Geodesic
Duality and free measures in vector spaces; spectral theory and the actions of non locally compact groups
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 25, Tome 457 (2017), pp. 74-100 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The paper presents a general duality theory for vector measure spaces taking its origin in the author's papers written in the 60-s. The main result establishes the direct correspondence between the geometry of a measure in a vector space and the properties of the space of measurable linear functionals on this space viewed upon as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of free measure as well as its applications. 
@article{ZNSL_2017_457_a5,
     author = {A. M. Vershik},
     title = {Duality and free measures in vector spaces; spectral theory and the actions of non locally compact groups},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {74--100},
     year = {2017},
     volume = {457},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a5/}
}
TY  - JOUR
AU  - A. M. Vershik
TI  - Duality and free measures in vector spaces; spectral theory and the actions of non locally compact groups
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2017
SP  - 74
EP  - 100
VL  - 457
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a5/
LA  - ru
ID  - ZNSL_2017_457_a5
ER  - 
%0 Journal Article
%A A. M. Vershik
%T Duality and free measures in vector spaces; spectral theory and the actions of non locally compact groups
%J Zapiski Nauchnykh Seminarov POMI
%D 2017
%P 74-100
%V 457
%U http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a5/
%G ru
%F ZNSL_2017_457_a5
A. M. Vershik. Duality and free measures in vector spaces; spectral theory and the actions of non locally compact groups. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 25, Tome 457 (2017), pp. 74-100. http://geodesic.mathdoc.fr/item/ZNSL_2017_457_a5/

[1] N. K. Bari, Trigonometricheskie ryady, FML, M., 1961 | MR

[2] A. M. Vershik, “Izmerimaya realizatsiya nepreryvnykh grupp avtomorfizmov unitarnogo koltsa”, Izv. AN SSSR, ser. matem., 29:1 (1965), 127–136 | MR | Zbl

[3] A. M. Vershik, “Dvoistvennost v teorii mery v lineinykh prostranstvakh”, DAN SSSR, 170 (1966), 497–500 | Zbl

[4] A. M. Vershik, “Dvoistvennost v teorii mery v lineinykh prostranstvakh”, Mezhdunar. matem. kongress, Sb., Tezisy nauchn. soobschenii. Sekts. 5, M., 1966, 39

[5] A. M. Vershik, “Aksiomatika teorii mery v lineinykh prostranstvakh”, DAN SSSR, 178:2 (1968), 278–281 | MR | Zbl

[6] A. M. Vershik, “Izmerimye realizatsii grupp avtomorfizmov i integralnye predstavleniya polozhitelnykh operatorov”, Sib. mat. zhurn., 28:1 (1987), 52–60 | MR | Zbl

[7] A. M. Vershik, “Teoriya filtratsii podalgebr, standartnost i nezavisimost”, UMN, 72:2 (2017), 67–146 | DOI | MR | Zbl

[8] A. M. Vershik, “Sluchainye i universalnye metricheskie prostranstva”, Fundamentalnaya matematika segodnya, Cb., K desyatiletiyu NMU, NMU, MTsNMO, M., 2003

[9] A. M. Vershik, V. N. Sudakov, “Veroyatnostnye mery v beskonechnomernykh prostranstvakh”, Zap. nauchn. semin. LOMI, 12, 1969, 7–67 | MR | Zbl

[10] I. M. Gelfand, M. I. Graev, N. Ya. Vilenkin, Integralnaya geometriya i svyazannye s nei voprosy teorii predstavlenii. Obobschennye funktsii, v. 5, Fizmatgiz, M., 1962 | MR

[11] L. N. Dovbysh, V. N. Sudakov, “O matritsakh Grama–de Finetti. Problemy teorii veroyatnostnykh raspredelenii”, Zap. nauchn. semin. LOMI, 119, 1982, 77–86 | MR | Zbl

[12] V. A. Rokhlin, “Ob osnovnykh ponyatiyakh teorii mery”, Matem. sb., 25(67):1 (1949), 107–150 | MR | Zbl

[13] A. V. Skorokhod, Integrirovanie v gilbertovom prostranstve, Nauka, M., 1975

[14] V. N. Sudakov, “Geometricheskie problemy teorii beskonechnomernykh veroyatnostnykh raspredelenii”, Tr. MIAN SSSR, 141, 1976, 3–191 | MR | Zbl

[15] G. E. Shilov, Fan Dyk Tin, Integral, mera, proizvodnaya na lineinykh prostranstvakh, Nauka, 1967 | MR

[16] E. Glasner, B. Tsirelson, B. Weiss, “The automorphism group of the Gaussian measure cannot act pointwise”, Israel Journal of Mathematics, 148 (2005), 305–329 | DOI | MR | Zbl

[17] S. Albeverio, R. Hoegh-Krohn, D. Testard, A. Vershik, “Factorial representations of path groups”, J. Funct. Anal., 51:1 (1983), 115–131 | DOI | MR | Zbl

[18] I. M. Gelfand, M. I. Graev, A. M. Vershik, “Representations of the group of functions taking values in a compact Lie group”, Compos. Math., 42:2 (1980), 217–243 | MR | Zbl

[19] G. W. Mackey, “Point realization of transformation groups”, Ill. J. Math., 6:2 (1962), 1927–1933 | MR

[20] A. Connes, J. Feldman, B. Weiss, “An amenable equivalence reltion is generated by single transformation”, Erg. Theor. Dyn. Syst., 1:4 (1981), 431–450 | DOI | MR | Zbl

[21] S. Thomas, “A descriptive view of unitary group representations”, J. European Math. Soc., 17:7 (2015), 1761–1787 | DOI | MR | Zbl

[22] S. Albeverio, B. K. Driver, M. Gordina, A. Vershik, “Equivalence of the Brownian and energy representations”, Zap. Nauchn. Semin. POMI, 441, 2015, 17–44 | MR

[23] I. M. Gelfand, M. I. Graev, A. M. Vershik, “Models of representations of current grups”, Representations of Lie groups and Lie algebras (Budapest, 1971), Akad. Kiadó, Budapest, 1985, 121–179 | MR