群的酉表示及相关的C~*-代数
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摘要
群的酉表示是代数学、几何学、泛函分析等数学分支的重要研究课题。近年来,群与C*-代数交叉的理论和应用的研究也日趋活跃。本文主要从群的酉表示入手,在(T)-性质群的几乎酉表示、自由群F_2的受限酉表示和相应的群C*-代数的刻画等方面展开了研究,同时也讨论了几乎收敛性在C*-代数和离散拓扑半群中的应用。所得的主要结果如下:
1.将Z˙uk关于(T)性质的充分条件推广到了几乎酉表示的情形。这使得对(T)-性质群在几乎酉表示π下的平均算子π(χ)谱的研究不再依赖于渐进酉表示成为可能,从而将我们研究对象的范围拓展到非AGA的(T)-性质群。
2.给出了自由群F_2的μ-受限酉表示的概念,指出了该类表示的大量存在性以及对参数μ的依赖性。利用μ-受限酉表示诱导出了对应的群C*-代数A_μ,并通过证明(A_μ,I,A/B)是一个C*-代数的连续丛,刻画了M_T?关于参数K的连续性。将A_0与一个C*-代数的融合积等同起来,应用Cuntz关于融合自由积的K-理论正合列计算出了A_0的K-群。再以K4和A_0的K-群相同作为基础,通过建立K2泛表示和平凡表示的同伦,借助于函子M_T?的同伦不变性计算出了M_T的K-群。
3.分别从理论和应用两个方面对几乎收敛性展开了研究。理论方面,在N上定义了有限可加的概率测度M_TM_T,指出M_TM_T可测的序列,即恰当分布列都是几乎收敛的,而且其(唯一的)Banach极限可表示成一个形式积分。通过引入Banach极限泛函的概念,定义了赋范向量空间中有界序列的强几乎收敛性,指出现有文献中的几乎收敛和准几乎收敛与我们的强几乎收敛是等价的,从而对向量值序列几乎收敛的概念进行了总结和统一。应用方面,从几乎收敛列的空间M_T∞M_T(N,C)中提取出一个含幺交换C*-代数――M_T.通过Gelfand变换,证明M_T的极大理想空间KN是N的一个紧化,而且包含KN作为闭子集,M_T则为Hilbert C*-模理论提供了一个非C*-自反C*-代数的例子。最后还指出上述几乎收敛性的应用对于可数无限的左顺从可消半群同样适用。
Unitary representation of group is an important research subject of algebra, geome-try, functional analysis and other branches of mathematics. Recently, research on interdis-ciplinary theory and application of group and C*-algebra has become more and more ac-tive. Starting with unitary representation of group, this dissertation studies almost unitaryrepresentation of Property (T) group, constrained unitary representation of free group F_2and corresponding group C*-algebras, applications of almost convergence to C*-algebraand discrete topological semi-group, etc. The main results obtained are as follows:
1. Z˙uk’s sufficient condition for Property (T) is generalized to the case of almostunitary representation. This makes the study on the spectrum of averaging operatorπ(χ)of Property (T) group with almost unitary representationμpossibly independent fromasymptotic unitary representation, hence extends our research range to non-AGA Property(T) groups.
2. The concept ofμ-constrained unitary representation is defined for free group F_2,and we show its existence, abundance and dependence on parameterμ. Correspondinggroup C*-algebraμμis induced byμ-constrained unitary representations. By provingthat (Aμ,I,A/B) is a continuous bundle of C*-algebras, it is shown that A_μs possesscertain continuity with respect toμ. We calculate theμ-groups of A0, by identifyingA_0 with some amalgamated free product of C*-algebras and applying Cuntz’sμ-theoryexact sequences about amalgamated free product. Based on the fact thatμ-groups ofA_0 and A_4 are the same, by constructing homotopy between universal representation andtrivial representation of F_2, we calculate the K-groups of Aμfrom homotopy invarianceof functor K_i.
3. Both theory and application of almost convergence are studied respectively. Intheory, finitely additive probability measureμ_(ac) is defined on N, and it is pointed out thatμ_(ac)-measurable sequence, i.e., properly distributed sequence, is almost convergent withtheir (unique) Banach limit in the form of formal integral. By introducing the concept ofBanach limit functional, the concept of strong almost convergence is defined for boundedsequences in normed vector space, and it is shown that almost convergence and quasi-almost convergence in some papers are equivalent to our strong almost convergence here, hence we summarize and unify the theory of almost convergence of vector-valued se-quences. In application, we extract a unital commutative C*-algebra–M_T from the spacel~∞_(ac)(N,C) of almost convergent sequences. By Gelfand transform, it is proved that themaximal ideal space t_N of M_T is a compactification of N containingβN as a closedsubset, while M_T provides an example of non-C*-re?exive C*-algebra for Hilbert C*-module theory. Finally, we assert that applications of almost convergence above could begeneralized to countably infinite left amenable cancellative semi-groups.
