Abstract
Let \(G=K < imes A\) be the semi-direct product group of a compact group K acting on an abelian locally compact group A. We describe the \(C^*\)-algebra \(C^*(G)\) of G in terms of an algebra of operator fields defined over the spectrum of G, generalizing previous results obtained for some special classes of such groups.
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References
Leptin, H., Ludwig, J.: Unitary representation theory of exponential Lie groups, De Gruyter Expositions in Mathematics, 18, (1994)
Lipsman, R.L.: Orbit theory and harmonic analysis on Lie groups with co-compact nilradical. J. Math. Pures et Appl. 59, 337–374 (1980)
Baggett, W.: A description of the topology on the dual spaces of certain locally compact groups. Trans. Am. Math. Soc. 132, 175–215 (1968)
Abdelmoula, F., Elloumi, M., Ludwig, J.: The \(C^*\)-algebra of the motion group \(SO(n) < imes {{\mathbb{R}}}^n\). Bull. Sci. Math. 135, 166–177 (2011)
Halima, Ben, Rahali, M.: A On the dual topology of a class of Cartan motion groups. J. Lie Theory 22, 491–503 (2012)
Rahali, A., Regeiba, H.: The \(C^*\)-algebras of the Cartan motion groups. https://arxiv.org/submit/2551412/view
Fell, J.M.G.: Weak containment and induced representations of groups II. Trans. Am. Math. Soc. 110, 424–447 (1964)
Mackey, G.W.: The Theory of Unitary Group Representations. Chicago University Press, Chicago (1976)
Mackey, G.W.: Unitary group representations in physics, probability and number theory. Benjamin-Cummings, San Francisco (1978)
Mackey, G.W.: Induced representations of locally compact groups. I Ann. Math. 55, 101–139 (1952)
Dixmier, J.: Les \(C^*\)-algèbres et leurs représentations, Gauthier-Villars
Arveson, W.B.: Subalgebras of \(C^*\)-algebras. Acta Math. 123, 141–224 (1969)
Reiter, Hans, Stegeman, Jan D.: Classical Harmonic Analysis and Locally Compact Groups. Mathematical Society Monographs, vol. 22, 2nd edn. Oxford University Press, New York (2000)
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Communicated by Karlheinz Gröchenig.
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Regeiba, H., Ludwig, J. The \(C^*\)-algebra of the semi-direct product \(K < imes A\). Monatsh Math 192, 915–934 (2020). https://doi.org/10.1007/s00605-020-01418-3
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DOI: https://doi.org/10.1007/s00605-020-01418-3