Abstract
Roughly the first half of this paper is an introductory exposition of some of the main ideas of the theory of induced representations with special emphasis on the influence of the work of M. H. Stone. The second half deals with applications and with certain extensions and refinements of known results demanded by these applications.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ambrose, W.: Spectral resolution of groups of unitary operators. Duke Math. J. 11, 589–595 (1944).
Bargmann, V.: Irreducible unitary representations of the Lorentz group. Ann. of Math. (2) 48, 568–640 (1947).
Bargmann, V.: On a Hilbert space of analytic functions and an associated integral transform. Comm. Pure Appl. Math. 14, 187–214 (1961).
Birkhoff, G. D.: Proof of a recurrence theorem for strongly transitive systems. Proof of the ergodic theorem. Proc. Nat. Acad. Sci. US 17, 650–660 (1931).
Bradley, C. J.: Space groups and selection rules. J. Math. Phys. 7, 1145–1152 (1966).
Cartier, P.: Quantum mechanical commutation relations and theta functions. Algebraic groups and discontinuous subgroups. Proc. Symp. Pure Math. Boulder Colorado 1965, p. 361–383. Providence, R. I.: Amer. Math. Soc. 1966.
Dixmier, J.: Les C* algebres et leurs représentations. Cahiers Scientifiques, Fasc. XXIX. Paris: Gauthier-Villars 1964.
Gelfand, I., and M. Neumark: Unitary representations of the group of linear transformations of the straight line. Dokl. Akad. Nauk SSSR 11, 411–504 (1947) [Russian].
Godement, R.: Sur une generalization d’un théorème de Stone. Compt. Rend. 218, 901–903 (1944).
Halmos, P. R.: Introduction to Hilbert space and the theory of spectral multiplicity. New York: Chelsea Publ. Co. 1951.
Knox, R. S., and A. Gold: Symmetry in the solid state. New York: W. A. Benjamin Inc. 1964.
Koopman, B. O.: Hamiltonian systems and Hilbert space. Proc. Nat. Acad. Sci. US 17, 315–318 (1931).
Langlands, R. P.: The dimension of spaces of automorphic forms. Am. J. Math. 85, 99–125 (1963).
Lomont, J. S.: Decomposition of direct products of representations of the inhomogeneous Lorentz group. J. Math. Phys. 1, 237–243 (1960).
Loomis, L. H.: Note on a theorem of Mackey. Duke Math. J. 19, 641–645 (1952).
Mackey, G. W.: A theorem of Stone and von Neumann. Duke Math. J. 16, 313–326 (1949).
Mackey, G. W.: Imprimitivity for representations of locally compact groups I. Proc. Nat. Acad. Sci. US 35, 537–545 (1949).
Mackey, G. W.: On induced representations of groups. Am. J. Math. 73, 576–592 (1951).
Mackey, G. W.: Induced representations of locally compact groups. I. Ann. of Math. (2) 55, 101–139 (1952).
Mackey, G. W.: The theory of group representations (mimeographed notes by J. M. G. Fell and D. Lowdenslager). Chicago: Chicago University Press 1955.
Mackey, G. W.: Borel structure in groups and their duals. Trans. Am. Math. Soc. 85, 134–165 (1957).
Mackey, G. W.: Unitary representations of group extensions I. Acta Math. 99, 265–311 (1958).
Mackey, G.W.: Infinite dimensional group representations. Bull. Am. Math. Soc. 69, 628–686 (1963).
Mackey, G.W.: Induced representations of groups and quantum mechanics. New York: W. A. Benjamin Inc. 1968.
Mautner, F. I.: Unitary representations of locally compact groups. I. Ann. of Math. (2) 51, 1–25 (1950).
Neumann, J. v.: Die Eindeutigkeit der Schrodingerschen Operatoren. Math. Ann. 104, 570–578 (1931).
Neumann, J. v.: Proof of the quasi ergodic hypothesis. Proc. Nat. Acad. Sci. US 18, 70–82 (1932).
Neumann, J. v.: On rings of operators. Reduction theory. Ann. of Math. (2) 50, 401–485 (1949).
Neumark, M. A.: Positive definite operator functions on a commutative group. Izv. Akad. Nauk SSSR 7, 237–244 (1943). [Russian].
Petersson, H.: Über eine Metrisierung der Automorphen Formen und die Theorie der Poincareschen Reihen. Math. Ann. 117, 453–537 (1940).
Pukanszky, L.: On the Kronecker product of irreducible unitary representations of the inhomogeneous Lorentz group. J. Math. Mech. 10, 475–491 (1961).
Segal, I.E.: Mathematical problems of relativistic physics. Lectures in applied mathematics (Proceedings of the Summer Seminar, Boulder Colorado, 1960), vol.11. Providence, R. I.: Amer. Math. Soc. 1963.
Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20, 47–87 (1956).
Stone, M. H.: Linear transformations in Hubert space. III. Operational methods and group theory. Proc. Nat. Acad. Sci. US 16, 172–175 (1930).
Stone, M. H.: Linear transformations in Hilbert space and their applications to analysis. Am. Math. Soc. Coll. Publ. XV. New York 1932.
Weil, A.: Sur certains groupes d’opérateurs unitaires. Acta Math. 111, 143–211 (1964).
Wightman, A. S.: On the localizability of quantum mechanical systems. Rev. Mod. Phys. 34, 845–872 (1962).
Wigner, E. P.: On unitary representations of the inhomogeneous Lorentz group. Ann. of Math. (2) 40, 149–204 (1939).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1970 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Mackey, G.W. (1970). Induced Representations of Locally Compact Groups and Applications. In: Browder, F.E. (eds) Functional Analysis and Related Fields. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-48272-4_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-48272-4_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-48274-8
Online ISBN: 978-3-642-48272-4
eBook Packages: Springer Book Archive