1.将Z˙uk关于(T)性质的充分条件推广到了几乎酉表示的情形。这使得对(T)-性质群在几乎酉表示π下的平均算子π(χ)谱的研究不再依赖于渐进酉表示成为可能,从而将我们研究对象的范围拓展到非AGA的(T)-性质群。
2.给出了自由群F_2的μ-受限酉表示的概念,指出了该类表示的大量存在性以及对参数μ的依赖性。利用μ-受限酉表示诱导出了对应的群C*-代数A_μ,并通过证明(A_μ,I,A/B)是一个C*-代数的连续丛,刻画了M_T?关于参数K的连续性。将A_0与一个C*-代数的融合积等同起来,应用Cuntz关于融合自由积的K-理论正合列计算出了A_0的K-群。再以K4和A_0的K-群相同作为基础,通过建立K2泛表示和平凡表示的同伦,借助于函子M_T?的同伦不变性计算出了M_T的K-群。
3.分别从理论和应用两个方面对几乎收敛性展开了研究。理论方面,在N上定义了有限可加的概率测度M_TM_T,指出M_TM_T可测的序列,即恰当分布列都是几乎收敛的,而且其(唯一的)Banach极限可表示成一个形式积分。通过引入Banach极限泛函的概念,定义了赋范向量空间中有界序列的强几乎收敛性,指出现有文献中的几乎收敛和准几乎收敛与我们的强几乎收敛是等价的,从而对向量值序列几乎收敛的概念进行了总结和统一。应用方面,从几乎收敛列的空间M_T∞M_T(N,C)中提取出一个含幺交换C*-代数――M_T.通过Gelfand变换,证明M_T的极大理想空间KN是N的一个紧化,而且包含KN作为闭子集,M_T则为Hilbert C*-模理论提供了一个非C*-自反C*-代数的例子。最后还指出上述几乎收敛性的应用对于可数无限的左顺从可消半群同样适用。
Unitary representation of group is an important research subject of algebra, geome-try, functional analysis and other branches of mathematics. Recently, research on interdis-ciplinary theory and application of group and C*-algebra has become more and more ac-tive. Starting with unitary representation of group, this dissertation studies almost unitaryrepresentation of Property (T) group, constrained unitary representation of free group F_2and corresponding group C*-algebras, applications of almost convergence to C*-algebraand discrete topological semi-group, etc. The main results obtained are as follows:
1. Z˙uk’s sufficient condition for Property (T) is generalized to the case of almostunitary representation. This makes the study on the spectrum of averaging operatorπ(χ)of Property (T) group with almost unitary representationμpossibly independent fromasymptotic unitary representation, hence extends our research range to non-AGA Property(T) groups.
2. The concept ofμ-constrained unitary representation is defined for free group F_2,and we show its existence, abundance and dependence on parameterμ. Correspondinggroup C*-algebraμμis induced byμ-constrained unitary representations. By provingthat (Aμ,I,A/B) is a continuous bundle of C*-algebras, it is shown that A_μs possesscertain continuity with respect toμ. We calculate theμ-groups of A0, by identifyingA_0 with some amalgamated free product of C*-algebras and applying Cuntz’sμ-theoryexact sequences about amalgamated free product. Based on the fact thatμ-groups ofA_0 and A_4 are the same, by constructing homotopy between universal representation andtrivial representation of F_2, we calculate the K-groups of Aμfrom homotopy invarianceof functor K_i.
3. Both theory and application of almost convergence are studied respectively. Intheory, finitely additive probability measureμ_(ac) is defined on N, and it is pointed out thatμ_(ac)-measurable sequence, i.e., properly distributed sequence, is almost convergent withtheir (unique) Banach limit in the form of formal integral. By introducing the concept ofBanach limit functional, the concept of strong almost convergence is defined for boundedsequences in normed vector space, and it is shown that almost convergence and quasi-almost convergence in some papers are equivalent to our strong almost convergence here, hence we summarize and unify the theory of almost convergence of vector-valued se-quences. In application, we extract a unital commutative C*-algebra–M_T from the spacel~∞_(ac)(N,C) of almost convergent sequences. By Gelfand transform, it is proved that themaximal ideal space t_N of M_T is a compactification of N containingβN as a closedsubset, while M_T provides an example of non-C*-re?exive C*-algebra for Hilbert C*-module theory. Finally, we assert that applications of almost convergence above could begeneralized to countably infinite left amenable cancellative semi-groups.
引文
1 A. Valette, B. Bekka, P. de la Harpe. Kazhdan’s Property (T), New MathematicalMonographs, 11[M]. Cambridge University Press, 2008.
2 H. Lin. An Introduction to the Classification of Amenable C*-algebras[M]. 1st ed.,World Scientific Publishing Company, 2001.
3 K. Davidson. C*-algebras by Example, Fields Institute Monographs, 6[M]. Provi-dence: American Mathematical Society, 1996.
4 J. M. Cohen, A. Figa`-Talamanca. Idempotents in the Reduced C*-algebra of a FreeGroup[J]. Proc. Amer. Math. Soc., 1988, 103:779–782.
5 M. Pimsner, D. V. Voiculescu. K-groups of Reduced Crossed Products by FreeGroups[J]. J. Operator Theory, 1982, 8:131–156.
6 R. T. Powers. Simplicity of the C*-algebra Associated with the Free Group on TwoGenerators[J]. Duke J. Math., 1975, 42:151–156.
7 M. A. Rieffel. C*-algebras Associated with Irrational Rotations,[J]. Pacific J.Math., 1981, 93:415–429.
8 M. A. Rieffel. The Cancellation Theorem for Projective Modules Over IrrationalRotation Algebras[J]. Proc. London Math. Soc., 1983, 47(3):285–302.
