## Recursive subhomogeneous algebras

HTML articles powered by AMS MathViewer

- by N. Christopher Phillips PDF
- Trans. Amer. Math. Soc.
**359**(2007), 4595-4623 Request permission

## Abstract:

We introduce and characterize a particularly tractable class of unital type 1 C*-algebras with bounded dimension of irreducible representations. Algebras in this class are called recursive subhomogeneous algebras, and they have an inductive description (through iterated pullbacks) which allows one to carry over from algebras of the form $C (X, M_n)$ many of the constructions relevant in the study of the stable rank and K-theory of simple direct limits of homogeneous C*-algebras. Our characterization implies, in particular, that if $A$ is a separable C*-algebra whose irreducible representations all have dimension at most $N < \infty ,$ and if for each $n$ the space of $n$-dimensional irreducible representations has finite covering dimension, then $A$ is a recursive subhomogeneous algebra. We demonstrate the good properties of this class by proving subprojection and cancellation theorems in it. Consequences for simple direct limits of recursive subhomogeneous algebras, with applications to the transformation group C*-algebras of minimal homeomorphisms, will be given in separate papers.## References

- Charles A. Akemann, Gert K. Pedersen, and Jun Tomiyama,
*Multipliers of $C^*$-algebras*, J. Functional Analysis**13**(1973), 277–301. MR**0470685**, DOI 10.1016/0022-1236(73)90036-0 - Bruce Blackadar,
*$K$-theory for operator algebras*, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR**859867**, DOI 10.1007/978-1-4613-9572-0 - Bruce Blackadar,
*Matricial and ultramatricial topology*, Operator algebras, mathematical physics, and low-dimensional topology (Istanbul, 1991) Res. Notes Math., vol. 5, A K Peters, Wellesley, MA, 1993, pp. 11–38. MR**1259056** - B. Blackadar,
*Operator algebras*, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Theory of $C^*$-algebras and von Neumann algebras; Operator Algebras and Non-commutative Geometry, III. MR**2188261**, DOI 10.1007/3-540-28517-2 - Bruce Blackadar, Marius Dădărlat, and Mikael Rørdam,
*The real rank of inductive limit $C^*$-algebras*, Math. Scand.**69**(1991), no. 2, 211–216 (1992). MR**1156427**, DOI 10.7146/math.scand.a-12379 - Ola Bratteli, George A. Elliott, David E. Evans, and Akitaka Kishimoto,
*Finite group actions on AF algebras obtained by folding the interval*, $K$-Theory**8**(1994), no. 5, 443–464. MR**1310287**, DOI 10.1007/BF00961400 - M. Dadarlat and G. Gong,
*A classification result for approximately homogeneous $C^*$-algebras of real rank zero*, Geom. Funct. Anal.**7**(1997), no. 4, 646–711. MR**1465599**, DOI 10.1007/s000390050023 - Marius Dădărlat, Gabriel Nagy, András Némethi, and Cornel Pasnicu,
*Reduction of topological stable rank in inductive limits of $C^*$-algebras*, Pacific J. Math.**153**(1992), no. 2, 267–276. MR**1151561**, DOI 10.2140/pjm.1992.153.267 - Jacques Dixmier,
*$C^*$-algebras*, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR**0458185** - P. Donovan and M. Karoubi,
*Graded Brauer groups and $K$-theory with local coefficients*, Inst. Hautes Études Sci. Publ. Math.**38**(1970), 5–25. MR**282363**, DOI 10.1007/BF02684650 - Samuel Eilenberg and Norman Steenrod,
*Foundations of algebraic topology*, Princeton University Press, Princeton, N.J., 1952. MR**0050886**, DOI 10.1515/9781400877492 - George A. Elliott,
*On the classification of $C^*$-algebras of real rank zero*, J. Reine Angew. Math.**443**(1993), 179–219. MR**1241132**, DOI 10.1515/crll.1993.443.179 - Ryszard Engelking,
*Teoria wymiaru*, Biblioteka Matematyczna, Tom 51. [Mathematics Library, Vol. 51], Państwowe Wydawnictwo Naukowe, Warsaw, 1977 (Polish). MR**0482696** - J. M. G. Fell,
*The structure of algebras of operator fields*, Acta Math.**106**(1961), 233–280. MR**164248**, DOI 10.1007/BF02545788 - K. R. Goodearl,
*Riesz decomposition in inductive limit $C^*$-algebras*, Rocky Mountain J. Math.**24**(1994), no. 4, 1405–1430. MR**1322235**, DOI 10.1216/rmjm/1181072345 - Dale Husemoller,
*Fibre bundles*, 3rd ed., Graduate Texts in Mathematics, vol. 20, Springer-Verlag, New York, 1994. MR**1249482**, DOI 10.1007/978-1-4757-2261-1 - Xinhui Jiang and Hongbing Su,
*A classification of simple limits of splitting interval algebras*, J. Funct. Anal.**151**(1997), no. 1, 50–76. MR**1487770**, DOI 10.1006/jfan.1997.3120 - Xinhui Jiang and Hongbing Su,
*On a simple unital projectionless $C^*$-algebra*, Amer. J. Math.**121**(1999), no. 2, 359–413. MR**1680321**, DOI 10.1353/ajm.1999.0012 - John L. Kelley,
