No Access

MINIMAL DYNAMICS AND -STABLE CLASSIFICATION

KAREN R. STRUNG

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

Search for more papers by this author

and

WILHELM WINTER

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom

Search for more papers by this author

https://doi.org/10.1142/S0129167X10006665Cited by:10 (Source: Crossref)

Abstract

Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary that implements α in the transformation group C*-algebra C(X) ⋊_α ℤ, and a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in . Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra , we prove that (C(X) ⋊_α ℤ) ⊗ M_q^∞ is tracially approximately if there exists a y ∈ X such that the C*-subalgebra (C*(C(X), uC₀(X\{y}))) ⊗ M_q^∞ is tracially approximately . If the class consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either C(X) ⋊_α ℤ or C*(C(X), uC₀(X\{y})), nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces.

Keywords:

AMSC: 46L85, 46L35

References

I. D. Berg, Michigan Math. J. 21, 377 (1975). Web of Science, Google Scholar
B. Blackadar, A. Kumjian and M. Rørdam, J. K-Theory 6, 267 (1992). Crossref, Web of Science, Google Scholar
L. G. Brown, Pacific J. Math. 71, 335 (1977). Crossref, Web of Science, Google Scholar
A. Connes, Adv. Math. 39, 31 (1981). Crossref, Web of Science, Google Scholar
J. Cuntz, Comm. Math. Phys. 57, 173 (1977), DOI: 10.1007/BF01625776. Crossref, Web of Science, Google Scholar
G. A. Elliott, J. Reine Angew. Math. 443, 179 (1993). Web of Science, Google Scholar
G. A. Elliott and D. E. Evans, Ann. Math. 138(2), 477 (1993), DOI: 10.2307/2946553. Crossref, Web of Science, Google Scholar
G. A. Elliott and Z. Niu, J. Funct. Anal. 254(2), 396 (2008). Crossref, Web of Science, Google Scholar
J. Giol and D. Kerr, J. Reine Angew. Math. 639, 107 (2010). Web of Science, Google Scholar
T. Giordiano, I. F. Putnam and C. F. Skau, J. Reine Angew. Math. 469, 51 (1995). Web of Science, Google Scholar
U. Haagerup, Every quasi-trace on an exact C* -algebra is a trace, preprint, 1991 . Google Scholar
X. Jiang and H. Su, Amer. J. Math. 121(2), 359 (1999). Crossref, Web of Science, Google Scholar
H. Lin , An Introduction to the Classification of Amenable C* -Algebras ( World Scientific , Singapore , 2001 ) . Crossref, Google Scholar
H. Lin, Proc. London Math. Soc. (3) 83(1), 199 (2001), DOI: 10.1112/plms/83.1.199. Crossref, Web of Science, Google Scholar
H. Lin, Trans. Amer. Math. Soc. 353, 693 (2001), DOI: 10.1090/S0002-9947-00-02680-5. Crossref, Web of Science, Google Scholar
H. Lin, Localizing the Elliott conjecture at strongly self-absorbing C* -algebras — An appendix, preprint, 2007 , arXiv:0709.1654 . Google Scholar
H. Lin, J. Funct. Anal. 251(2), 601 (2007). Crossref, Web of Science, Google Scholar
H. Lin, Asymptotically unitary equivalence and classification of simple amenable C* -algebras, preprint, 2009 , arXiv:0806.0636 . Google Scholar
H. Lin and Z. Niu, Adv. Math. 219(5), 1729 (2008). Crossref, Web of Science, Google Scholar
H. Lin and N. C. Phillips, J. Reine Angew. Math. (2004). Google Scholar
Q. Lin and N. C. Phillips, Operator Algebras and Operator Theory (Shanghai, 1997), Contemp. Math 228 (Amer. Math. Soc., Providence, RI, 1998) pp. 289–314. Crossref, Google Scholar
Q. Lin and N. C. Phillips, Operator Algebras and Applications, Advanced Studies in Pure Mathematics 38 (Math. Soc. Japan, Tokyo, 2004) pp. 107–133. Crossref, Google Scholar
P. W. Ng and W. Winter, C. R. Acad. Sci. Canada 28, 91 (2006). Google Scholar
N. C. Phillips, Trans. Amer. Math. Soc. 359(10), 4625 (2007), DOI: 10.1090/S0002-9947-07-03849-4. Crossref, Web of Science, Google Scholar
I. F. Putnam, Pacific J. Math. 136(2), 329 (1989). Crossref, Web of Science, Google Scholar
M. Rieffel, Proc. London Math. Soc. 46(3), 301 (1983). Web of Science, Google Scholar
M. Rørdam, J. Funct. Anal. 100, 1 (1991). Crossref, Web of Science, Google Scholar
M. Rørdam, J. Funct. Anal. 107, 255 (1992). Crossref, Web of Science, Google Scholar
M. Rørdam, Classification of Nuclear C*-algebras: Entropy in Operator Algebras, Encyclopaedia Math. Sci 126 (Springer, Berlin, 2002) pp. 1–145. Crossref, Google Scholar
M. Rørdam and W. Winter, J. Reine Angew. Math. 642, 129 (2010). Web of Science, Google Scholar
A. S. Toms, J. Reine Angew. Math. 578, 185 (2005). Web of Science, Google Scholar
A. S. Toms, Comm. Math. Phys. 289(2), 401 (2009), DOI: 10.1007/s00220-008-0665-4. Crossref, Web of Science, Google Scholar
A. S. Toms and W. Winter, Minimal dynamics and K-theoretic rigidity: Elliott's conjecture, preprint, 2009 , arXiv:0903.4133 . Google Scholar
A. S. Toms and W. Winter, Proc. Natl. Acad. Sci. USA 106(40), 16942 (2009), DOI: 10.1073/pnas.0903629106. Crossref, Web of Science, Google Scholar
W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C* -algebras, preprint, with an appendix by H. Lin, 2007 , arXiv:0708.0283v3 . Google Scholar
W. Winter, J. Funct. Anal. 243, 394 (2007). Crossref, Web of Science, Google Scholar
W. Winter, Strongly self-absorbing C* -algebras are 𝒵-stable, preprint, to appear in J. Noncomm. Geom., 2009 , arXiv:0905.0583 . Google Scholar
W. Winter, Invent. Math. 179(2), 229 (2010), DOI: 10.1007/s00222-009-0216-4. Crossref, Web of Science, Google Scholar