9 J. Rosenberg. Quasidiagonality and Nuclearity[J]. J. Operator Theory, 1987, 18:15–18.
10 I. M. Gelfand. Normierte Ringe[J]. Mat. Sbornik, 1941, 9:3–24.
11 I. Kaplansky. Normed Algebras[J]. Duke J. Math., 1949, 16:399–419.
12 I. Kaplansky. A Theorem on Rings of Operators[J]. Pacific J. Math., 1951, 1:227–232.
13 F. J. Murray, J. von Neumann. On Rings of Operators[J]. Ann. Math., 1936,37:116–229.
14 E. C. Lance. Hilbert C*-modules: A Toolkit for Operator Algebraists, LondonMathematical Society Lecture Note Series 210[M]. Cambridge University Press,1995.
15 V. M. Manuilov, E. V. Troitsky. Hilbert C*-modules, Translations of Mathemati-cal Monographs, Vol.226[M]. Providence, R.I.: American Mathematical Society,2005.
16 K. K. Jensen, K. Thomsen. Elements of ????-theory, Mathematics Theory andApplications[M]. Basel-Boston-Berlin: Birkhauser, 1990.
17 I. M. Gelfand, M. Naimark. On the Imbedding of Normed Rings Into the Ring ofOperators in Hilbert Space[J]. Mat. Sbornik, 1943, 12:197–213.
18 J. Roe. Lectures on Coarse Geometry, University Lecture Series Vol. 31[M]. Prov-idence, R.I.: American Mathematical Society, 2003.
19 B. E. Blackadar. A Simple C*-algebra with No Non-trivial Projections[J]. Proc.Amer. Math. Soc., 1980, 78:504–508.
20 B. E. Blackadar. Symmetries of the Car Algebra[J]. Ann. Math., 1990, 131:589–623.
21 L. G. Brown, G. K. Pedersen. C*-algebras of Real Rank Zero[J]. J. Func Anal.,1991, 99:131–149.
22 L. Cobum. The C*-algebra of an Isometry[J]. Bull. Amer. Math. Soc., 1967,73:722–726.
23 J. M. Cohen. C*-algebras without Idempotents[J]. J. Func. Anal., 1979, 33:211–216.
24 K. R. Davidson. Berg’s Technique and Irrational Rotation Algebras[J]. Proc. RoyalIrish Acad., 1984, 84A:117–123.
25 R. V. Kadison. Irreducible Operator Algebras[J]. Proc. Nat. Acad. Sci. U.S.A.,1957, 43:273–278.
26 M. Pimsner. Embedding some Transformation Groups C*-algebras Into Af Alge-bras,[J]. Ergodic Thy. Dynam. Sys., 1983, 3:613–626.
27 I. Putnam. The C*-algebras Associated with Minimal Homeomorphisms of theCantor Set[J]. Pacific J. Math., 1989, 136:329–353.
28 I. Putnam. The Invertibles Are Dense in the Irrational Rotation C*-algebras[J]. J.Reine Angew. Math., 1990, 410:160–166.
29 R. Ji. Smooth Dense Subalgebras of Reduced C*-algebras, Schwartz Cohomologyof Groups, and Cyclic Cohomology[J]. J. Func. Anal., 1992, 107:1–33.
30 G. A. Elliott, D. E. Evans. The Structure of the Irrational Rotation C*-algebra[J].Ann. Math., 1993, 138:477–501.
31 M. D. Choi. A Simple C*-algebra Generated by Two Finite Order Unitaries[J].Can. J. Math., 1979, 31:867–880.
32 W. B. Arveson. A Note on Essentially Normal Operators[J]. Proc. Royal IrishAcad., 1974, 74:143–146.
33 B. Blackadar, A. Kumjian. Skew Products of Relations and the Structure of SimpleC*-algebras[J]. Math. Zeit., 1985, 189:55–63.
34 O. Bratteli. Inductive Limits of Finite Dimensional C*-algebras[J]. Trans. Amer.Math. Soc., 1972, 171:195–234.
35 J. Bunce, J. Deddens. A Family of Simple C*-algebras Related to Weighted ShiftOperators[J]. J.Func. Anal., 1975, 19:12–34.
36 J. Bunce, N. Salinas. Completely Positive Maps on C*-algebras and the Left Es-sential Matricial Spectrum of an Operator[J]. Duke Math. J., 1976, 43:747–774.
37 G. J. Murphy. C*-algebras and Operator Theory[M]. New York: Academic Press,Inc., 1990.
38 W. B. Arveson. An Invitation to C*-algebras, Grad. Texts Math. 39[M]. Berlin,New York: Springer-Verlag, 1976.
39 R. G. Douglas. Banach Algebra Techniques in Operator Theory[M]. New York andLondon: Academic Press, 1972.
40 J. Dixmier. C*-algebras. Translated from the French by Francis Jellett. North-holland Mathematical Library, Vol. 15.[M]. Amsterdam-New York-Oxford: North-Holland Publishing Co., 1977.
41 D. Kazhdan. Connection of the Dual Space of a Group with the Structure of itsClosed Subgroups[J]. Funct. Anal. and its Appl., 1967, 1:63–65.
42 P. de la Harpe, A. Valette. La Proprie′te′(T) De Kazhdan Pour Les Groupes Locale-ment Compacts[J]. Aste′risque, 1989, 175:1–157.
43 T. Gelander, A. Z˙uk. Dependence of Kazhdan Constants on Generating Subsets[J].Israel J. Math., 2002, 129:93–99.
44 P. de la Harpe, G. Robertson, A. Valette. On the Spectrum of the Sum of Generatorsof a Finitely Generated Group[J]. Israel J. Math., 1993, 81:65–96.
