The ranges of 𝐾-theoretic invariants for nonsimple graph algebras

Eilers, Søren; Katsura, Takeshi; Tomforde, Mark; West, James

doi:10.1090/tran/6443

The ranges of $K$-theoretic invariants for nonsimple graph algebras
HTML articles powered by AMS MathViewer

by Søren Eilers, Takeshi Katsura, Mark Tomforde and James West PDF

Trans. Amer. Math. Soc. 368 (2016), 3811-3847 Request permission

Abstract:

There are many classes of nonsimple graph $C^*$-algebras that are classified by the six-term exact sequence in $K$-theory. In this paper we consider the range of this invariant and determine which cyclic six-term exact sequences can be obtained by various classes of graph $C^*$-algebras. To accomplish this, we establish a general method that allows us to form a graph with a given six-term exact sequence of $K$-groups by splicing together smaller graphs whose $C^*$-algebras realize portions of the six-term exact sequence. As rather immediate consequences, we obtain the first permanence results for extensions of graph $C^*$-algebras.

We are hopeful that the results and methods presented here will also prove useful in more general cases, such as situations where the $C^*$-algebras under investigation have more than one ideal and where there are currently no relevant classification theories available.

References

P. Ara, M. A. Moreno, and E. Pardo, Nonstable $K$-theory for graph algebras, Algebr. Represent. Theory 10 (2007), no. 2, 157–178. MR 2310414, DOI 10.1007/s10468-006-9044-z
Sara E. Arklint, Rasmus Bentmann, and Takeshi Katsura, The K-theoretical range of Cuntz-Krieger algebras, J. Funct. Anal. 266 (2014), no. 8, 5448–5466. MR 3177344, DOI 10.1016/j.jfa.2014.01.020
Sara E. Arklint, Rasmus Bentmann, and Takeshi Katsura, Reduction of filtered K-theory and a characterization of Cuntz-Krieger algebras, J. K-Theory 14 (2014), no. 3, 570–613. MR 3349327, DOI 10.1017/is014009013jkt281
Sara E. Arklint and Efren Ruiz, Corners of Cuntz-Krieger algebras, Trans. Amer. Math. Soc., to appear, DOI 10.1090/S0002-9947-2015-06283-7.
Teresa Bates, Jeong Hee Hong, Iain Raeburn, and Wojciech Szymański, The ideal structure of the $C^*$-algebras of infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159–1176. MR 1988256
Bruce Blackadar, $K$-theory for operator algebras, 2nd ed., Mathematical Sciences Research Institute Publications, vol. 5, Cambridge University Press, Cambridge, 1998. MR 1656031
Lawrence G. Brown and Gert K. Pedersen, $C^*$-algebras of real rank zero, J. Funct. Anal. 99 (1991), no. 1, 131–149. MR 1120918, DOI 10.1016/0022-1236(91)90056-B
L. G. Brown, Extensions of $AF$-algebras: the projection lifting problem, Operator Algebras and Applications: Symp. Pure Math. 38 (1982), 175–176.
Toke Meier Carlsen, Søren Eilers, and Mark Tomforde, Index maps in the $K$-theory of graph algebras, J. K-Theory 9 (2012), no. 2, 385–406. MR 2922394, DOI 10.1017/is011004017jkt156
Klaus Deicke, Jeong Hee Hong, and Wojciech Szymański, Stable rank of graph algebras. Type I graph algebras and their limits, Indiana Univ. Math. J. 52 (2003), no. 4, 963–979. MR 2001940, DOI 10.1512/iumj.2003.52.2350
D. Drinen, Viewing AF-algebras as graph algebras, Proc. Amer. Math. Soc. 128 (2000), no. 7, 1991–2000. MR 1657723, DOI 10.1090/S0002-9939-99-05286-7
D. Drinen and M. Tomforde, The $C^*$-algebras of arbitrary graphs, Rocky Mountain J. Math. 35 (2005), no. 1, 105–135. MR 2117597, DOI 10.1216/rmjm/1181069770
D. Drinen and M. Tomforde, Computing $K$-theory and $\textrm {Ext}$ for graph $C^*$-algebras, Illinois J. Math. 46 (2002), no. 1, 81–91. MR 1936076, DOI 10.1215/ijm/1258136141
Edward G. Effros, David E. Handelman, and Chao Liang Shen, Dimension groups and their affine representations, Amer. J. Math. 102 (1980), no. 2, 385–407. MR 564479, DOI 10.2307/2374244
Søren Eilers and Gunnar Restorff, On Rørdam’s classification of certain $C^*$-algebras with one non-trivial ideal, Operator Algebras: The Abel Symposium 2004, Abel Symp., vol. 1, Springer, Berlin, 2006, pp. 87–96. MR 2265044, DOI 10.1007/978-3-540-34197-0_{4}
Søren Eilers, Gunnar Restorff, and Efren Ruiz, Classifying $C^*$-algebras with both finite and infinite subquotients, J. Funct. Anal. 265 (2013), no. 3, 449–468. MR 3056712, DOI 10.1016/j.jfa.2013.05.006
