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Cocycles and representations of groups of CAR type

Part of: Topological and differentiable algebraic systems

Published online by Cambridge University Press:  09 April 2009

A. L. Carey  and
William Moran
A. L. Carey
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra A.C.T., Australia
William Moran
Affiliation:
Department of Mathematics, University of Adelaide, Adelaide S.A., Australia
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Abstract

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Representations of non-type I groups G which may be expressed as an increasing union of type I normal subgroups are considered. Groups with this structure are natural generalisations of the CAR algebra (viewed as a twisted group C*-algebra) and are also group theoretic analogues of AF algebras. This paper gives a systematic account of their representation theory based on a canonical construction of one-cocycles for the G-action on the dual of a normal subgroup. Some examples are considered showing how to construct inquivalent irreducible representations (non-cohomologous cocycles) and also factor representations by a method which generalises the well-known construction of non-isomorphic factors for the CAR algebra.

MSC classification

Secondary: 22A25: Representations of general topological groups and semigroups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 1 , February 1986 , pp. 20 - 33
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Auslander, L. and Moore, C. C., ‘Unitary representations of solvable Lie groups’, Mem. Amer. Math. Soc. 62 (1966).Google Scholar
[2]Baggett, L. and Kleppner, A., ‘Multiplier representations of abelian groups’, J. Funct. Anal. 14 (1973), 299–324.CrossRefGoogle Scholar
[3]Baggett, L., Mitchell, W. E. and Ramsay, A., ‘Representations of the discrete Heisenberg group and cocycles of an irrational rotation’, preprint.Google Scholar
[4]Brown, E., ‘Bloch electrons in unicform magnetic field’, Phys. Rev. A 133 (1964), 1038–1044.CrossRefGoogle Scholar
[5]Busby, R. C. and Smith, H. A., ‘Representations of twisted group algebras’, Trans. Amer. Math. Soc. 149 (1970), 503–537.CrossRefGoogle Scholar
[6]Carey, A. L. and Morn, W., ‘On the group theoretic approach to the CAR’, Publ. Res. Inst. Math. Sci. Kyoto 19 (1983).CrossRefGoogle Scholar
[7]Carey, A. L. and Moran, W., ‘Non-monomial unitary representations of nilpotent groups I: the abelian case’, Proc. Lodon Math. Soc. 46 (1983), 53–82.Google Scholar
[8]Evans, D. E. and Lewis, J. T., ‘Dilations of irreversible evolutions in algebraic quantum theory, Communication of Dublin Inst. for Advanced Studies, Series A, 24 (1977).Google Scholar
[9]Garding, L. and Wightman, A. S., ‘Representations of the anti-communiation relations’, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 617–621.Google Scholar
[10]Gootman, E. C. and Rosenberg, J., ‘The structure of crossed product C*-algebras: a proof of the generalized Effros-Hahn conjecture’, Invent. Math. 52 (1979), 283–298.CrossRefGoogle Scholar
[11]Golodets, V. J., ‘Factor representations of the anticommuntation relations,’ Trudy Moskov. Mat. Obšč. 22 (1970), 3–62.Google Scholar
[12]Hannabuss, K. C., ‘Representations of nipotent locally compact groups’, J. Funct. Anal. 34 (1979), 146–165.Google Scholar
[13]Kreiger, W., ‘On constructing non-*isomorphic hyperfinite factors of type III’ J. Funct. Anal. 6 (1970), 97–109.CrossRefGoogle Scholar
[14]Mackey, G. W., ‘Unitary representations of group extensions I’, Acta Math. 99 (1958), 256–311.Google Scholar
[15]Olsen, D., Pedersen, G. K. and Takesaki, M., ‘Ergodic actions of compact adelian groups’, J. Operator Theory 3 (1980), 237–269.Google Scholar
[16]Powers, R. T., ‘Representations of uniformly hyperfinite algebras ans their associated von Neumann rings’, Ann. of Math. 86 (1967), 138–171.CrossRefGoogle Scholar
[17]Slawny, J., ‘On factor representations of the C*-algebras of the CCR’, Comm. Math. Phys. 24 (1972), 151–170.CrossRefGoogle Scholar
[18]Stratila, S. and Voiculescu, D., Representations of AF-algebras and the group U(∞) (Lecture Notes in Math. 486, Springer, Berlin, 1975).CrossRefGoogle Scholar
[19]Sutherland, C., ‘On a construction of unitary cocycles and the representation theory of amenable groups’, Ergodic Theory Dynamical Systems 3 (1983), 129–135.Google Scholar