45 A. Z˙uk. Property (T) and Kazhdan Constants for Discrete Groups[J]. Geom. Funct.Anal., 2003, 13:643–670.
46 V. M.Manuilov. Almost Representations and Asymptotic Representations of Dis-crete Groups[J]. Izv. Ross. Akad. Nauk Ser. Mat., 1999, 63:159–178.
47 D. Voiculescu. Asymptotically Commuting Finite Rank Unitary Operators withoutCommuting Approximants[J]. Acta Sci. Math. (Szeged), 1983, 45:429–431.
48 P. R. Halmos. Some Unsolved Problems of Unknown Depth about Operators onHilbert Space[J]. Proc. Roy. Soc. Edinburgh Sect. A, 1976, 76:67–76.
49 H. Lin. Almost Commuting Selfadjoint Matrices and Applications. Operator Al-gebras and Their Applications (waterloo)[J]. Fields Inst. Commun., 1997, 13:193–233.
50 A. Connes, M. Gromov, H. Moscovici. Conjecture De Novikov Et Fibre′s PresquePlats[J]. C. R. Acad. Sci. Paris, se′rie I, 1990, 310:273–277.
51 P. Halmos. Normal Dilations and Extension of Operators[J]. Summa Brasil Math.,1950, 2:125–134.
52 M. D. Choi. The Full C*-algebra of the Free Group on Two Generators[J]. PacificJ. Math., 1980, 87(1):41–48.
53 M. D. Choi, C. K. Li. Constrained Unitary Dilations and Numerical Ranges[J]. J.Operator Theory, 2001, 46:435–477.
54 D. Hajdukovic′. Quasi-almost Convergence in a Normed Space[J]. Univ. Beograd.Publ. Elektrotehn. Fak. Ser. Mat., 2002, 13:36–41.
55 A. Valette. Minimal Projections, Integrable Representations and Property (T)[J].Arch. Math. (Basel), 1984, 43(5):397–406.
56 A. I. Shtern. Quasirepresentations and Pseudorepresentations[J]. Funktsional. Anal.i Prilozhen., 1991, 25(2):70–73.
57 D. Fisher, G.Margulis. Almost Isometric Actions, Property (T), and Local Rigid-ity[J]. Invent.Math., 2005, 162(1):19–80.
58 H. Kesten. Symmetric Random Walks on Groups[J]. Trans. AMS, 1959, 92:336–354.
59 R. Eliasen. K K-theory of Amalgamated Free Products of C*-algebras[J]. Math.Nachr., 2007, 280(3):271–280.
60 C. A. Akemann, G. K. Pedersen. Ideal Perturbations of Elements in C*-algebras[J].Math. Scand., 1977, 41:117–139.
61 I. D. Berg. An Extension of the Weyl-von Neumann Theorem to Normal Opera-tors[J]. Trans. Amer. Math. Soc., 1971, 160:365–371.
62 B. Blackadar, O. Bratteli, G. A. Elliott, et al. Reduction of Real Rank in InductiveLimit C*-algebras[J]. Math. Ann., 1992, 292:111–126.
63 B. Blackadar, M. Dadarlat, M. Rordam. The Real Rank of Inductive Limit C*-algebras[J]. Math. Scand., 1991, 69:211–216.
64 G. Corach, A. R. Larotonda. Stable Range in Banach Algebras[J]. J. Pure Appl.Algebra, 1984, 32:289–300.
65 E. G. Effros, D. E. Handelman, C. L. Shen. Dimension Groups and Their AffineTransformations[J]. Amer. J. Math., 1980, 102:385–402.
66 M. Pimsner, D. V. Voiculescu. Exact Sequences for ??-groups and Ext-groups ofCertain Crossed Products of C*-algebras,[J]. J. Operator Theory, 1980, 4:93–118.
67 M. F. Atiyah. K-theory[M]. New York, Amsterdam: W.A. Benjamin Inc., 1967.
68 G. A. Elliott. On the Classification of Inductive Limits of Sequences of Semi-simpleFinite Dimensional Algebras[J]. J. Algebra, 1976, 38:29–44.
69 A. M. Davie. Classification of Essentially Normal Operators[J]. Lect. Notes Math.,1976, 512:31–55.
70 I. D. Berg, K. R. Davidson. A Quantitative Version of the Brown-douglas-fillmoreTheorem[J]. Acta. Math., 1991, 166:121–161.
71 L. G. Brown, R. G. Douglas, P. A. Fillmore. Extensions of C*-algebras and K-homology[J]. Ann. Math., 1977, 105:265–324.
72 W. B. Arveson. Notes on Extensions of C*-algebras[J]. Duke Math. J., 1977,44:329–355.
73 M. D. Choi. Lifting Projections from Quotient C*-algebras,[J]. J. Operator Theory,1983, 10:21–30.
74 M. D. Choi, E. G. Effros. The Completely Positive Lifting Problem for C*-algebras[J]. Ann. Math., 1976, 104:585–609.
75 M. R?rdam, F. Larsen, N. Laustsen. An Introduction to K-theory for C*-algebras,London Mathematical Society Student Texts 49[M]. Cambridge University Press,2000.
