Recasting the Elliott conjecture

Abstract Let A be a simple, unital, finite, and exact $ C^{*} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial ma...
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Gespeichert in:

Autor*in:	Perera, Francesc [verfasserIn] Toms, Andrew S.

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2007

Schlagwörter:	Dimension Function Positive Element Compact Hausdorff Space Stable Rank Real Rank

Anmerkung:	© Springer-Verlag 2007

Übergeordnetes Werk:	Enthalten in: Mathematische Annalen - Berlin : Springer, 1869, 338(2007), 3 vom: 24. Feb., Seite 669-702
Übergeordnetes Werk:	volume:338 ; year:2007 ; number:3 ; day:24 ; month:02 ; pages:669-702

Links:	Volltext

DOI / URN:	10.1007/s00208-007-0093-3

Katalog-ID:	SPR001889125

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520			\|a Abstract Let A be a simple, unital, finite, and exact $ C^{} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that %${\mathrm{K}}%$ -theoretic invariants will classify separable and nuclear $ C^{*} $-algebras—with the recent appearance of counterexamples to its strongest concrete form.
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912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_267
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2018
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
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912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
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912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
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912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2070
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912			\|a GBV_ILN_2093
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912			\|a GBV_ILN_2108
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912			\|a GBV_ILN_2111
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912			\|a GBV_ILN_2113
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matchkey_str	article:14321807:2007----::eatntelitc
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publishDate	2007
allfields	10.1007/s00208-007-0093-3 doi (DE-627)SPR001889125 (SPR)s00208-007-0093-3-e DE-627 ger DE-627 rakwb eng Perera, Francesc verfasserin aut Recasting the Elliott conjecture 2007 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2007 Abstract Let A be a simple, unital, finite, and exact $ C^{} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that %${\mathrm{K}}%$ -theoretic invariants will classify separable and nuclear $ C^{*} $-algebras—with the recent appearance of counterexamples to its strongest concrete form. Dimension Function (dpeaa)DE-He213 Positive Element (dpeaa)DE-He213 Compact Hausdorff Space (dpeaa)DE-He213 Stable Rank (dpeaa)DE-He213 Real Rank (dpeaa)DE-He213 Toms, Andrew S. aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 338(2007), 3 vom: 24. Feb., Seite 669-702 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:338 year:2007 number:3 day:24 month:02 pages:669-702 https://dx.doi.org/10.1007/s00208-007-0093-3 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_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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_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_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_4393 GBV_ILN_4700 AR 338 2007 3 24 02 669-702
spelling	10.1007/s00208-007-0093-3 doi (DE-627)SPR001889125 (SPR)s00208-007-0093-3-e DE-627 ger DE-627 rakwb eng Perera, Francesc verfasserin aut Recasting the Elliott conjecture 2007 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2007 Abstract Let A be a simple, unital, finite, and exact $ C^{} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that %${\mathrm{K}}%$ -theoretic invariants will classify separable and nuclear $ C^{*} $-algebras—with the recent appearance of counterexamples to its strongest concrete form. Dimension Function (dpeaa)DE-He213 Positive Element (dpeaa)DE-He213 Compact Hausdorff Space (dpeaa)DE-He213 Stable Rank (dpeaa)DE-He213 Real Rank (dpeaa)DE-He213 Toms, Andrew S. aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 338(2007), 3 vom: 24. Feb., Seite 669-702 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:338 year:2007 number:3 day:24 month:02 pages:669-702 https://dx.doi.org/10.1007/s00208-007-0093-3 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_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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_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_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_4393 GBV_ILN_4700 AR 338 2007 3 24 02 669-702
allfields_unstemmed	10.1007/s00208-007-0093-3 doi (DE-627)SPR001889125 (SPR)s00208-007-0093-3-e DE-627 ger DE-627 rakwb eng Perera, Francesc verfasserin aut Recasting the Elliott conjecture 2007 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2007 Abstract Let A be a simple, unital, finite, and exact $ C^{} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that %${\mathrm{K}}%$ -theoretic invariants will classify separable and nuclear $ C^{*} $-algebras—with the recent appearance of counterexamples to its strongest concrete form. Dimension Function (dpeaa)DE-He213 Positive Element (dpeaa)DE-He213 Compact Hausdorff Space (dpeaa)DE-He213 Stable Rank (dpeaa)DE-He213 Real Rank (dpeaa)DE-He213 Toms, Andrew S. aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 338(2007), 3 vom: 24. Feb., Seite 669-702 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:338 year:2007 number:3 day:24 month:02 pages:669-702 https://dx.doi.org/10.1007/s00208-007-0093-3 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_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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_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_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_4393 GBV_ILN_4700 AR 338 2007 3 24 02 669-702
