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ISSN 1088-6850(e) ISSN 0002-9947(p)

An infinite family of non-isomorphic C-algebras with identical $\mathrm{K}$ -theory

Author(s): Andrew S. Toms
Journal: Trans. Amer. Math. Soc. 360 (2008), 5343-5354.
MSC (2000): Primary 46L35; Secondary 46L80
Posted: May 21, 2008
MathSciNet review: 2415076
Retrieve article in: PDF

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Abstract: We exhibit a countably infinite family of simple, separable, nuclear, and mutually non-isomorphic C-algebras which agree on $\mathrm{K}$ -theory and traces. The algebras do not absorb the Jiang-Su algebra $\mathcal{Z}$ tensorially, answering a question of N. C. Phillips. They are also pairwise shape and Morita equivalent, confirming a conjecture from our earlier work. The distinguishing invariant is the radius of comparison, a non-stable invariant of the Cuntz semigroup.

References:

1.

Blackadar, B.: Traces on simple AF C-algebras, J. Funct. Anal. 38 (1980), 156-168 MR 587906 (82a:46062)

2.

Blackadar, B., and Handelman, D.:Dimension Functions and Traces on C-algebras, J. Funct. Anal. 45 (1982), 297-340 MR 650185 (83g:46050)

3.

Brown, L. G. and Pedersen, G. K.: C-algebras of real rank zero, J. Funct. Anal. 99 (1991), 131-149 MR 1120918 (92m:46086)

4.

Brown, N., Perera, F., and Toms, A. S.: The Cuntz semigroup, the Elliott conjecture, and dimension functions on C-algebras, to appear in J. Reine Angew. Math., arXiv preprint math.OA/0609182 (2006)

5.

Cuntz, J.: Dimension Functions on Simple C-algebras, Math. Ann. 233 (1978), 145-153 MR 0467332 (57:7191)

6.

Elliott, G. A.: The classification problem for amenable C-algebras, Proc. ICM '94, Zurich, Switzerland, Birkhauser Verlag, Basel, Switzerland, 922-932 MR 1403992 (97g:46072)

7.

Gong, G., Jiang, X. and Su, H.: Obstructions to $\mathcal{Z}$ -stability for unital simple $C^{*}$ -algebras, Canad. Math. Bull. 43 (2000), 418-426 MR 1793944 (2001k:46086)

8.

Husemoller, D.: Fibre Bundles, McGraw-Hill, New York, 1966 MR 0229247 (37:4821)

9.

Jiang, X. and Su, H.: On a simple unital projectionless $C^{*}$ -algebra, Amer. J. Math. 121 (1999), 359-413 MR 1680321 (2000a:46104)

10.

Kirchberg, E. and Rørdam, M.: Non-simple purely infinite C-algebras, Amer. J. Math. 122 (2000), 637-666 MR 1759891 (2001k:46088)

11.

Perera, F. and Toms, A. S.: Recasting the Elliott conjecture, Math. Ann. 338 (2007), 669-702 MR 2317934

12.

Rieffel, M. A.: Dimension and stable rank in the -theory of C-algebras, Proc. London Math. Soc. (3) 46 (1983), 301-333 MR 693043 (84g:46085)

13.

Rørdam, M.: A simple C-algebra with a finite and an infinite projection, Acta Math. 191 (2003), 109-142 MR 2020420 (2005m:46096)

14.

Rørdam, M.: The stable and the real rank of $\mathcal{Z}$ -absorbing $C^{*}$ -algebras, Int. J. Math. 15 (2004), 1065-1084 MR 2106263 (2005k:46164)

15.

Rørdam, M., private communication

16.

Toms, A. S.: On the independence of $\mathrm{K}$ -theory and stable rank for simple C-algebras, J. Reine Angew. Math. 578 (2005), 185-199 MR 2113894 (2005k:46189)

17.

Toms, A. S.: On the classification problem for nuclear C-algebras, Ann. of Math. (2) 167 (2008),

1059-1074.

18.

Toms, A. S.: Flat dimension growth for C-algebras, J. Funct. Anal. 238 (2006), 678-708 MR 2253738 (2007j:46098)

19.

Toms, A. S.: Stability in the Cuntz semigroup of a commutative C-algebra, Proc. London Math. Soc. 96 (2008), 1-25.

20.

Villadsen, J.: Simple C-algebras with perforation, J. Funct. Anal. 154 (1998), 110-116 MR 1616504 (99j:46069)

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