*General topology*, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR**0070144** - H. Lin and N. C. Phillips,
*Crossed products by minimal homeomorphisms*, in preparation. - Q. Lin,
*Analytic structure of the transformation group $C^*$-algebra associated with minimal dynamical systems*, preprint. - Q. Lin and N. Christopher Phillips,
*Ordered $K$-theory for $C^\ast$-algebras of minimal homeomorphisms*, Operator algebras and operator theory (Shanghai, 1997) Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 289–314. MR**1667666**, DOI 10.1090/conm/228/03293 - Qing Lin and N. Christopher Phillips,
*Direct limit decomposition for $C^*$-algebras of minimal diffeomorphisms*, Operator algebras and applications, Adv. Stud. Pure Math., vol. 38, Math. Soc. Japan, Tokyo, 2004, pp. 107–133. MR**2059804**, DOI 10.2969/aspm/03810107 - Q. Lin and N. C. Phillips,
*The structure of C*-algebras of minimal diffeomorphisms*, in preparation. - Terry A. Loring,
*Lifting solutions to perturbing problems in $C^*$-algebras*, Fields Institute Monographs, vol. 8, American Mathematical Society, Providence, RI, 1997. MR**1420863**, DOI 10.1090/fim/008 - Mircea Martin and Cornel Pasnicu,
*Some comparability results in inductive limit $C^*$-algebras*, J. Operator Theory**30**(1993), no. 1, 137–147. MR**1302612** - J. Milnor,
*Differential Topology*, mimeographed notes, Princeton University, 1958. - James R. Munkres,
*Topology: a first course*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1975. MR**0464128** - J. Mygind,
*Classification of simple inductive limits of interval algebras with dimension drops*, preprint 1998. - Jesper Mygind,
*Classification of certain simple $C^*$-algebras with torsion in $K_1$*, Canad. J. Math.**53**(2001), no. 6, 1223–1308. MR**1863849**, DOI 10.4153/CJM-2001-046-2 - A. R. Pears,
*Dimension theory of general spaces*, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975. MR**0394604** - Gert K. Pedersen,
*Pullback and pushout constructions in $C^*$-algebra theory*, J. Funct. Anal.**167**(1999), no. 2, 243–344. MR**1716199**, DOI 10.1006/jfan.1999.3456 - N. Christopher Phillips,
*Representable $K$-theory for $\sigma$-$C^*$-algebras*, $K$-Theory**3**(1989), no. 5, 441–478. MR**1050490**, DOI 10.1007/BF00534137 - N. Christopher Phillips,
*The $C^*$ projective length of $n$-homogeneous $C^*$-algebras*, J. Operator Theory**31**(1994), no. 2, 253–276. MR**1331776** - N. C. Phillips,
*Cancellation and stable rank for direct limits of recursive subhomogeneous algebras*, Trans. Amer. Math. Soc., this issue. - N. C. Phillips,
*Real rank and property (SP) for direct limits of recursive subhomogeneous algebras*, Trans. Amer. Math. Soc., to appear. - 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,
*Dimension and stable rank in the $K$-theory of $C^{\ast }$-algebras*, Proc. London Math. Soc. (3)**46**(1983), no. 2, 301–333. MR**693043**, DOI 10.1112/plms/s3-46.2.301 - Marc A. Rieffel,
*The homotopy groups of the unitary groups of noncommutative tori*, J. Operator Theory**17**(1987), no. 2, 237–254. MR**887221** - Edwin H. Spanier,
*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112** - Richard G. Swan,
*Vector bundles and projective modules*, Trans. Amer. Math. Soc.**105**(1962), 264–277. MR**143225**, DOI 10.1090/S0002-9947-1962-0143225-6 - Klaus Thomsen,
*Limits of certain subhomogeneous $C^*$-algebras*, Mém. Soc. Math. Fr. (N.S.)**71**(1997), vi+125 pp. (1998) (English, with English and French summaries). MR**1649315**, DOI 10.24033/msmf.385 - Jun Tomiyama and Masamichi Takesaki,
*Applications of fibre bundles to the certain class of $C^{\ast }$-algebras*, Tohoku Math. J. (2)**13**(1961), 498–522. MR**139025**, DOI 10.2748/tmj/1178244253 - N. B. Vasil′ev,
*$C^{\ast }$-algebras with finite-dimensional irreducible representations*, Uspehi Mat. Nauk**21**(1966), no. 1 (127), 135–154 (Russian). MR**0201994**

## Additional Information

**N. Christopher Phillips**- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
- Received by editor(s): January 22, 2001
- Received by editor(s) in revised form: August 2, 2004
- Published electronically: May 11, 2007
- Additional Notes: This research was partially supported by NSF grants DMS 9400904 and DMS 9706850
- © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**359**(2007), 4595-4623 - MSC (2000): Primary 46L05; Secondary 19A13, 19B14, 19K14, 46L80
- DOI: https://doi.org/10.1090/S0002-9947-07-03850-0
- MathSciNet review: 2320643