On systems of imprimitivity on locally compact abelian groups with dense actions
Annales de l'Institut Fourier, Tome 28 (1978) no. 2, pp. 1-23.

Nous considérons quatre paires de groupes (Γ,R), (Γ/Γ 0 ,R/Γ 0 ), (KS,P) et (S,B), où Γ et R sont des groupes abéliens localement compacts à base dénombrable, Γ apparaissant comme un sous-groupe dense de R de sorte que l’inclusion soit continue ; Γ 0 est un sous-groupe de Γ fermé dans R ; S et B sont les duaux de R et Γ respectivement, et K est l’annihilateur de Γ 0 dans B. Dans chaque paire, le premier terme agit sur le second par translation. En partant d’un système d’imprimitivité sur une des quatre paires nous obtenons, dans une façon naturelle, un système d’imprimitivité sur chacune des paires considérées. Nous établissons un diagramme commutatif (voir section 1) reliant les quatre systèmes d’imprimitivité.

Consider the four pairs of groups (Γ,R), (Γ/Γ 0 ,R/Γ 0 ), (KS,P) and (S,B), where Γ, R are locally compact second countable abelian groups, Γ is a dense subgroup of R with inclusion map from Γ to R continuous; Γ 0 ΓR is a closed subgroup of R; S, B are the duals of R and Γ respectively, and K is the annihilator of Γ 0 in B. Let the first co-ordinate of each pair act on the second by translation. We connect, by a commutative diagram, the systems of imprimitivity which arise in a natural fashion on each pair, starting with a system of imprimitivity on one of the pairs (see section 1 for details).

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     author = {Mathew, J. and Nadkarni, M. G.},
     title = {On systems of imprimitivity on locally compact abelian groups with dense actions},
     journal = {Annales de l'Institut Fourier},
     pages = {1--23},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {28},
     number = {2},
     year = {1978},
     doi = {10.5802/aif.687},
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     zbl = {0365.22005},
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Mathew, J.; Nadkarni, M. G. On systems of imprimitivity on locally compact abelian groups with dense actions. Annales de l'Institut Fourier, Tome 28 (1978) no. 2, pp. 1-23. doi : 10.5802/aif.687. http://www.numdam.org/articles/10.5802/aif.687/

[1] S.C. Bagchi, J. Mathew and M.G. Nadkarni, On systems of imprimitivity on locally compact Abelian groups with dense actions, Acta Mathematica (Uppsala), 133 (1974), 287-304. | MR | Zbl

[2] T.W. Gamelin, Uniform Algebras, Prentice Hall N.J. (U.S.A.), (1969). | MR | Zbl

[3] V.S. Varadarajan, Geometry of Quantum Theory, Vol. 2, Van Nostrand Reinhold Co., (1970). | MR | Zbl

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