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

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).

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.

@article{AIF_1978__28_2_1_0,
     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},
     zbl = {0365.22005},
     mrnumber = {81j:22002},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.687/}
}
TY  - JOUR
AU  - Mathew, J.
AU  - Nadkarni, M. G.
TI  - On systems of imprimitivity on locally compact abelian groups with dense actions
JO  - Annales de l'Institut Fourier
PY  - 1978
DA  - 1978///
SP  - 1
EP  - 23
VL  - 28
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.687/
UR  - https://zbmath.org/?q=an%3A0365.22005
UR  - https://www.ams.org/mathscinet-getitem?mr=81j:22002
UR  - https://doi.org/10.5802/aif.687
DO  - 10.5802/aif.687
LA  - en
ID  - AIF_1978__28_2_1_0
ER  - 
%0 Journal Article
%A Mathew, J.
%A Nadkarni, M. G.
%T On systems of imprimitivity on locally compact abelian groups with dense actions
%J Annales de l'Institut Fourier
%D 1978
%P 1-23
%V 28
%N 2
%I Institut Fourier
%C Grenoble
%U https://doi.org/10.5802/aif.687
%R 10.5802/aif.687
%G en
%F AIF_1978__28_2_1_0
Mathew, J.; Nadkarni, M. G. On systems of imprimitivity on locally compact abelian groups with dense actions. Annales de l'Institut Fourier, Volume 28 (1978) no. 2, pp. 1-23. doi : 10.5802/aif.687. https://aif.centre-mersenne.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

Cited by Sources: