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Reciprocity theorems in the theory of representations of groups and algebras

Autorzy

Antoni Wawrzyńczyk

Seria

Rozprawy Matematyczne tom/nr w serii: 126 wydano: 1975

Zawartość

Pełne teksty:
rm126_01.pdf

Warianty tytułu

Abstrakty

EN


Contents
Introduction .............................................................................................................................................................................. 5
1. Regular representations of algebras with approximate unit. A duality theorem..................................................... 9
2. Induced representations of algebras. The main duality............................................................................................. 20
3. Specialization; differentiable induced representations of Yamabe groups............................................................ 24
4. Unitary induced representations of groups................................................................................................................... 31
5. Induced representations of Banach algebras............................................................................................................... 39
6. The Frobenius reciprocity in the theory of square-integrable representations of Hilbert algebras.................... 45
References............................................................................................................................................................................... 59

Słowa kluczowe

Tematy

Wydawca

Instytut Matematyczny Polskiej Akademii Nauk

Miejsce publikacji

Warszawa

Copyright

Seria

Rozprawy Matematyczne tom/nr w serii: 126

Liczba stron

60

Liczba rozdzia³ów

Opis fizyczny

Dissertationes Mathematicae, Tom CXXVI

Daty

wydano
1975

Twórcy

autor
Antoni Wawrzyńczyk
  • Department of Mathematical Methods of Physics, Warsaw University, Warsaw

