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单C*-代数α-比较性的一种等价刻画
An Equivalent Characterization for the α-comparison Property of Simple C*-algebras
给出C*-代数α-比较性的等价刻画：对于单的含单位元的稳定有限的C*-代数A而言，A具有α-比较性，当且仅当对于任意的<a>，<b>∈W（A），若α·dτ（a）<dτ（b）（∀τ∈QT（A）），则<a>≤<b>在Cuntz半群W（A）中成立.利用此刻画，证明了具有α-比较性的C*-代数一定具有弱比较性；若A具有α-比较性，其中α=m+1，则A具有正元的强迹m-比较性；对于满足Kirchberg-Rørdam条件的C*-代数，Z-稳定、严格比较、α-比较性（α=m+1）、强迹m-比较性、弱比较性以及局部弱比较性彼此等价；若α：=inf{α'∈（1，∞）|A具有α'-比较}<∞，则A具有α-比较性.
We give an equivalent characterization for the α-comparison property of C*-algebras:any simple unital stably finite C*-algebra A has the α-comparison property, if and only if, for any <a>, <b>∈W(A), α·dτ(a) <dτ(b)(∀τ∈QT(A)) implies that <a>≤<b> holds in W(A).Using this characterization, we prove the following results:C*-algebras with α-comparison property have weak comparison;C*-algebras with α-comparison property for α=m+1 have strong tracial m-comparison of positive elements;Z-stability strict comparison α-comparison property for α=m+1 strong tracial m-comparison weak comparison and local weak comparison all agree for the C*-algebras satisfying the conditions given by Kirchberg-Rørdam;if α:=inf{α'∈(1, ∞)|A has the α-comparison property}<∞, then A has the α-comparison property.
&alpha / -比较性 / Cuntz半群 / Z-稳定 {{custom_keyword}} /
α-comparison property / Cuntz semigroup / Z-stability {{custom_keyword}} /
[1] Ara P., Perera F., Toms A. S., K-theory for operator algebras, Classification of C*-algebra, Aspects of Operator Algebras and Applications, Contemp. Math., Providence, RI:Amer. Math. Soc., 2011, 534:1-71.
[2] Blackadar B., Handelman D., Dimension functions and traces on C*-algebras, J. Funct. Anal., 1982, 45(3):297-340.
[3] Elliott G. A., Gong G., Li L., Approximate divisibility of simple inductive limit C*-algebras, Contemp. Math., 1998, 228:87-97.
[4] Elliott G. A., Gong G., Li L., On the classification of simple inductive limit C*-algebras, Ⅱ:The isomorphism theorem, Invent. Math., 2007, 168(2):249-320.
[5] Jiang X., Su H., On a simple unital projectionless C*-algebra, Amer. J. Math., 1999, 121(2):359-413.
[6] Kirchberg E., The classification of purely infinite C*-algebras using Kasparov's theory, Preliminary Preprint(3rd draft), 1994.
[7] Kirchberg E., Rørdam M., Non-simple purely infinite C*-algebras, Amer. J. Math., 2000, 122(3):637-666.
[8] Kirchberg E., Rørdam M., Central sequence C*-algebras and tensorial absorption of the Jiang-Su algebra, J. Reine Angew. Math., 2013, 2014(695):62-76.
[9] Lin H., Classification of simple tracially AF C*-algebras, Canad. J. Math., 2001, 53(1):161-194.
[10] Lin H., Classification of simple C*-algebras of tracial topological rank zero, Duke Math. J., 2004, 125(1):91-119.
[11] Matui H., Sato Y., Strict comparison and Z-absorption of nuclear C*-algebras, Acta Math., 2012, 209(1):179-196.
[12] Ortega E., Perera F., Rørdam M., The corona factorization property, stability, and the Cuntz semigroup of a C*-algebra, Int. Math. Res. Not., 2010, 2012(1):34-66.
[13] Perera F., Toms A. S., Recasting the Elliott Conjecture, Math. Ann., 2007, 338(3):669-702.
[14] Phillips N. C., A classification theorem for nuclear purely infinite simple C*-algebra, Doc. Math., 2000, 5:49-114.
[15] Rørdam M., On the structure of simple C*-algebras tensored with a UHF-algebra Ⅱ, J. Funct. Anal., 1992, 107(2):255-269.
[16] Rørdam M., Classification of nuclear, simple C*-algebras, Classification of nuclear C*-algebras, Entropy in Operator Algebras, J. Cuntz and V. Jones, eds., Encyclopaedia Math. Sci., Subseries:Operator Algebras and Non-commutative Geometry, no. VⅡ, Springer-Verlag, Berlin, 2001, 126:1-145.
[17] Rørdam M., A simple C*-algebra with a finite and an infinite projection, Acta Math., 2003, 191(1):109-142.
[18] Rørdam M., The stable and the real rank of Z-absorbing C*-algebras, Internat. J. Math., 2004, 15(10):1065-1084.
[19] Rørdam M., Winter W., The Jiang-Su algebra revisied, J. Reine Angew. Math., 2010, 642:129-155.
[20] Toms A. S., On the classification problem for nuclear C*-algebras, Ann. Math., 2008, 167(3):1029-1044.
[21] Toms A. S., On the independence of K-theory and stable rank for simple C*-algebras, J. Reine Angew. Math., 2005, 578:185-199.
[22] Toms A. S., Winter W., Strongly self-absorbing C*-algebras, Trans. Amer. Math. Soc., 2007, 359(8):3999-4029.
[23] Toms A. S., Winter W., Z-stable ASH algebras, Canad. J. Math., 2008, 60(3):703-720.
[24] Toms A. S., Winter W., The Elliott conjecture for Villadsen algebras of the first type, J. Funct. Anal., 2009, 256(5):1311-1340.
[25] Villadsen J., Simple C*-algebras with perforation, J. Funct. Anal., 1998, 154(1):110-116.
[26] Villadsen J., On the stable rank of simple C*-algebras, J. Amer. Math. Soc., 1999, 12(4):1091-1102.
[27] Winter W., On the classification of simple Z-stable C*-algebras with real rank zero and finite decomposition rank, J. London Math. Soc., 2006, 74(1):167-183.
[28] Winter W., Simple C*-algebras with locally finite decomposition rank, J. Funct. Anal., 2007, 243(2):394-425.
[29] Winter W., Zacharias J., The nuclear dimension of C*-algebras, Adv. Math., 2010, 224(2):461-498.
[30] Winter W., Nuclear dimension and Z-stability of pure C*-algebras, Invent. Math., 2012, 187(2):259-342.
国家自然科学基金资助项目（11371279）
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