Square-integrability modulo a subgroup

Cassinelli, G.; De Vito, E.

doi:10.1090/S0002-9947-02-03220-8

Square-integrability modulo a subgroup
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by G. Cassinelli and E. De Vito PDF

Trans. Amer. Math. Soc. 355 (2003), 1443-1465 Request permission

Abstract:

We prove a weak form of the Frobenius reciprocity theorem for locally compact groups. As a consequence, we propose a definition of square-integrable representation modulo a subgroup that clarifies the relations between coherent states, wavelet transforms and covariant localisation observables. A self-contained proof of the imprimitivity theorem for covariant positive operator-valued measures is given.

References

S. Twareque Ali, A general theorem on square-integrability: vector coherent states, J. Math. Phys. 39 (1998), no. 8, 3954–3964. MR 1633187, DOI 10.1063/1.532478
S. T. Ali, J.-P. Antoine, and J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, 2000.
S. T. Ali, H. Führ, and A. E. Krasowska, Plancherel inversion as unified approach to wavelet transform and Wigner function, preprint math-ph/0106014.
Robert J. Blattner, On induced representations, Amer. J. Math. 83 (1961), 79–98. MR 125456, DOI 10.2307/2372722
Armand Borel, Représentations de groupes localement compacts, Lecture Notes in Mathematics, Vol. 276, Springer-Verlag, Berlin-New York, 1972. MR 0414779
Paul Busch, Marian Grabowski, and Pekka J. Lahti, Operational quantum physics, Lecture Notes in Physics. New Series m: Monographs, vol. 31, Springer-Verlag, Berlin, 1995. MR 1356220
U. Cattaneo, On Mackey’s imprimitivity theorem, Comment. Math. Helv. 54 (1979), no. 4, 629–641. MR 552681, DOI 10.1007/BF02566297
D. P. L. Castrigiano and R. W. Henrichs, Systems of covariance and subrepresentations of induced representations, Lett. Math. Phys. 4 (1980), no. 3, 169–175. MR 583080, DOI 10.1007/BF00316670
E. B. Davies, On the repeated measurements of continuous observables in quantum mechanics, J. Functional Analysis 6 (1970), 318–346. MR 0368655, DOI 10.1016/0022-1236(70)90064-9
J. Dieudonné, Éléments d’analyse. Tome II: Chapitres XII à XV, Cahiers Scientifiques, Fasc. XXXI, Gauthier-Villars, Éditeur, Paris, 1968 (French). MR 0235946
J. Dieudonné, Éléments d’analyse. Tome VI. Chapitre XXII, Cahiers Scientifiques, Fasc. XXXIX, Gauthier-Villars Éditeur, Paris, 1975 (French). MR 0621690
M. Duflo and Calvin C. Moore, On the regular representation of a nonunimodular locally compact group, J. Functional Analysis 21 (1976), no. 2, 209–243. MR 0393335, DOI 10.1016/0022-1236(76)90079-3
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397028
H. Führ and M. Mayer,Continuous wavelet transforms from semidirect products: Cyclic representations and Plancherel measure, J. Fourier Anal. Appl. 8 (2002), 375–397.
H. Führ, Admissible vectors for the regular representation, Proc. Amer. Math. Soc. 130 (2002), 2959–2970.
Steven A. Gaal, Linear analysis and representation theory, Die Grundlehren der mathematischen Wissenschaften, Band 198, Springer-Verlag, New York-Heidelberg, 1973. MR 0447465
A. S. Kholevo, A generalization of canonical quantization, Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 1, 181–194, 208 (Russian). MR 835571
A. S. Holevo, Probabilistic and statistical aspects of quantum theory, North-Holland Series in Statistics and Probability, vol. 1, North-Holland Publishing Co., Amsterdam, 1982. Translated from the Russian by the author. MR 681693
M. Tomić, Sur un critère pour la convergence des séries de Fourier dans un point fixe, Acad. Serbe Sci. Arts Glas 249 (1961), 225–232 (Serbo-Croatian, with French summary). MR 152805
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
Calvin C. Moore, On the Frobenius reciprocity theorem for locally compact groups, Pacific J. Math. 12 (1962), 359–365. MR 141737
Henri Moscovici and Andrei Verona, Coherent states and square integrable representations, Ann. Inst. H. Poincaré Sect. A (N.S.) 29 (1978), no. 2, 139–156. MR 513686
H. Neumann, Transformation properties of observables, Helv. Phys. Acta 45 (1972/73), no. 5, 811–819. MR 403443
Bent Ørsted, Induced representations and a new proof of the imprimitivity theorem, J. Functional Analysis 31 (1979), no. 3, 355–359. MR 531137, DOI 10.1016/0022-1236(79)90009-0
A. Perelomov, Generalized coherent states and their applications, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1986. MR 858831, DOI 10.1007/978-3-642-61629-7
N. K. S. Poulsen, Regularity Aspects of Infinite Dimensional Representations of Lie Groups, Ph. D. Thesis, M.I.T. Cambridge, Mass., 1970.
Neils Skovhus Poulsen, On $C^{\infty }$-vectors and intertwining bilinear forms for representations of Lie groups, J. Functional Analysis 9 (1972), 87–120. MR 0310137, DOI 10.1016/0022-1236(72)90016-x
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
D. J. Rowe, G. Rosensteel, and R. Gilmore, Vector coherent state representation theory, J. Math. Phys. 26 (1985), no. 11, 2787–2791. MR 808492, DOI 10.1063/1.526702
Franklin E. Schroeck Jr., Quantum mechanics on phase space, Fundamental Theories of Physics, vol. 74, Kluwer Academic Publishers Group, Dordrecht, 1996. MR 1374789, DOI 10.1007/978-94-017-2830-0
H. Scutaru, Coherent states and induced representations, Lett. Math. Phys. 2 (1977/78), no. 2, 101–107. MR 507248, DOI 10.1007/BF00398574
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Die Grundlehren der mathematischen Wissenschaften, Band 188, Springer-Verlag, New York-Heidelberg, 1972. MR 0498999