76 B. Russo, H. A. Dye. A Note on Unitary Operators in C*-algebras[J]. Duke J.Math., 1966, 33:413–416.
77 R. Bott. The Stable Homotopy of the Classical Groups[J]. Ann. Math., 1959,70:313–337.
78 J. Cuntz. The K-groups for Free Products of C*-algebras[J]. Proc. of Symp. in PureMath., 1982, 38:81–84.
79 S. Wassermann. Exact C*-algebras and Related Topics, Lecture Notes Series19[M]. Seoul National University, 1994.
80 B. Blackadar. K-theory for Operator Algebras, Msri Publications, Vol. 5[M]. 2nded., Cambridge: Cambridge University Press, 1998.
81 B. Q. Feng, J. L. Li. Some Estimations of Banach Limits[J]. J. Math. Anal. Appl.,2006, 323(1):481–496.
82 G. G. Lorentz. A Contribution to the Theory of Divergent Sequences[J]. ActaMath., 1948, 80:167–190.
83 L. Sucheston. Banach Limits[J]. Amer. Math. Monthly, 1967, 74:308–311.
84 L. Kuipers, H. Niederreiter. Uniform Distribution of Sequences[M]. New York:Wiley, 1974.
85 S. Shaw, S. Lin. Strong and Absolute Almost-convergence of Vector Functions andRelated Tauberian Theorems[J]. J. Math. Anal. Appl., 2007, 334:1073–1087.
86 J. C. Kurtz. Almost Convergent Vector Sequences[J]. T(?)hoku Math. Journ., 1970,22:493–498.
87 C. Chou. The Multipliers of the Space of Almost Convergent Sequences[J]. IllinoisJ. Math., 1972, 16(4):687–694.
88 M. Friedberg, J. W. Stepp. A Note on the Bohr Compactification[J]. SemigroupForum, 1973, 6(1):362–364.
89 J. Kelley. General Topology (graduate Texts in Mathematics)[M]. Springer, 1975.
90 E. Kaniuth. A Course in Commutative Banach Algebras, Graduate Texts in Math-ematics 246[M]. Springer, 2008.
91 W. L. Paschke. The Double B-dual of an Inner Product Module Over a C*-algebra B[J]. Can. J. Math., 1974, 26(5):1272–1280.
92 M. Frank. Self-duality and C*-re?exivity of Hilbert C*-moduli[J]. Z. Anal. An-wendungen, 1990, 9(2):165–176.
93 V. A. Trofimov. Re?exivity of Hilbert Modules Over an Algebra of Compact Oper-ators with Associated Identity[J]. Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1986,(5):60–64.
94 G. G. Kasparov. Hilbert C*-modules: Theorems of Stinespring and Voiculescu[J].J. Operator Theory, 1980, 4(1):133–150.
95 W. L. Paschke. Inner Product Modules Over C*-algebras[J]. Trans. Amer. Math.Soc., 1973, 182:443–468.
96 A. S. Mishchenko. Representations of Compact Groups in Hilbert Modules OverC*-algebras[J]. Trudy Mat. Inst. Steklov, 1984, 166:161–176.
97 M. Frank, V. M. Manuilov, E. V. Troitzky. A Reflexivity Criterion for Hilbert C*-modules Over Commutative C*-algebras[J]. arXiv:0908.1414v2, 2010.
98 P. Komja′th, V. Totik. Problems and Theorems in Classical Set Theory[M]. NewYork: Springer-Verlag, 2006.
99 N. Hindman, D. Strauss. Algebra in the Stone-cech Compactification. Theory andApplications, Vol. 27, De Gruyter Expositions in Mathematics[M]. Berlin: Walterde Gruyter & Co., 1998.
100 C. Chou, J. P. Duran. Multipliers for the Space of Almost Convergent Functions ona Semigroup[J]. Proc. AMS, 1973, 39(1):125–128.
101 J. Glimm. On a Certain Class of Operator Algebras[J]. Trans. Amer. Math. Soc.,1960, 95:318–340.
102 J. Glimm. A Stone-weierstrass Theorem for C*-algebras[J]. Ann. Math., 1960,72:216–244.
103 D. Hadwin. An Operator Valued Spectrum[J]. Indiana Univ. Math. J., 1977,26:329–340.
104 D. Hadwin. Nonseparable Approximate Equivalence[J]. Trans. Amer. Math. Soc.,1980, 266:203–231.
105 R. V. Kadison, G. K. Pedersen. Means and Convex Combinations of Unitary Oper-ators[J]. Math. Scand., 1985, 57:249–266.
106 M. Pimsner, D. V. Voiculescu. Imbedding the Irrational Rotaation Algebras Into anAF Algebra[J]. J. Operator Theory, 1980, 4:201–210.
107 S. C. Power. Simplicity of C*-algebras of Minimal Dynamical Systems[J]. J.London Math. Soc., 1978, 18(2):534–538.
2 H. Lin. An Introduction to the Classification of Amenable C*-algebras[M]. 1st ed.,World Scientific Publishing Company, 2001.
3 K. Davidson. C*-algebras by Example, Fields Institute Monographs, 6[M]. Provi-dence: American Mathematical Society, 1996.
4 J. M. Cohen, A. Figa`-Talamanca. Idempotents in the Reduced C*-algebra of a FreeGroup[J]. Proc. Amer. Math. Soc., 1988, 103:779–782.
5 M. Pimsner, D. V. Voiculescu. K-groups of Reduced Crossed Products by FreeGroups[J]. J. Operator Theory, 1982, 8:131–156.