Søren Eilers, Gunnar Restorff, and Efren Ruiz, The ordered $K$-theory of a full extension, Canad. J. Math. 66 (2014), no. 3, 596–625. MR 3194162, DOI 10.4153/CJM-2013-015-7
Søren Eilers, Gunnar Restorff, and Efren Ruiz, Strong classification of extensions of classifiable $C^*$-algebras, submitted for publication.
Søren Eilers and Mark Tomforde, On the classification of nonsimple graph $C^*$-algebras, Math. Ann. 346 (2010), no. 2, 393–418. MR 2563693, DOI 10.1007/s00208-009-0403-z
George A. Elliott, On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), no. 1, 29–44. MR 397420, DOI 10.1016/0021-8693(76)90242-8
K. R. Goodearl and D. E. Handelman, Stenosis in dimension groups and AF $C^{\ast }$-algebras, J. Reine Angew. Math. 332 (1982), 1–98. MR 656856, DOI 10.1515/crll.1982.332.1
David Handelman, Extensions for AF $C^{\ast }$ algebras and dimension groups, Trans. Amer. Math. Soc. 271 (1982), no. 2, 537–573. MR 654850, DOI 10.1090/S0002-9947-1982-0654850-0
Danrun Huang, Flow equivalence of reducible shifts of finite type and Cuntz-Krieger algebras, J. Reine Angew. Math. 462 (1995), 185–217. MR 1329907, DOI 10.1515/crll.1995.462.185
Danrun Huang, The classification of two-component Cuntz-Krieger algebras, Proc. Amer. Math. Soc. 124 (1996), no. 2, 505–512. MR 1301504, DOI 10.1090/S0002-9939-96-03079-1
Takeshi Katsura, Aidan Sims, and Mark Tomforde, Realization of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras, J. Funct. Anal. 257 (2009), no. 5, 1589–1620. MR 2541281, DOI 10.1016/j.jfa.2009.05.002
E. Kirchberg, The classification of purely infinite $C^*$-algebras using Kasparov’s theory, to appear in the Fields Institute Communications series.
Eberhard Kirchberg and Mikael Rørdam, Non-simple purely infinite $C^\ast$-algebras, Amer. J. Math. 122 (2000), no. 3, 637–666. MR 1759891, DOI 10.1353/ajm.2000.0021
D. Kucerovsky and P. W. Ng, $S$-regularity and the corona factorization property, Math. Scand. 99 (2006), no. 2, 204–216. MR 2289022, DOI 10.7146/math.scand.a-15009
Hua Xin Lin, The simplicity of the quotient algebra $M(A)/A$ of a simple $C^*$-algebra, Math. Scand. 65 (1989), no. 1, 119–128. MR 1051828, DOI 10.7146/math.scand.a-12270
Saunders Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR 0156879, DOI 10.1007/978-3-642-62029-4
Ralf Meyer and Ryszard Nest, $\textrm {C}^*$-algebras over topological spaces: filtrated K-theory, Canad. J. Math. 64 (2012), no. 2, 368–408. MR 2953205, DOI 10.4153/CJM-2011-061-x
P. W. Ng, The corona factorization property, Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 97–110. MR 2277206, DOI 10.1090/conm/414/07802
N. Christopher Phillips, A classification theorem for nuclear purely infinite simple $C^*$-algebras, Doc. Math. 5 (2000), 49–114. MR 1745197
Iain Raeburn and Wojciech Szymański, Cuntz-Krieger algebras of infinite graphs and matrices, Trans. Amer. Math. Soc. 356 (2004), no. 1, 39–59. MR 2020023, DOI 10.1090/S0002-9947-03-03341-5
Gunnar Restorff, Classification of Cuntz-Krieger algebras up to stable isomorphism, J. Reine Angew. Math. 598 (2006), 185–210. MR 2270572, DOI 10.1515/CRELLE.2006.074
Mikael Rørdam, Classification of extensions of certain $C^*$-algebras by their six term exact sequences in $K$-theory, Math. Ann. 308 (1997), no. 1, 93–117. MR 1446202, DOI 10.1007/s002080050067
M. Rørdam, Classification of nuclear, simple $C^*$-algebras, Classification of nuclear $C^*$-algebras. Entropy in operator algebras, Encyclopaedia Math. Sci., vol. 126, Springer, Berlin, 2002, pp. 1–145. MR 1878882, DOI 10.1007/978-3-662-04825-2_{1}
M. Rørdam, F. Larsen, and N. Laustsen, An introduction to $K$-theory for $C^*$-algebras, London Mathematical Society Student Texts, vol. 49, Cambridge University Press, Cambridge, 2000. MR 1783408
Wojciech Szymański, The range of $K$-invariants for $C^*$-algebras of infinite graphs, Indiana Univ. Math. J. 51 (2002), no. 1, 239–249. MR 1896162, DOI 10.1512/iumj.2002.51.1920
Mark Tomforde, The ordered $K_0$-group of a graph $C^*$-algebra, C. R. Math. Acad. Sci. Soc. R. Can. 25 (2003), no. 1, 19–25 (English, with English and French summaries). MR 1962131
Shuang Zhang, A property of purely infinite simple $C^*$-algebras, Proc. Amer. Math. Soc. 109 (1990), no. 3, 717–720. MR 1010004, DOI 10.1090/S0002-9939-1990-1010004-X