272 |
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Enthalten in Archiv der Mathematik 77(2001), 3 vom: 01. Sept., Seite 265-272 volume:77 year:2001 number:3 day:01 month:09 pages:265-272 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR00004556X</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230327125421.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201001s2001 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/PL00000490</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR00004556X</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)PL00000490-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">Runde, V.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Banach space properties forcing a reflexive, amenable Banach algebra to be trivial</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2001</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">© Birkhäuser Verlag, Basel 2001</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Banach Space</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Open Problem</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Space Property</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Banach Algebra</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Approximation Property</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Archiv der Mathematik</subfield><subfield code="d">Berlin : Springer, 1948</subfield><subfield code="g">77(2001), 3 vom: 01. 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Runde, V. |
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Banach space properties forcing a reflexive, amenable Banach algebra to be trivial Banach Space (dpeaa)DE-He213 Open Problem (dpeaa)DE-He213 Space Property (dpeaa)DE-He213 Banach Algebra (dpeaa)DE-He213 Approximation Property (dpeaa)DE-He213 |
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banach space properties forcing a reflexive, amenable banach algebra to be trivial |
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Banach space properties forcing a reflexive, amenable Banach algebra to be trivial |
| abstract |
Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional. © Birkhäuser Verlag, Basel 2001 |
| abstractGer |
Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional. © Birkhäuser Verlag, Basel 2001 |
| abstract_unstemmed |
Abstract. It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. As an application, we show that, if %${\frak A}%$ is an amenable Banach algebra whose underlying Banach space is an $ L^{p} $-space with %$ p\in (1,\infty) %$ such that for each maximal left ideal L the quotient %$ {\frak A}/L %$ has the approximation property, then %$ {\frak A} %$ is finite-dimensional. © Birkhäuser Verlag, Basel 2001 |
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| title_short |
Banach space properties forcing a reflexive, amenable Banach algebra to be trivial |
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https://dx.doi.org/10.1007/PL00000490 |
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10.1007/PL00000490 |
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2024-07-03T13:41:14.990Z |
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It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a finite direct sum of full matrix algebras). If %$ {\frak A} %$ is a reflexive, amenable Banach algebra such that for each maximal left ideal L of %$ {\frak A} %$ (i) the quotient %$ {\frak A}/L %$ has the approximation property and (ii) the canonical map from %$ {\frak A} \check{\otimes} L^\perp $ to $({\frak A} / L) \check{\otimes} L^\perp %$ is open, then %$ {\frak A} %$ is finite-dimensional. 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| score |
7.3999414 |