6 R. T. Powers. Simplicity of the C*-algebra Associated with the Free Group on TwoGenerators[J]. Duke J. Math., 1975, 42:151–156.
7 M. A. Rieffel. C*-algebras Associated with Irrational Rotations,[J]. Pacific J.Math., 1981, 93:415–429.
8 M. A. Rieffel. The Cancellation Theorem for Projective Modules Over IrrationalRotation Algebras[J]. Proc. London Math. Soc., 1983, 47(3):285–302.
9 J. Rosenberg. Quasidiagonality and Nuclearity[J]. J. Operator Theory, 1987, 18:15–18.
10 I. M. Gelfand. Normierte Ringe[J]. Mat. Sbornik, 1941, 9:3–24.
11 I. Kaplansky. Normed Algebras[J]. Duke J. Math., 1949, 16:399–419.
12 I. Kaplansky. A Theorem on Rings of Operators[J]. Pacific J. Math., 1951, 1:227–232.
13 F. J. Murray, J. von Neumann. On Rings of Operators[J]. Ann. Math., 1936,37:116–229.
14 E. C. Lance. Hilbert C*-modules: A Toolkit for Operator Algebraists, LondonMathematical Society Lecture Note Series 210[M]. Cambridge University Press,1995.
15 V. M. Manuilov, E. V. Troitsky. Hilbert C*-modules, Translations of Mathemati-cal Monographs, Vol.226[M]. Providence, R.I.: American Mathematical Society,2005.
16 K. K. Jensen, K. Thomsen. Elements of ????-theory, Mathematics Theory andApplications[M]. Basel-Boston-Berlin: Birkhauser, 1990.
17 I. M. Gelfand, M. Naimark. On the Imbedding of Normed Rings Into the Ring ofOperators in Hilbert Space[J]. Mat. Sbornik, 1943, 12:197–213.
18 J. Roe. Lectures on Coarse Geometry, University Lecture Series Vol. 31[M]. Prov-idence, R.I.: American Mathematical Society, 2003.
19 B. E. Blackadar. A Simple C*-algebra with No Non-trivial Projections[J]. Proc.Amer. Math. Soc., 1980, 78:504–508.
20 B. E. Blackadar. Symmetries of the Car Algebra[J]. Ann. Math., 1990, 131:589–623.
21 L. G. Brown, G. K. Pedersen. C*-algebras of Real Rank Zero[J]. J. Func Anal.,1991, 99:131–149.
22 L. Cobum. The C*-algebra of an Isometry[J]. Bull. Amer. Math. Soc., 1967,73:722–726.
23 J. M. Cohen. C*-algebras without Idempotents[J]. J. Func. Anal., 1979, 33:211–216.
24 K. R. Davidson. Berg’s Technique and Irrational Rotation Algebras[J]. Proc. RoyalIrish Acad., 1984, 84A:117–123.
25 R. V. Kadison. Irreducible Operator Algebras[J]. Proc. Nat. Acad. Sci. U.S.A.,1957, 43:273–278.
26 M. Pimsner. Embedding some Transformation Groups C*-algebras Into Af Alge-bras,[J]. Ergodic Thy. Dynam. Sys., 1983, 3:613–626.
27 I. Putnam. The C*-algebras Associated with Minimal Homeomorphisms of theCantor Set[J]. Pacific J. Math., 1989, 136:329–353.
28 I. Putnam. The Invertibles Are Dense in the Irrational Rotation C*-algebras[J]. J.Reine Angew. Math., 1990, 410:160–166.
29 R. Ji. Smooth Dense Subalgebras of Reduced C*-algebras, Schwartz Cohomologyof Groups, and Cyclic Cohomology[J]. J. Func. Anal., 1992, 107:1–33.
30 G. A. Elliott, D. E. Evans. The Structure of the Irrational Rotation C*-algebra[J].Ann. Math., 1993, 138:477–501.
31 M. D. Choi. A Simple C*-algebra Generated by Two Finite Order Unitaries[J].Can. J. Math., 1979, 31:867–880.
32 W. B. Arveson. A Note on Essentially Normal Operators[J]. Proc. Royal IrishAcad., 1974, 74:143–146.
33 B. Blackadar, A. Kumjian. Skew Products of Relations and the Structure of SimpleC*-algebras[J]. Math. Zeit., 1985, 189:55–63.
34 O. Bratteli. Inductive Limits of Finite Dimensional C*-algebras[J]. Trans. Amer.Math. Soc., 1972, 171:195–234.
35 J. Bunce, J. Deddens. A Family of Simple C*-algebras Related to Weighted ShiftOperators[J]. J.Func. Anal., 1975, 19:12–34.
36 J. Bunce, N. Salinas. Completely Positive Maps on C*-algebras and the Left Es-sential Matricial Spectrum of an Operator[J]. Duke Math. J., 1976, 43:747–774.
37 G. J. Murphy. C*-algebras and Operator Theory[M]. New York: Academic Press,Inc., 1990.
38 W. B. Arveson. An Invitation to C*-algebras, Grad. Texts Math. 39[M]. Berlin,New York: Springer-Verlag, 1976.
39 R. G. Douglas. Banach Algebra Techniques in Operator Theory[M]. New York andLondon: Academic Press, 1972.
40 J. Dixmier. C*-algebras. Translated from the French by Francis Jellett. North-holland Mathematical Library, Vol. 15.[M]. Amsterdam-New York-Oxford: North-Holland Publishing Co., 1977.
