On group 𝐶*-algebras of bounded representation dimension

Raeburn, Iain

doi:10.1090/S0002-9947-1982-0662057-6

On group $C^{\ast }$-algebras of bounded representation dimension
HTML articles powered by AMS MathViewer

by Iain Raeburn

Trans. Amer. Math. Soc. 272 (1982), 629-644

DOI: https://doi.org/10.1090/S0002-9947-1982-0662057-6

PDF | 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

The mathematical theory of symmetry in solids: Representation theory for point groups and space groups

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

21

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