## Classification of crossed-product $C^ *$-algebras associated with characters on free groups

HTML articles powered by AMS MathViewer

- by Hong Sheng Yin
- Trans. Amer. Math. Soc.
**320**(1990), 105-143 - DOI: https://doi.org/10.1090/S0002-9947-1990-0962286-2
- PDF | Request permission

## Abstract:

We study the classification problem of crossed-product ${C^ * }$-algebras of the form $C_r^ * (G){ \times _{{\alpha _\chi }}}{\mathbf {Z}}$, where $G$ is a discrete group, $\chi$ is a one-dimensional character of $G$, and ${\alpha _\chi }$ is the unique $*$-automorphism of $C_r^ * (G)$ such that if $U$ is the left regular representation of $G$, then ${\alpha _{\chi }(U_{g})=\chi (g)U_{g}}$, $g \in G$. When ${G = F_{n}}$, the free group on $n$ generators, we have a complete classification of these crossed products up to $*$-isomorphism which generalizes the classification of irrational and rational rotation ${C^ * }$-algebras. We show that these crossed products are determined by two $K$-theoretic invariants, that these two invariants correspond to two orbit invariants of the characters under the natural $\operatorname {Aut} ({F_n})$-action, and that these two orbit invariants completely classify the characters up to automorphisms of ${F_n}$. The classification of crossed products follows from these results. We also consider the same problem for $G$ some other groups.## References

