## On group $C^{\ast }$-algebras of bounded representation dimension

HTML articles powered by AMS MathViewer

- by Iain Raeburn PDF
- Trans. Amer. Math. Soc.
**272**(1982), 629-644 Request permission

## Abstract:

We consider the structure of group $C^{\ast }$-algebras whose irreducible representations have bounded dimension. We give some general results, including a description of the topology on the spectrum, and then calculate explicitly the $C^{\ast }$-algebras of two specific groups, one of them a nonsymmorphic space group.## References

- Lawrence Baggett,
*A description of the topology on the dual spaces of certain locally compact groups*, Trans. Amer. Math. Soc.**132**(1968), 175–215. MR**409720**, DOI 10.1090/S0002-9947-1968-0409720-2 - Larry Baggett and Adam Kleppner,
*Multiplier representations of abelian groups*, J. Functional Analysis**14**(1973), 299–324. MR**0364537**, DOI 10.1016/0022-1236(73)90075-x - Larry Baggett and Terje Sund,
*The Hausdorff dual problem for connected groups*, J. Functional Analysis**43**(1981), no. 1, 60–68. MR**639797**, DOI 10.1016/0022-1236(81)90037-9
C. J. Bradley and A. P. Cracknell, - Charles W. Curtis and Irving Reiner,
*Representation theory of finite groups and associative algebras*, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR**0144979** - 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** - Maurice J. Dupré,
*The classification and structure of $C^{\ast }$-algebra bundles*, Mem. Amer. Math. Soc.**21**(1979), no. 222, i–ix, 1–77. MR**539224**, DOI 10.1090/memo/0222 - J. M. G. Fell,
*The dual spaces of $C^{\ast }$-algebras*, Trans. Amer. Math. Soc.**94**(1960), 365–403. MR**146681**, DOI 10.1090/S0002-9947-1960-0146681-0 - J. M. G. Fell,
*The structure of algebras of operator fields*, Acta Math.**106**(1961), 233–280. MR**164248**, DOI 10.1007/BF02545788 - J. M. G. Fell,
*Weak containment and induced representations of groups. II*, Trans. Amer. Math. Soc.**110**(1964), 424–447. MR**159898**, DOI 10.1090/S0002-9947-1964-0159898-X - Siegfried Grosser and Martin Moskowitz,
*Harmonic analysis on central topological groups*, Trans. Amer. Math. Soc.**156**(1971), 419–454. MR**276418**, DOI 10.1090/S0002-9947-1971-0276418-3 - A. A. Kirillov,
*Elements of the theory of representations*, Grundlehren der Mathematischen Wissenschaften, Band 220, Springer-Verlag, Berlin-New York, 1976. Translated from the Russian by Edwin Hewitt. MR**0412321**, DOI 10.1007/978-3-642-66243-0 - John R. Liukkonen,
*Dual spaces of groups with precompact conjugacy classes*, Trans. Amer. Math. Soc.**180**(1973), 85–108. MR**318390**, DOI 10.1090/S0002-9947-1973-0318390-5 - George W. Mackey,
*Unitary representations of group extensions. I*, Acta Math.**99**(1958), 265–311. MR**98328**, DOI 10.1007/BF02392428 - Calvin C. Moore,
*Groups with finite dimensional irreducible representations*, Trans. Amer. Math. Soc.**166**(1972), 401–410. MR**302817**, DOI 10.1090/S0002-9947-1972-0302817-8 - B. H. Neumann,
*Groups with finite classes of conjugate elements*, Proc. London Math. Soc. (3)**1**(1951), 178–187. MR**43779**, DOI 10.1112/plms/s3-1.1.178 - I. Schochetman,
*The dual topology of certain group extensions*, Adv. in Math.**35**(1980), no. 2, 113–128. MR**560131**, DOI 10.1016/0001-8708(80)90044-4 - Terje Sund,
*Duality theory for locally compact groups with precompact conjugacy classes. II. The dual space*, Trans. Amer. Math. Soc.**224**(1976), no. 2, 313–321. MR**439982**, DOI 10.1090/S0002-9947-1976-0439982-1 - Elmar Thoma,
*Eine Charakterisierung diskreter Gruppen vom Typ I*, Invent. Math.**6**(1968), 190–196 (German). MR**248288**, DOI 10.1007/BF01404824 - 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** - Robert B. Warfield Jr.,
*Nilpotent groups*, Lecture Notes in Mathematics, Vol. 513, Springer-Verlag, Berlin-New York, 1976. MR**0409661**, DOI 10.1007/BFb0080152

*The mathematical theory of symmetry in solids: Representation theory for point groups and space groups*, Oxford Univ. Press, London, 1972. A. P. Cracknell,

*Tables of the irreducible representations of the 17 two-dimensional space groups and their relevance to quantum mechanical eigenstates for surfaces and thin films*, Thin Solid Films

**21**(1974), 107-127.

## Additional Information

- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**272**(1982), 629-644 - MSC: Primary 22D25; Secondary 46L05
- DOI: https://doi.org/10.1090/S0002-9947-1982-0662057-6
- MathSciNet review: 662057