The Elliott conjecture for Villadsen algebras of the first type. (English) Zbl 1184.46061
The Jiang-Su \(C^*\)-algebra \(\mathcal{Z}\), constructed as a simple unital inductive limit of dimension drop intervals, has the same \(K\)-theory as the complex numbers [X.-H.Jiang and H.-B.Su, “On a simple unital projectionless \(C^*\)-algebra”, Am.J.Math.121, No.2, 359–413 (1999; Zbl 0923.46069)]. A \(C^*\)-algebra \(A\) is said to be \(\mathcal{Z}\)-stable, or to tensorially absorb \(\mathcal{Z}\), when \(A\cong A \otimes \mathcal{Z}\). It is widely agreed [cf., e.g., A.S.Toms and W.Winter, “\(\mathcal{Z}\)-stable ASH algebras”, Can.J.Math.60, No.3, 703–720 (2008; Zbl 1157.46034)] that the largest class for which Elliott’s classification conjecture [G.A.Elliott, “The classification problem for amenable \(C^*\)-algebras”, in Proc.ICM ’94, Zürich, Birkhäuser, 922–932 (1995; Zbl 0946.46050)] can hold consists of \(\mathcal Z\)-stable algebras.
In the paper under review, a broad class \({\mathcal V}\text{I}\) of AH algebras, called Villadsen algebras of the first type, is defined. This includes Villadsen’s example of a simple separable nuclear \(C^*\)-algebra with perforated ordered \(K_0\)-group, Goodearl algebras, as well as many AF algebras, \(A\mathbf{T}\) algebras, and AH algebras of slow dimension growth. All topological spaces involved in the definition of a \({\mathcal V}\text{I}\) AH algebra are certain powers of the same space \(X\), called the seed space of \(A\).
Assuming that such a \(C^*\)-algebra \(A\) is simple with a finite-dimensional CW complex as seed space, the authors prove the equivalence of the following six properties: (i) \(A\) is \(\mathcal{Z}\)-stable; (ii) \(A\) has strict comparison of positive elements; (iii) \(A\) has finite decomposition rank; (iv) \(A\) has slow dimension growth; (v) \(A\) has bounded dimension growth; (vi) \(A\) is approximately divisible.
Furthermore, it is shown that these conditions are fulfilled when \(A\) has real rank zero. Finally, a large family of non-isomorphic simple \({\mathcal V}\text{I}\) algebras is exhibited, with the same topological \(K\)-theory and tracial state space.
In the paper under review, a broad class \({\mathcal V}\text{I}\) of AH algebras, called Villadsen algebras of the first type, is defined. This includes Villadsen’s example of a simple separable nuclear \(C^*\)-algebra with perforated ordered \(K_0\)-group, Goodearl algebras, as well as many AF algebras, \(A\mathbf{T}\) algebras, and AH algebras of slow dimension growth. All topological spaces involved in the definition of a \({\mathcal V}\text{I}\) AH algebra are certain powers of the same space \(X\), called the seed space of \(A\).
Assuming that such a \(C^*\)-algebra \(A\) is simple with a finite-dimensional CW complex as seed space, the authors prove the equivalence of the following six properties: (i) \(A\) is \(\mathcal{Z}\)-stable; (ii) \(A\) has strict comparison of positive elements; (iii) \(A\) has finite decomposition rank; (iv) \(A\) has slow dimension growth; (v) \(A\) has bounded dimension growth; (vi) \(A\) is approximately divisible.
Furthermore, it is shown that these conditions are fulfilled when \(A\) has real rank zero. Finally, a large family of non-isomorphic simple \({\mathcal V}\text{I}\) algebras is exhibited, with the same topological \(K\)-theory and tracial state space.