allfieldsGer	10.1007/s00208-007-0093-3 doi (DE-627)SPR001889125 (SPR)s00208-007-0093-3-e DE-627 ger DE-627 rakwb eng Perera, Francesc verfasserin aut Recasting the Elliott conjecture 2007 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2007 Abstract Let A be a simple, unital, finite, and exact $ C^{} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that %${\mathrm{K}}%$ -theoretic invariants will classify separable and nuclear $ C^{*} $-algebras—with the recent appearance of counterexamples to its strongest concrete form. Dimension Function (dpeaa)DE-He213 Positive Element (dpeaa)DE-He213 Compact Hausdorff Space (dpeaa)DE-He213 Stable Rank (dpeaa)DE-He213 Real Rank (dpeaa)DE-He213 Toms, Andrew S. aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 338(2007), 3 vom: 24. Feb., Seite 669-702 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:338 year:2007 number:3 day:24 month:02 pages:669-702 https://dx.doi.org/10.1007/s00208-007-0093-3 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_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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_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_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_4393 GBV_ILN_4700 AR 338 2007 3 24 02 669-702
allfieldsSound	10.1007/s00208-007-0093-3 doi (DE-627)SPR001889125 (SPR)s00208-007-0093-3-e DE-627 ger DE-627 rakwb eng Perera, Francesc verfasserin aut Recasting the Elliott conjecture 2007 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Springer-Verlag 2007 Abstract Let A be a simple, unital, finite, and exact $ C^{} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that %${\mathrm{K}}%$ -theoretic invariants will classify separable and nuclear $ C^{*} $-algebras—with the recent appearance of counterexamples to its strongest concrete form. Dimension Function (dpeaa)DE-He213 Positive Element (dpeaa)DE-He213 Compact Hausdorff Space (dpeaa)DE-He213 Stable Rank (dpeaa)DE-He213 Real Rank (dpeaa)DE-He213 Toms, Andrew S. aut Enthalten in Mathematische Annalen Berlin : Springer, 1869 338(2007), 3 vom: 24. Feb., Seite 669-702 (DE-627)254630715 (DE-600)1462120-4 1432-1807 nnns volume:338 year:2007 number:3 day:24 month:02 pages:669-702 https://dx.doi.org/10.1007/s00208-007-0093-3 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_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_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 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_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_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_4393 GBV_ILN_4700 AR 338 2007 3 24 02 669-702
language	English
source	Enthalten in Mathematische Annalen 338(2007), 3 vom: 24. Feb., Seite 669-702 volume:338 year:2007 number:3 day:24 month:02 pages:669-702
sourceStr	Enthalten in Mathematische Annalen 338(2007), 3 vom: 24. Feb., Seite 669-702 volume:338 year:2007 number:3 day:24 month:02 pages:669-702
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Dimension Function Positive Element Compact Hausdorff Space Stable Rank Real Rank
isfreeaccess_bool	false
container_title	Mathematische Annalen
authorswithroles_txt_mv	Perera, Francesc @@aut@@ Toms, Andrew S. @@aut@@
publishDateDaySort_date	2007-02-24T00:00:00Z
hierarchy_top_id	254630715
id	SPR001889125
language_de	englisch
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We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. 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author	Perera, Francesc
spellingShingle	Perera, Francesc misc Dimension Function misc Positive Element misc Compact Hausdorff Space misc Stable Rank misc Real Rank Recasting the Elliott conjecture
authorStr	Perera, Francesc
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topic_title	Recasting the Elliott conjecture Dimension Function (dpeaa)DE-He213 Positive Element (dpeaa)DE-He213 Compact Hausdorff Space (dpeaa)DE-He213 Stable Rank (dpeaa)DE-He213 Real Rank (dpeaa)DE-He213
topic	misc Dimension Function misc Positive Element misc Compact Hausdorff Space misc Stable Rank misc Real Rank
topic_unstemmed	misc Dimension Function misc Positive Element misc Compact Hausdorff Space misc Stable Rank misc Real Rank
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title	Recasting the Elliott conjecture
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title_full	Recasting the Elliott conjecture
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title_sort	recasting the elliott conjecture
title_auth	Recasting the Elliott conjecture
abstract	Abstract Let A be a simple, unital, finite, and exact $ C^{} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that %${\mathrm{K}}%$ -theoretic invariants will classify separable and nuclear $ C^{*} $-algebras—with the recent appearance of counterexamples to its strongest concrete form. © Springer-Verlag 2007
abstractGer	Abstract Let A be a simple, unital, finite, and exact $ C^{} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that %${\mathrm{K}}%$ -theoretic invariants will classify separable and nuclear $ C^{*} $-algebras—with the recent appearance of counterexamples to its strongest concrete form. © Springer-Verlag 2007
abstract_unstemmed	Abstract Let A be a simple, unital, finite, and exact $ C^{} $-algebra which absorbs the Jiang–Su algebra %${\mathcal{Z}}%$ tensorially. We prove that the Cuntz semigroup of A admits a complete order embedding into an ordered semigroup which is obtained from the Elliott invariant in a functorial manner. We conjecture that this embedding is an isomorphism, and prove the conjecture in several cases. In these same cases—%${\mathcal{Z}}%$ -stable algebras all—we prove that the Elliott conjecture in its strongest form is equivalent to a conjecture which appears much weaker. Outside the class of %${\mathcal{Z}}%$ -stable $ C^{} $-algebras, this weaker conjecture has no known counterexamples, and it is plausible that none exist. Thus, we reconcile the still intact principle of Elliott’s classification conjecture—that %${\mathrm{K}}%$ -theoretic invariants will classify separable and nuclear $ C^{*} $-algebras—with the recent appearance of counterexamples to its strongest concrete form. © Springer-Verlag 2007
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container_issue	3
title_short	Recasting the Elliott conjecture
url	https://dx.doi.org/10.1007/s00208-007-0093-3
remote_bool	true
author2	Toms, Andrew S.
author2Str	Toms, Andrew S.
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doi_str	10.1007/s00208-007-0093-3
up_date	2024-07-04T00:52:29.604Z
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score	7.402524

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Recasting the Elliott conjecture

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Vorhandene Bände

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