41 D. Kazhdan. Connection of the Dual Space of a Group with the Structure of itsClosed Subgroups[J]. Funct. Anal. and its Appl., 1967, 1:63–65.
42 P. de la Harpe, A. Valette. La Proprie′te′(T) De Kazhdan Pour Les Groupes Locale-ment Compacts[J]. Aste′risque, 1989, 175:1–157.
43 T. Gelander, A. Z˙uk. Dependence of Kazhdan Constants on Generating Subsets[J].Israel J. Math., 2002, 129:93–99.
44 P. de la Harpe, G. Robertson, A. Valette. On the Spectrum of the Sum of Generatorsof a Finitely Generated Group[J]. Israel J. Math., 1993, 81:65–96.
45 A. Z˙uk. Property (T) and Kazhdan Constants for Discrete Groups[J]. Geom. Funct.Anal., 2003, 13:643–670.
46 V. M.Manuilov. Almost Representations and Asymptotic Representations of Dis-crete Groups[J]. Izv. Ross. Akad. Nauk Ser. Mat., 1999, 63:159–178.
47 D. Voiculescu. Asymptotically Commuting Finite Rank Unitary Operators withoutCommuting Approximants[J]. Acta Sci. Math. (Szeged), 1983, 45:429–431.
48 P. R. Halmos. Some Unsolved Problems of Unknown Depth about Operators onHilbert Space[J]. Proc. Roy. Soc. Edinburgh Sect. A, 1976, 76:67–76.
49 H. Lin. Almost Commuting Selfadjoint Matrices and Applications. Operator Al-gebras and Their Applications (waterloo)[J]. Fields Inst. Commun., 1997, 13:193–233.
50 A. Connes, M. Gromov, H. Moscovici. Conjecture De Novikov Et Fibre′s PresquePlats[J]. C. R. Acad. Sci. Paris, se′rie I, 1990, 310:273–277.
51 P. Halmos. Normal Dilations and Extension of Operators[J]. Summa Brasil Math.,1950, 2:125–134.
52 M. D. Choi. The Full C*-algebra of the Free Group on Two Generators[J]. PacificJ. Math., 1980, 87(1):41–48.
53 M. D. Choi, C. K. Li. Constrained Unitary Dilations and Numerical Ranges[J]. J.Operator Theory, 2001, 46:435–477.
54 D. Hajdukovic′. Quasi-almost Convergence in a Normed Space[J]. Univ. Beograd.Publ. Elektrotehn. Fak. Ser. Mat., 2002, 13:36–41.
55 A. Valette. Minimal Projections, Integrable Representations and Property (T)[J].Arch. Math. (Basel), 1984, 43(5):397–406.
56 A. I. Shtern. Quasirepresentations and Pseudorepresentations[J]. Funktsional. Anal.i Prilozhen., 1991, 25(2):70–73.
57 D. Fisher, G.Margulis. Almost Isometric Actions, Property (T), and Local Rigid-ity[J]. Invent.Math., 2005, 162(1):19–80.
58 H. Kesten. Symmetric Random Walks on Groups[J]. Trans. AMS, 1959, 92:336–354.
59 R. Eliasen. K K-theory of Amalgamated Free Products of C*-algebras[J]. Math.Nachr., 2007, 280(3):271–280.
60 C. A. Akemann, G. K. Pedersen. Ideal Perturbations of Elements in C*-algebras[J].Math. Scand., 1977, 41:117–139.
61 I. D. Berg. An Extension of the Weyl-von Neumann Theorem to Normal Opera-tors[J]. Trans. Amer. Math. Soc., 1971, 160:365–371.
62 B. Blackadar, O. Bratteli, G. A. Elliott, et al. Reduction of Real Rank in InductiveLimit C*-algebras[J]. Math. Ann., 1992, 292:111–126.
63 B. Blackadar, M. Dadarlat, M. Rordam. The Real Rank of Inductive Limit C*-algebras[J]. Math. Scand., 1991, 69:211–216.
64 G. Corach, A. R. Larotonda. Stable Range in Banach Algebras[J]. J. Pure Appl.Algebra, 1984, 32:289–300.
65 E. G. Effros, D. E. Handelman, C. L. Shen. Dimension Groups and Their AffineTransformations[J]. Amer. J. Math., 1980, 102:385–402.
66 M. Pimsner, D. V. Voiculescu. Exact Sequences for ??-groups and Ext-groups ofCertain Crossed Products of C*-algebras,[J]. J. Operator Theory, 1980, 4:93–118.
67 M. F. Atiyah. K-theory[M]. New York, Amsterdam: W.A. Benjamin Inc., 1967.
68 G. A. Elliott. On the Classification of Inductive Limits of Sequences of Semi-simpleFinite Dimensional Algebras[J]. J. Algebra, 1976, 38:29–44.
69 A. M. Davie. Classification of Essentially Normal Operators[J]. Lect. Notes Math.,1976, 512:31–55.
70 I. D. Berg, K. R. Davidson. A Quantitative Version of the Brown-douglas-fillmoreTheorem[J]. Acta. Math., 1991, 166:121–161.
71 L. G. Brown, R. G. Douglas, P. A. Fillmore. Extensions of C*-algebras and K-homology[J]. Ann. Math., 1977, 105:265–324.