Reviewer: Florin P. Boca (Urbana-Champaign)
MSC:
| 46L35 | Classifications of \(C^*\)-algebras |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
Keywords:
nuclear C*-algebras; classification theory; Elliott’s invariant; Jiang-Su algebra; \({\mathcal Z}\)-stability; AH algebras; Villadsen algebras of the first type; dimension growth; strict comparison of positive elements; finite decomposition rank; real rank zeroReferences:
| [1] | Blackadar, B.; Bratteli, O.; Elliott, G. A.; Kumjian, A., Reduction of real rank in inductive limits of \(C^\ast \)-algebras, Math. Ann., 292, 111-126 (1992) · Zbl 0738.46027 |
| [2] | Blackadar, B.; Handelman, D., Dimension functions and traces on \(C^\ast \)-algebras, J. Funct. Anal., 45, 297-340 (1982) · Zbl 0513.46047 |
| [3] | Blackadar, B.; Kumjian, A.; Rørdam, M., Approximately central matrix units and the structure of noncommutative tori, K-Theory, 6, 267-284 (1992) · Zbl 0813.46064 |
| [4] | Bratteli, O., Inductive limits of finite dimensional \(C^\ast \)-algebras, Trans. Amer. Math. Soc., 171, 195-234 (1972) · Zbl 0264.46057 |
| [5] | Dadarlat, M., Reduction to dimension three of local spectra of real rank zero \(C^\ast \)-algebras, J. Reine Angew. Math., 460, 189-212 (1995) · Zbl 0815.46065 |
| [6] | Elliott, G. A., On the classification of inductive limits of sequences of semi-simple finite-dimensional algebras, J. Algebra, 38, 29-44 (1976) · Zbl 0323.46063 |
| [7] | Elliott, G. A., On the classification of \(C^\ast \)-algebras of real rank zero, J. Reine Angew. Math., 443, 179-219 (1993) · Zbl 0809.46067 |
| [8] | Elliott, G. A., An invariant for simple \(C^\ast \)-algebras, (Canadian Math. Soc. 1945-1995, vol. 3 (1996), Canadian Math. Soc.: Canadian Math. Soc. Ottawa, ON), 61-90 · Zbl 1206.46046 |
| [9] | G.A. Elliott, The classification problem for amenable \(C^{\ast;}\)-algebras, in: Proc. ICM ’94, Zurich, Switzerland, Birkhäuser, Basel, Switzerland, 1995 pp. 922-932; G.A. Elliott, The classification problem for amenable \(C^{\ast;}\)-algebras, in: Proc. ICM ’94, Zurich, Switzerland, Birkhäuser, Basel, Switzerland, 1995 pp. 922-932 · Zbl 0946.46050 |
| [10] | Elliott, G. A.; Gong, G., On the classification of \(C^\ast \)-algebras of real rank zero. II, Ann. of Math. (2), 144, 497-610 (1996) · Zbl 0867.46041 |
| [11] | Elliott, G. A.; Gong, G.; Li, L., Approximate divisibility of simple inductive limit \(C^\ast \)-algebras, Contemp. Math., 228, 87-97 (1998) · Zbl 0933.46054 |
| [12] | Elliott, G. A.; Gong, G.; Li, L., On the classification of simple inductive limit \(C^\ast \)-algebras, II: The isomorphism theorem, Invent. Math., 168, 249-320 (2007) · Zbl 1129.46051 |
| [13] | Glimm, J., On a certain class of operator algebras, Trans. Amer. Math. Soc., 95, 318-340 (1960) · Zbl 0094.09701 |
| [14] | Gong, G., On inductive limits of matrix algebras over higher-dimensional spaces. I, II, Math. Scand., 80, 41-55 (1997), 56-100 · Zbl 0901.46053 |
| [15] | Gong, G., On the classification of simple inductive limit \(C^\ast \)-algebras I. The reduction theorem, Doc. Math., 7, 255-461 (2002) · Zbl 1024.46018 |
| [16] | Jiang, X.; Su, H., On a simple unital projectionless \(C^\ast \)-algebra, Amer. J. Math., 121, 359-413 (1999) · Zbl 0923.46069 |
| [17] | E. Kirchberg, The classification of Purely Infinite \(C^{\ast;}\)-algebras using Kasparov’s theory, Fields Inst. Commun., in press; E. Kirchberg, The classification of Purely Infinite \(C^{\ast;}\)-algebras using Kasparov’s theory, Fields Inst. Commun., in press |
| [18] | Kirchberg, E.; Winter, W., Covering dimension and quasidiagonality, Internat. J. Math., 15, 63-85 (2004) · Zbl 1065.46053 |
| [19] | Lin, H., Classification of simple \(C^\ast \)-algebras with tracial topological rank zero, Duke Math. J., 125, 91-119 (2004) · Zbl 1068.46032 |
| [20] | Lin, H., Simple nuclear \(C^\ast \)-algebras of tracial topological rank one, J. Funct. Anal., 251, 601-679 (2007) · Zbl 1206.46052 |
| [21] | Phillips, N. C., A classification theorem for nuclear purely infinite simple \(C^\ast \)-algebras, Doc. Math., 5, 49-114 (2000) · Zbl 0943.46037 |
| [22] | Rørdam, M., On the structure of simple \(C^\ast \)-algebras tensored with a UHF-algebra, II, J. Funct. Anal., 107, 255-269 (1992) · Zbl 0810.46067 |
| [23] | Rørdam, M., Classification of Nuclear \(C^\ast \)-Algebras, Encyclopaedia Math. Sci., vol. 126 (2002), Springer-Verlag: Springer-Verlag Berlin-Heidelberg · Zbl 1016.46037 |
| [24] | Rørdam, M., A simple \(C^\ast \)-algebra with a finite and an infinite projection, Acta Math., 191, 109-142 (2003) · Zbl 1072.46036 |
| [25] | Rørdam, M., The stable and the real rank of \(Z\)-absorbing \(C^\ast \)-algebras, Internat. J. Math., 15, 1065-1084 (2004) · Zbl 1077.46054 |
| [26] | Toms, A. S., On the classification problem for nuclear \(C^\ast \)-algebras, Ann. of Math. (2), 167, 1029-1044 (2008) · Zbl 1181.46047 |
| [27] | Toms, A. S., On the independence of K-theory and stable rank for simple \(C^\ast \)-algebras, J. Reine Angew. Math., 578, 185-199 (2005) · Zbl 1073.46045 |
| [28] | Toms, A. S., Flat dimension growth for \(C^\ast \)-algebras, J. Funct. Anal., 238, 678-708 (2006) · Zbl 1111.46041 |
| [29] | Toms, A. S., Stability in the Cuntz semigroup of a commutative \(C^\ast \)-algebra, Proc. London Math. Soc., 96, 1-25 (2008) · Zbl 1143.46037 |
| [30] | A.S. Toms, An infinite family of non-isomorphic \(C^{\ast;}\)-algebras with identical K-theory, preprint, 2006, arXiv: math.OA/0609214, Trans. Amer. Math. Soc., in press; A.S. Toms, An infinite family of non-isomorphic \(C^{\ast;}\)-algebras with identical K-theory, preprint, 2006, arXiv: math.OA/0609214, Trans. Amer. Math. Soc., in press |
| [31] | Toms, A. S.; Winter, W., Strongly self-absorbing \(C^\ast \)-algebras, Trans. Amer. Math. Soc., 359, 3999-4029 (2007) · Zbl 1120.46046 |
| [32] | Toms, A. S.; Winter, W., \(Z\)-stable ASH algebras, Canad. J. Math., 60, 703-720 (2008) · Zbl 1157.46034 |
| [33] | Villadsen, J., Simple \(C^\ast \)-algebras with perforation, J. Funct. Anal., 154, 110-116 (1998) · Zbl 0915.46047 |
| [34] | Villadsen, J., On the stable rank of simple \(C^\ast \)-algebras, J. Amer. Math. Soc., 12, 1091-1102 (1999) · Zbl 0937.46052 |
| [35] | Winter, W., Covering dimension for nuclear \(C^\ast \)-algebras, J. Funct. Anal., 199, 535-556 (2003) · Zbl 1026.46049 |
| [36] | Winter, W., On topologically finite-dimensional simple \(C^\ast \)-algebras, Math. Ann., 332, 843-878 (2005) · Zbl 1089.46039 |
| [37] | Winter, W., On the classification of simple \(Z\)-stable \(C^\ast \)-algebras with real rank zero and finite decomposition rank, J. London Math. Soc., 74, 167-183 (2006) · Zbl 1104.46034 |
| [38] | Winter, W., Simple \(C^\ast \)-algebras with locally finite decomposition rank, J. Funct. Anal., 243, 394-425 (2007) · Zbl 1121.46047 |
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