Bibliografia

  • [1] W. Ambrose, Structure theorems for a special class of Banach algebras, Trans. Amer. Math. Soc. 57 (1945), pp. 364-386.
  • [2] N. Bourbaki, Éléments de mathématique, V, Espaces vectoriels topologiques, Paris 1955.
  • [3] N. Bourbaki, Éléments de mathématique, VI, Intégration, Chap. VII-VIII, Paris 1963.
  • [4] F. Bruhat, Sur les représentations induites des groupes de Lie, Bull. Soc. Math, france 84 (1956), pp. 07-205.
  • [5] F. Bruhat, Distributions sur un groupe localement compact et applications à l'étude des représentations des groupes p-adiques, Bull. Soc. Math. France 86 (1961), pp. 43-75.
  • [6] J. Dixmier, Les algèbres d'operateurs dans l'espace Hübertien (algébres de von Neumann), Paris 1957.
  • [7] J. Dixmier, Les C*-algébres et leurs représentations, Paris 1964.
  • [8] I. M. G. Fell, A new look at Mackey imprimitivity theorem, Proc. of the Internat. Conf. of Harm. Analysis, 1971.
  • [9] I. M. G. Fell, An extension of Mackey’s method to Banach *-algebraic boundles, Memoirs Amer. Math. Soc., n° 90, Providence, Rhode Island, 1969.
  • [10] I. M. Gelfand and I.I. Piateckiǐ-Šapiro, Theory of representations and automorphic functions, Uspiehi Mat. Nauk 14 (1959), pp. 171-194.
  • [11] R. Godement, Théorie des charactères; Algèbres unitaires, Ann. of Math. 59 (1954), pp. 47-62.
  • [12] A. Grothendieck, Produits tensoriel topologiques et espaces nucléaires, Memoirs Amer. Math. Soc., n° 16, Providence, Rhode Island, 1955.
  • [13] G. I. Kac, Generalized functions on a locally compact group and decompositions of unitary representations (Russian), Trudy Mosk. Mat. Obsc. 10 (1961), pp. 3-40.
  • [14] R. A. Kunze, A note on square-integrable representations, J. Funct. Analysis 6 (1970), pp. 454-459.
  • [15] L. H. Loomis, An Introduction to Abstract Harmonic Analysis, Princeton 1963.
  • [16] G. W. Mackey, Induced representations of locally compact groups. I, II, Ann. of Math. 55 (1952), pp. 101-139; 58 (1953), pp. 193-221.
  • [17] K. Maurin, Distributionen auf Yamabe-Gruppen. Harmonische Analyse auf einer Abelschen l. k. Gruppe, Bull. Acad. Polon. Sci., Ser. sci. math., 7 (1959), pp. 471-479.
  • [18] K. Maurin and L. Maurin, A generalization of the duality theorem of Gelfand-Piateckiǐ-Šapiro and Tamagava automorphic forms, J. Fac. Sci. Univ. Tokyo 17 (1970), pp. 331-339.
  • [19] K. Maurin and L. Maurin, General duality theorem for automorphic forms and reciprocity theorem of Frobenius-Bruhat type, Preprint No. 25, Inst. Math. Pol. Acad. Sci., 1971.
  • [20] K. Maurin and L. Maurin, Duality, imprimitivity, reciprocity, Ann. Polon. Math, (in print).
  • [21] K. Maurin and L. Maurin, Automorphic forms and Bruhat-Blattner distributions. Bull. Acad. Polon. Sci. (in print).
  • [22] L. Maurin, Dualitätsatz fur automorphe Funktionen auf einer lokalkompakten unimodularen Gruppe, I, II, Bull. Acad. Polon. Sci., Ser. sci. math., 14 (1906), pp. 493—496; 15 (1967), pp. 465-472.
  • [23] D. Montgomery and L. Zippin, Topological Transformations Groups, New York 1955.
  • [24] C. C. Moore, On the Frobenius reciprocity theorem for locally compact groups, Pacific J. of Math. 12 (1962), pp. 359-365.
  • [25] G. I. Olšanskiĭ, On the Frobenius duality theorem (Russian), Funct. Anal, i Priloz. 3 (1969), pp. 49-58.
  • [26] N. S. Poulsen, On $C^∞$-vectors and intertwining bilinear forms for representations of Lie groups, J. of Funct. Anal. 9 (1972), pp. 87-120.
  • [27] M. A. Riefell, Induced Banach representations of Banach algebras and locally compact groups, J. of Funct. Anal. 1 (1967), pp. 443—491.
  • [28] M. A. Riefell, Square-integrable representations of Hilbert algebras, J. of Funct. Anal. 3 (1969), pp. 265-300.
  • [29] M. A. Riefell, Unitary representations induced from compact subgroups, Stndia Math. 42 (1972), pp. 145-175.
  • [30] M. A. Riefell, Induced representations of C*-algebras, preprint.
  • [31] H. H. Schaefer, Topological Vector Spaces, New York 1966.
  • [32] L. Schwartz, Théorie des distributions, vol. 1, 2, Paris 1957.
  • [33] I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math. 57 (1953), pp. 401-457.
  • [34] W. F. Stinespring, Integration theorems for gages and duality for unimodular groups, Trans. Amer. Math. Soc. 90 (1959), pp. 15-56.
  • [35] A. Strasburger and A. Wawrzyńczyk, Reciprocity theorems in the theory of representations of groups, Studia Math. 42 (1972), pp. 1-14.
  • [36] A. Strasburger and A. Wawrzyńczyk, Automorphic forms for conical representations. Bull. Acad. Polon. Sci., Ser. sci. math., 20 (1972), pp. 921-927.
  • [37] J. Szmidt, On Frobenius reciprocity theorem for representations induced in Hilbert module tensor products, Bull. Acad. Polon. Sci., Ser. sci. math., 21 (1973), pp. 35-39.
  • [38] J. Szmidt and A. Wawrzyńczyk, On $C^∞$-vectors and intertwining operators for Banach space representations of Lie groups, Bull. Acad. Polon. Sci., Ser. sci, math., 6 (1974) (in print).
  • [39] G. Warner, Harmonic Analysis on Semi-simple Lie Groups, I, Springer Verlag, 1972.
  • [40] A. Wawrzyńczyk, A duality in the theory of group representations, Studia Math. 36 (1970), pp. 227-257.
  • [41] A. Wawrzyńczyk, On the Frobenius-Mautner reciprocity theorem. Bull. Acad. Polon. Sci, Ser. Sci. math., 20 (1970), pp. 555-559.
  • [42] A. Weil, L'integration dans les groupes topologiques et ses applications, Paris 1940.
  • [43] J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. of Math. 2 (1952), pp. 251-261.
  • [44] M. de Wilde, H. G. Garnir, I. Schmets, Analyse Fonctionelle, Birkhàser Verlag, 1968.

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