A note on trace scaling actions and fundamental groups of C*-algebras

Nawata, Norio

doi:10.1090/S0002-9939-2014-12346-X

A note on trace scaling actions and fundamental groups of C$^*$-algebras
HTML articles powered by AMS MathViewer

by Norio Nawata PDF

Proc. Amer. Math. Soc. 142 (2014), 3903-3908 Request permission

Abstract:

Using the Effros-Handelman-Shen theorem and Elliott’s classification theorem of AF algebras, we show that there exists a unital simple AF algebra $A$ with unique trace such that $A\otimes \mathbb {K}$ admits no trace scaling action of the fundamental group of $A$.

References

Ola Bratteli and Akitaka Kishimoto, Trace scaling automorphisms of certain stable AF algebras. II, Q. J. Math. 51 (2000), no. 2, 131–154. MR 1765786, DOI 10.1093/qjmath/51.2.131
Lawrence G. Brown, Philip Green, and Marc A. Rieffel, Stable isomorphism and strong Morita equivalence of $C^*$-algebras, Pacific J. Math. 71 (1977), no. 2, 349–363. MR 463928
A. Connes, A factor of type $\textrm {II}_{1}$ with countable fundamental group, J. Operator Theory 4 (1980), no. 1, 151–153. MR 587372
Alain Connes and Masamichi Takesaki, The flow of weights on factors of type $\textrm {III}$, Tohoku Math. J. (2) 29 (1977), no. 4, 473–575. MR 480760, DOI 10.2748/tmj/1178240493
Giulio Della Rocca and Masamichi Takesaki, The role of groupoids in classification theory: a new approach. The UHF algebra case, Groupoids in analysis, geometry, and physics (Boulder, CO, 1999) Contemp. Math., vol. 282, Amer. Math. Soc., Providence, RI, 2001, pp. 47–65. MR 1855242, DOI 10.1090/conm/282/04678
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
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
George A. Elliott, David E. Evans, and Akitaka Kishimoto, Outer conjugacy classes of trace scaling automorphisms of stable UHF algebras, Math. Scand. 83 (1998), no. 1, 74–86. MR 1662080, DOI 10.7146/math.scand.a-13843
David E. Evans and Akitaka Kishimoto, Trace scaling automorphisms of certain stable AF algebras, Hokkaido Math. J. 26 (1997), no. 1, 211–224. MR 1432548, DOI 10.14492/hokmj/1351257815
Michael J. Jacobson Jr. and Hugh C. Williams, Solving the Pell equation, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2009. MR 2466979
Kazunori Kodaka, Full projections, equivalence bimodules and automorphisms of stable algebras of unital $C^*$-algebras, J. Operator Theory 37 (1997), no. 2, 357–369. MR 1452282
Kazunori Kodaka, Picard groups of irrational rotation $C^*$-algebras, J. London Math. Soc. (2) 56 (1997), no. 1, 179–188. MR 1462834, DOI 10.1112/S0024610797005243
Kazunori Kodaka, Projections inducing automorphisms of stable UHF-algebras, Glasg. Math. J. 41 (1999), no. 3, 345–354. MR 1720450, DOI 10.1017/S0017089599000336
F. J. Murray and J. von Neumann, On rings of operators. IV, Ann. of Math. (2) 44 (1943), 716–808. MR 9096, DOI 10.2307/1969107
Norio Nawata, Fundamental group of simple $C^*$-algebras with unique trace III, Canad. J. Math. 64 (2012), no. 3, 573–587. MR 2962317, DOI 10.4153/CJM-2011-052-0
Norio Nawata and Yasuo Watatani, Fundamental group of simple $C^*$-algebras with unique trace, Adv. Math. 225 (2010), no. 1, 307–318. MR 2669354, DOI 10.1016/j.aim.2010.02.019
Norio Nawata and Yasuo Watatani, Fundamental group of simple $C^*$-algebras with unique trace II, J. Funct. Anal. 260 (2011), no. 2, 428–435. MR 2737407, DOI 10.1016/j.jfa.2010.09.014
Sorin Popa, Strong rigidity of $\rm II_1$ factors arising from malleable actions of $w$-rigid groups. I, Invent. Math. 165 (2006), no. 2, 369–408. MR 2231961, DOI 10.1007/s00222-006-0501-4
Sorin Popa and Stefaan Vaes, Actions of $\Bbb F_\infty$ whose $\textrm {II}_1$ factors and orbit equivalence relations have prescribed fundamental group, J. Amer. Math. Soc. 23 (2010), no. 2, 383–403. MR 2601038, DOI 10.1090/S0894-0347-09-00644-4
Sorin Popa and Stefaan Vaes, On the fundamental group of $\textrm {II}_1$ factors and equivalence relations arising from group actions, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 519–541. MR 2732063
Florin Rădulescu, The fundamental group of the von Neumann algebra of a free group with infinitely many generators is $\textbf {R}_+\sbs 0$, J. Amer. Math. Soc. 5 (1992), no. 3, 517–532. MR 1142260, DOI 10.1090/S0894-0347-1992-1142260-1
Florin Rădulescu, A one-parameter group of automorphisms of ${\scr L}(F_\infty )\otimes B(H)$ scaling the trace, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 13, 1027–1032 (English, with French summary). MR 1168529
Masamichi Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973), 249–310. MR 438149, DOI 10.1007/BF02392041
Dan Voiculescu, Circular and semicircular systems and free product factors, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris, 1989) Progr. Math., vol. 92, Birkhäuser Boston, Boston, MA, 1990, pp. 45–60. MR 1103585