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ON THE STABILITY OF A FIXED POINT ALGEBRA C*(E)γ OF A GAUGE ACTION ON A GRAPH C*-ALGEBRA
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 Title & Authors
ON THE STABILITY OF A FIXED POINT ALGEBRA C*(E)γ OF A GAUGE ACTION ON A GRAPH C*-ALGEBRA
Jeong, Ja-A.;
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 Abstract
The fixed point algebra of a gauge action on a graph -algebra and its AF subalgebras associated to each vertex v do play an important role for the study of dynamical properties of . In this paper, we consider the stability of (an AF algebra is either stable or equipped with a (nonzero bounded) trace). It is known that is stably isomorphic to a graph -algebra which we observe being stable. We first give an explicit isomorphism from to a full hereditary -subalgebra of and then show that is stable whenever is so. Thus cannot be stable if admits a trace. It is shown that this is the case if the vertex matrix of E has an eigenvector with an eigenvalue > 1. The AF algebras are shown to be nonstable whenever E is irreducible. Several examples are discussed.
 Keywords
graph -algebra;stable -algebra;fixed point algebra;full hereditary -subalgebra;
 Language
English
 Cited by
 References
1.
T. Bates, J. H. Hong, I. Raeburn, and W. Szymanski, The ideal structure of the $C^*$-algebras of infinite graphs, Illinois J. Math. 46 (2002), no. 4, 1159–1176

2.
T. Bates and D. Pask, Flow equivalence of graph algebras, Ergodic Theory Dynam. Systems 24 (2004), no. 2, 367–382 crossref(new window)

3.
T. Bates, D. Pask, I. Raeburn, and W. Szymanski, The $C^*$-algebras of row-finite graphs, New York J. Math. 6 (2000), 307–324

4.
B. Blakadar, Traces on simple AF $C^*$-algebras, J. Funct. Anal. 38 (1980), no. 2, 156-168 crossref(new window)

5.
B. Blakadar, The stable rank of full corners in $C^*$-algebras, Proc. Amer. Math. Soc. 132 (2004), no. 10, 2945–2950 crossref(new window)

6.
B. Blakadar, Operator Algebras, Theory of $C^*$-algebras and von Neumann algebras, Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry, III. Springer-Verlag, Berlin, 2006

7.
L. G. Brown, Stable isomorphism of hereditary subalgebras of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 335–348

8.
N. P. Brown, Topological entropy in exact $C^*$-algebras, Math. Ann. 314 (1999), no. 2, 347–367 crossref(new window)

9.
J. Cuntz and W. Krieger, A class of $C^*$-algebras and topological Markov chains, Invent. Math. 56 (1980), no. 3, 251–268 crossref(new window)

10.
P. A. Fillmore, A User'S Guide to Operator Algebras, Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996

11.
J. V. B. Hjelmborg, Purely infinite and stable $C^*$-algebras of graphs and dynamical systems, Ergodic Theory Dynam. Systems 21 (2001), no. 6, 1789–1808 crossref(new window)

12.
J. V. B. Hjelmborg and M. Rordam, On stability of $C^*$-algebras, J. Funct. Anal. 155 (1998), no. 1, 153–170 crossref(new window)

13.
J. A Jeong and G. H. Park, Dynamical systems in graph $C^*$-algebras, Internat. J. Math. 16 (2005), no. 9, 999–1015

14.
J. A Jeong and G. H. Park, Topological entropy and AF subalgebras of graph $C^*$-algebras, Proc. Amer. Math. Soc. 134 (2006), no. 1, 215–228 crossref(new window)

15.
J. A Jeong and G. H. Park, Topological entropy and graph $C^*-algebras C^*$(E) of irreducible infinite graphs E, submitted, 2007

16.
J. A Jeong and G. H. Park, Saturated actions by finite-dimensional Hopf *-algebras on $C^*$-algebras, Internat. J. Math. 19 (2008), no. 2, 125–144

17.
A. Kumjian and D. Pask, $C^*$-algebras of directed graphs and group actions, Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1503–1519 crossref(new window)

18.
A. Kumjian, D. Pask, and I. Raeburn, Cuntz-Krieger algebras of directed graphs, Pacific J. Math. 184 (1998), no. 1, 161–174

19.
A. Kumjian, D. Pask, I. Raeburn, and J. Renault, Graphs, groupoids, and Cuntz-Krieger algebras, J. Funct. Anal. 144 (1997), no. 2, 505–541 crossref(new window)

20.
D. Pask and S.-J. Rho, Some intrinsic properties of simple graph $C^*$-algebras, Operator algebras and mathematical physics (Constanta, 2001), 325–340, Theta, Bucharest, 2003

21.
G. K. Pedersen, $C^*$-Algebras and Their Automorphism Groups, London Mathematical Society Monographs, 14. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979

22.
I. Raeburn, Graph Algebras, CBMS Regional Conference Series in Mathematics, 103. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2005

23.
M. A. Rieffel, Dimension and stable rank in the K-theory of $C^*$-algebras, Proc. London Math. Soc. (3) 46 (1983), no. 2, 301–333 crossref(new window)

24.
M. Rordam, Stable $C^*$-algebras, Operator algebras and applications, 177–199, Adv. Stud. Pure Math., 38, Math. Soc. Japan, Tokyo, 2004

25.
J. Renault, A groupoid approach to $C^*$-algebras, Lecture Notes in Mathematics, 793. Springer, Berlin, 1980

26.
J. Rosenberg, Appendix to: 'Crossed products of UHF algebras by product type actions' by O. Bratteli Duke Math. J. 46 (1979), no. 1, 25–26 crossref(new window)

27.
I. A. Salama, Topological entropy and recurrence of countable chains, Pacific J. Math. 134 (1988), no. 2, 325–341

28.
D. Voiculescu, Dynamical approximation entropies and topological entropy in operator algebras, Comm. Math. Phys. 170 (1995), no. 2, 249–281 crossref(new window)