72 W. B. Arveson. Notes on Extensions of C*-algebras[J]. Duke Math. J., 1977,44:329–355.
73 M. D. Choi. Lifting Projections from Quotient C*-algebras,[J]. J. Operator Theory,1983, 10:21–30.
74 M. D. Choi, E. G. Effros. The Completely Positive Lifting Problem for C*-algebras[J]. Ann. Math., 1976, 104:585–609.
75 M. R?rdam, F. Larsen, N. Laustsen. An Introduction to K-theory for C*-algebras,London Mathematical Society Student Texts 49[M]. Cambridge University Press,2000.
76 B. Russo, H. A. Dye. A Note on Unitary Operators in C*-algebras[J]. Duke J.Math., 1966, 33:413–416.
77 R. Bott. The Stable Homotopy of the Classical Groups[J]. Ann. Math., 1959,70:313–337.
78 J. Cuntz. The K-groups for Free Products of C*-algebras[J]. Proc. of Symp. in PureMath., 1982, 38:81–84.
79 S. Wassermann. Exact C*-algebras and Related Topics, Lecture Notes Series19[M]. Seoul National University, 1994.
80 B. Blackadar. K-theory for Operator Algebras, Msri Publications, Vol. 5[M]. 2nded., Cambridge: Cambridge University Press, 1998.
81 B. Q. Feng, J. L. Li. Some Estimations of Banach Limits[J]. J. Math. Anal. Appl.,2006, 323(1):481–496.
82 G. G. Lorentz. A Contribution to the Theory of Divergent Sequences[J]. ActaMath., 1948, 80:167–190.
83 L. Sucheston. Banach Limits[J]. Amer. Math. Monthly, 1967, 74:308–311.
84 L. Kuipers, H. Niederreiter. Uniform Distribution of Sequences[M]. New York:Wiley, 1974.
85 S. Shaw, S. Lin. Strong and Absolute Almost-convergence of Vector Functions andRelated Tauberian Theorems[J]. J. Math. Anal. Appl., 2007, 334:1073–1087.
86 J. C. Kurtz. Almost Convergent Vector Sequences[J]. T(?)hoku Math. Journ., 1970,22:493–498.
87 C. Chou. The Multipliers of the Space of Almost Convergent Sequences[J]. IllinoisJ. Math., 1972, 16(4):687–694.
88 M. Friedberg, J. W. Stepp. A Note on the Bohr Compactification[J]. SemigroupForum, 1973, 6(1):362–364.
89 J. Kelley. General Topology (graduate Texts in Mathematics)[M]. Springer, 1975.
90 E. Kaniuth. A Course in Commutative Banach Algebras, Graduate Texts in Math-ematics 246[M]. Springer, 2008.
91 W. L. Paschke. The Double B-dual of an Inner Product Module Over a C*-algebra B[J]. Can. J. Math., 1974, 26(5):1272–1280.
92 M. Frank. Self-duality and C*-re?exivity of Hilbert C*-moduli[J]. Z. Anal. An-wendungen, 1990, 9(2):165–176.
93 V. A. Trofimov. Re?exivity of Hilbert Modules Over an Algebra of Compact Oper-ators with Associated Identity[J]. Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1986,(5):60–64.
94 G. G. Kasparov. Hilbert C*-modules: Theorems of Stinespring and Voiculescu[J].J. Operator Theory, 1980, 4(1):133–150.
95 W. L. Paschke. Inner Product Modules Over C*-algebras[J]. Trans. Amer. Math.Soc., 1973, 182:443–468.
96 A. S. Mishchenko. Representations of Compact Groups in Hilbert Modules OverC*-algebras[J]. Trudy Mat. Inst. Steklov, 1984, 166:161–176.
97 M. Frank, V. M. Manuilov, E. V. Troitzky. A Reflexivity Criterion for Hilbert C*-modules Over Commutative C*-algebras[J]. arXiv:0908.1414v2, 2010.
98 P. Komja′th, V. Totik. Problems and Theorems in Classical Set Theory[M]. NewYork: Springer-Verlag, 2006.
99 N. Hindman, D. Strauss. Algebra in the Stone-cech Compactification. Theory andApplications, Vol. 27, De Gruyter Expositions in Mathematics[M]. Berlin: Walterde Gruyter & Co., 1998.
100 C. Chou, J. P. Duran. Multipliers for the Space of Almost Convergent Functions ona Semigroup[J]. Proc. AMS, 1973, 39(1):125–128.
101 J. Glimm. On a Certain Class of Operator Algebras[J]. Trans. Amer. Math. Soc.,1960, 95:318–340.
102 J. Glimm. A Stone-weierstrass Theorem for C*-algebras[J]. Ann. Math., 1960,72:216–244.
103 D. Hadwin. An Operator Valued Spectrum[J]. Indiana Univ. Math. J., 1977,26:329–340.
104 D. Hadwin. Nonseparable Approximate Equivalence[J]. Trans. Amer. Math. Soc.,1980, 266:203–231.
105 R. V. Kadison, G. K. Pedersen. Means and Convex Combinations of Unitary Oper-ators[J]. Math. Scand., 1985, 57:249–266.
106 M. Pimsner, D. V. Voiculescu. Imbedding the Irrational Rotaation Algebras Into anAF Algebra[J]. J. Operator Theory, 1980, 4:201–210.
107 S. C. Power. Simplicity of C*-algebras of Minimal Dynamical Systems[J]. J.London Math. Soc., 1978, 18(2):534–538.