Abstract
This is a brief survey of M. G. Krein's papers in the theory of representations and harmonic analysis on topological groups. These papers are known to be classical and form the basis of numerous contemporary researches into these fields.
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L. I. Vainerman, “Hypercomplex systems with compact and discrete bases,”Dokl. Acad. Nauk SSSR,278, No. 1, 16–20 (1984).
L. I. Vainerman, “Duality of algebras with involution and operators of generalized shift,” in:VINITI Series in Mathematical Analysis [in Russian], Vol 24, VINITI, Moscow (1986), pp. 165–205.
L. I. Vainerman, “On the abstract Plancherel formula and inversion formula,”Ukr. Mat. Zh.,41, No. 8, 1041–1047 (1989).
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S. G. Gindikin and F. I. Karpelevich, “Plancherel measure on Riemannian symmetric spaces of nonpositive curvature,”Dokl. Acad. Nauk SSSR,145, 252–255 (1962).
R. Ya. Grabovskaya and S. G. Krein, “On a representation of the algebra of differential operators and on the related differential equations,”Dokl. Acad. Nauk SSSR,212, No. 2, 280–284 (1973).
P. Deligne and J. S. Milne, “Tannaka categories,” in: P. Deligne, J. S. Milne, A. Ogus, and Kuang-yen Shih,Hodge Cycles, Motives, and Shimura Varieties, Springer Lect. Notes Math.,900, 101–228 (1982).
J. Dixmier,Les C *-Algebres et leurs Representations, Gauthier-Villars, Paris (1969).
V. G. Drinfel'd, “Quantum groups,”Zap. Leningr. Otd. Mat. Inst. Acad. Nauk SSSR,155, 18–49 (1986).
A. A. Kalyuzhnyi, “A theorem on the existence of a multiplicative measure,”Ukr. Mat. Zh.,35, No. 3, 369–371 (1983).
G. I. Kats, “A generalization of the group duality principle,”Dokl. Acad. Nauk SSSR,138, No. 2, 275–278 (1961).
G. I. Kats, “Representations of compact ring groups,”Dokl. Acad. Nauk SSSR,145, No. 5, 989–992 (1962).
G. I. Kats, “Compact and discrete ring groups,”Ukr. Mat. Zh.,14, No. 3, 260–270 (1962).
G. I. Kats, “Ring groups and the duality principle. I,”Tr. Mosk. Mat. Obshch.,12, 259–301 (1963); II,Ibid.,13, 84–113 (1965).
G. I. Kats and V. G. Palyutkin, “Finite ring groups,”Tr. Mosk. Mat. Obshch.,15, 224–261 (1966).
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M. G. Krein, “On a ring of functions defined on a topological group,”Dokl. Acad. Nauk SSSR,29, No. 4, 275–280 (1940).
M. G. Krein, “On a special ring of functions,”Dokl. Acad. Nauk SSSR,29, No. 5–6, 355–359 (1940).
M. G. Krein, “On the theory of almost periodic functions on a topological group,”Dokl. Acad. Nauk SSSR,30, No. 1, 5–8 (1941).
M. G. Krein, “Positive functionals on almost periodic functions,”Dokl. Acad. Nauk SSSR,30, No. 1, 9–12 (1941).
M. G. Krein, “On a generalization of the Plancherel theorem to the case of Fourier integrals on a commutative topological group,”Dokl. Acad. Nauk SSSR,30, No. 6, 482–486 (1941).
M. G. Krein, “On a general method of expansion of positive definite kernels in primary products,”Dokl. Acad. Nauk SSSR,53, No. 1, 3–6 (1946).
M. G. Krein, “The duality principle for a bicompact group and a square block algebra,”Dokl. Acad. Nauk SSSR,69, No. 6, 725–728 (1949).
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Yu. M. Berezanskii and A. A. Kalyuzhnyi,Harmonic Analysis in Hypercomplex Systems [in Russian], Naukova Dumka, Kiev (1992).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 198–211, March, 1994.
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Vainerman, L.I. On M. G. Krein's works in the theory of representations and harmonic analysis on topological groups. Ukr Math J 46, 204–218 (1994). https://doi.org/10.1007/BF01062235
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DOI: https://doi.org/10.1007/BF01062235