- Charles A. Akemann and Phillip A. Ostrand,
*Computing norms in group $C^*$-algebras*, Amer. J. Math.**98**(1976), no. 4, 1015–1047. MR**442698**, DOI 10.2307/2374039 - M. F. Atiyah,
*$K$-theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1967. Lecture notes by D. W. Anderson. MR**0224083** - Marc De Brabanter,
*The classification of rational rotation $C^{\ast }$-algebras*, Arch. Math. (Basel)**43**(1984), no. 1, 79–83. MR**758343**, DOI 10.1007/BF01193614 - Marc De Brabanter and Heinrich H. Zettl,
*$C^{\ast }$-algebras associated with rotation groups and characters*, Manuscripta Math.**47**(1984), no. 1-3, 153–174. MR**744317**, DOI 10.1007/BF01174591 - H. S. M. Coxeter and W. O. J. Moser,
*Generators and relations for discrete groups*, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14, Springer-Verlag, New York-Heidelberg, 1972. MR**0349820**, DOI 10.1007/978-3-662-21946-1 - Joachim Cuntz,
*$K$-theoretic amenability for discrete groups*, J. Reine Angew. Math.**344**(1983), 180–195. MR**716254**, DOI 10.1515/crll.1983.344.180 - Joachim Cuntz,
*The internal structure of simple $C^{\ast }$-algebras*, Operator algebras and applications, Part 1 (Kingston, Ont., 1980) Proc. Sympos. Pure Math., vol. 38, Amer. Math. Soc., Providence, R.I., 1982, pp. 85–115. MR**679697** - Joachim Cuntz,
*$K$-theory and $C^{\ast }$-algebras*, Algebraic $K$-theory, number theory, geometry and analysis (Bielefeld, 1982) Lecture Notes in Math., vol. 1046, Springer, Berlin, 1984, pp. 55–79. MR**750677**, DOI 10.1007/BFb0072018 - Shaun Disney, George A. Elliott, Alexander Kumjian, and Iain Raeburn,
*On the classification of noncommutative tori*, C. R. Math. Rep. Acad. Sci. Canada**7**(1985), no. 2, 137–141. MR**781813** - Shaun Disney and Iain Raeburn,
*Homogeneous $C^\ast$-algebras whose spectra are tori*, J. Austral. Math. Soc. Ser. A**38**(1985), no. 1, 9–39. MR**765447**, DOI 10.1017/S1446788700022576 - Shaun Disney, George A. Elliott, Alexander Kumjian, and Iain Raeburn,
*On the classification of noncommutative tori*, C. R. Math. Rep. Acad. Sci. Canada**7**(1985), no. 2, 137–141. MR**781813** - G. A. Elliott,
*On the $K$-theory of the $C^{\ast }$-algebra generated by a projective representation of a torsion-free discrete abelian group*, Operator algebras and group representations, Vol. I (Neptun, 1980) Monogr. Stud. Math., vol. 17, Pitman, Boston, MA, 1984, pp. 157–184. MR**731772**
P. de la Harpe, - Raphael Høegh-Krohn and Tor Skjelbred,
*Classification of $C^{\ast }$-algebras admitting ergodic actions of the two-dimensional torus*, J. Reine Angew. Math.**328**(1981), 1–8. MR**636190**, DOI 10.1515/crll.1981.328.1 - Shigeru Itoh,
*Conditional expectations in $C^{\ast }$-crossed products*, Trans. Amer. Math. Soc.**267**(1981), no. 2, 661–667. MR**626496**, DOI 10.1090/S0002-9947-1981-0626496-0 - Max Karoubi,
*$K$-theory*, Grundlehren der Mathematischen Wissenschaften, Band 226, Springer-Verlag, Berlin-New York, 1978. An introduction. MR**0488029** - E. Christopher Lance,
*$K$-theory for certain group $C^{\ast }$-algebras*, Acta Math.**151**(1983), no. 3-4, 209–230. MR**723010**, DOI 10.1007/BF02393207 - A. Włodzimierz Mostowski,
*On automorphisms of relatively free groups*, Fund. Math.**50**(1961/62), 403–411. MR**137755**, DOI 10.4064/fm-50-4-403-411 - Dorte Olesen,
*Inner$^{\ast }$-automorphisms of simple $C^{\ast }$-algebras*, Comm. Math. Phys.**44**(1975), no. 2, 175–190. MR**388113**, DOI 10.1007/BF01608830 - Judith A. Packer,
*$C^*$-algebras generated by projective representations of the discrete Heisenberg group*, J. Operator Theory**18**(1987), no. 1, 41–66. MR**912812** - William L. Paschke,
*Inner product modules arising from compact automorphism groups of von Neumann algebras*, Trans. Amer. Math. Soc.**224**(1976), 87–102. MR**420294**, DOI 10.1090/S0002-9947-1976-0420294-7 - William L. Paschke and Norberto Salinas,
*$C^{\ast }$-algebras associated with free products of groups*, Pacific J. Math.**82**(1979), no. 1, 211–221. MR**549845**, DOI 10.2140/pjm.1979.82.211 - Gert K. Pedersen,
*$C^{\ast }$-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006** - John Phillips,
*Automorphisms of full $II_1$ factors, with applications to factors of type III*, Duke Math. J.**43**(1976), no. 2, 375–385. MR**402518** - John Phillips,
*Automorphisms of full $\textrm {II}_{1}$ factors. II*, Canad. Math. Bull.**21**(1978), no. 3, 325–328. MR**511580**, DOI 10.4153/CMB-1978-056-0 - Mihai V. Pimsner,
*Ranges of traces on $K_0$ of reduced crossed products by free groups*, Operator algebras and their connections with topology and ergodic theory (Buşteni, 1983) Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 374–408. MR**799581**, DOI 10.1007/BFb0074897 - M. Pimsner and D. Voiculescu,
*Imbedding the irrational rotation $C^{\ast }$-algebra into an AF-algebra*, J. Operator Theory**4**(1980), no. 2, 201–210. MR**595412** - M. Pimsner and D. Voiculescu,
*Exact sequences for $K$-groups and Ext-groups of certain cross-product $C^{\ast }$-algebras*, J. Operator Theory**4**(1980), no. 1, 93–118. MR**587369** - M. Pimsner and D. Voiculescu,
*$K$-groups of reduced crossed products by free groups*, J. Operator Theory**8**(1982), no. 1, 131–156. MR**670181** - Robert T. Powers,
*Simplicity of the $C^{\ast }$-algebra associated with the free group on two generators*, Duke Math. J.**42**(1975), 151–156. MR**374334** - Norbert Riedel,
*Classification of the $C^{\ast }$-algebras associated with minimal rotations*, Pacific J. Math.**101**(1982), no. 1, 153–161. MR**671848**, DOI 10.2140/pjm.1982.101.153 - Marc A. Rieffel,
*$C^{\ast }$-algebras associated with irrational rotations*, Pacific J. Math.**93**(1981), no. 2, 415–429. MR**623572**, DOI 10.2140/pjm.1981.93.415 - Marc A. Rieffel,
*The cancellation theorem for projective modules over irrational rotation $C^{\ast }$-algebras*, Proc. London Math. Soc. (3)**47**(1983), no. 2, 285–302. MR**703981**, DOI 10.1112/plms/s3-47.2.285
—, $K$ - J. L. Taylor,
*Banach algebras and topology*, Algebras in analysis (Proc. Instructional Conf. and NATO Advanced Study Inst., Birmingham, 1973) Academic Press, London, 1975, pp. 118–186. MR**0417789** - Yasuo Watatani,
*Toral automorphisms on irrational rotation algebras*, Math. Japon.**26**(1981), no. 4, 479–484. MR**634924** - Hong Sheng Yin,
*A simple proof of the classification of rational rotation $C^\ast$-algebras*, Proc. Amer. Math. Soc.**98**(1986), no. 3, 469–470. MR**857943**, DOI 10.1090/S0002-9939-1986-0857943-7
—, - George W. Mackey,
*Harmonic analysis as the exploitation of symmetry—a historical survey*, Bull. Amer. Math. Soc. (N.S.)**3**(1980), no. 1, 543–698. MR**571370**, DOI 10.1090/S0273-0979-1980-14783-7 - Pratibha G. Ghatage and William J. Phillips,
*$C^{\ast }$-algebras generated by weighted shifts. II*, Indiana Univ. Math. J.**30**(1981), no. 4, 539–546. MR**620266**, DOI 10.1512/iumj.1981.30.30044 - Berndt A. Brenken,
*Representations and automorphisms of the irrational rotation algebra*, Pacific J. Math.**111**(1984), no. 2, 257–282. MR**734854**, DOI 10.2140/pjm.1984.111.257 - Hong Sheng Yin,
*Classification of $C^\ast$-crossed products associated with characters on free groups*, C. R. Math. Rep. Acad. Sci. Canada**9**(1987), no. 1, 1–4. MR**873399**

*Reduced*${C^ * }$

*-algebras of discrete groups which are simple with unique trace*, Lecture Notes in Math, vol. 1132, Springer-Verlag, New York, pp. 230-253.

*-theory of crossed products of*${C^ * }$

*-algebras by discrete groups*, (preprint).

*Classification of*${C^ * }$

*-crossed products associated with characters on free groups*, thesis, Dalhousie University, 1985.

## Bibliographic Information

- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**320**(1990), 105-143 - MSC: Primary 46L55; Secondary 46L80
- DOI: https://doi.org/10.1090/S0002-9947-1990-0962286-2
- MathSciNet review: 962286