Geodesic
The Problem of Describing Central Measures on the Path Spaces of Graded Graphs
Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 4, pp. 26-46.

Voir la notice de l'article provenant de la source Math-Net.Ru

We suggest a new method for describing invariant measures on Markov compacta and on path spaces of graphs and, thereby, for describing characters of certain groups and traces of $AF$-algebras. The method relies on properties of filtrations associated with a graph and, in particular, on the notion of a standard filtration. The main tool is an intrinsic metric introduced on simplices of measures; this is an iterated Kantorovich metric, and the central result is that the relative compactness in this metric guarantees the possibility of a constructive enumeration of ergodic invariant measures. Applications include a number of classical theorems on invariant measures.
Keywords: invariant and central measures, projective limit of simplices, intrinsic metric, uniform compactness.
Mots-clés : filtrations 
@article{FAA_2014_48_4_a3,
     author = {A. M. Vershik},
     title = {The {Problem} of {Describing} {Central} {Measures} on the {Path} {Spaces} of {Graded} {Graphs}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {26--46},
     publisher = {mathdoc},
     volume = {48},
     number = {4},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2014_48_4_a3/}
}
TY  - JOUR
AU  - A. M. Vershik
TI  - The Problem of Describing Central Measures on the Path Spaces of Graded Graphs
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2014
SP  - 26
EP  - 46
VL  - 48
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2014_48_4_a3/
LA  - ru
ID  - FAA_2014_48_4_a3
ER  - 
%0 Journal Article
%A A. M. Vershik
%T The Problem of Describing Central Measures on the Path Spaces of Graded Graphs
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2014
%P 26-46
%V 48
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2014_48_4_a3/
%G ru
%F FAA_2014_48_4_a3
A. M. Vershik. The Problem of Describing Central Measures on the Path Spaces of Graded Graphs. Funkcionalʹnyj analiz i ego priloženiâ, Tome 48 (2014) no. 4, pp. 26-46. http://geodesic.mathdoc.fr/item/FAA_2014_48_4_a3/

[1] A. M. Vershik, “Intrinsic metric on graded graphs, standardness, and invariant measures”, Zap. nauchn. sem. POMI, 421 (2014), 58–67 | MR | Zbl

[2] A. Vershik, Smooth and non-smooth AF-algebras and a problem on invariant measures, arXiv: 1304.2193

[3] A. M. Vershik, “Ubyvayuschie posledovatelnosti izmerimykh razbienii i ikh prilozheniya”, Dokl. AN SSSR, 193 (1970), 748–751 | Zbl

[4] A. M. Bershik, “Teoriya ubyvayuschikh posledovatelnostei izmerimykh razbienii”, Algebra i analiz, 6:4 (1994), 1–68

[5] E. Thoma, “Die Unzerlegbaren positiv-definiten Klassenfunktionen der abzählbar unendlichen Simmetrischen Gruppe”, Math. Z., 85:1 (1964), 40–61 | DOI | MR | Zbl

[6] A. M. Vershik, C. V. Kerov, “Asimptoticheskaya teoriya kharakterov simmetricheskoi gruppy”, Funkts. analiz i ego pril., 15:4 (1981), 15–27 | MR

[7] A. Yu. Okunkov, “O predstavleniyakh beskonechnoi simmetricheskoi gruppy”, Zap. nauchn. sem. POMI, 240 (1997), 166–228 | Zbl

[8] G. I. Olshanskii, “Unitarnye predstavleniya (G,K)-par, svyazannykh s beskonechnoi simmetricheskoi gruppoi $S(\infty)$”, Algebra i analiz, 1:4 (1989), 178–209 | MR

[9] R. Felps, Lektsii o teoremakh Shoke, Mir, M., 1968

[10] E. T. Poulsen, “A simplex with dense extreme points”, Ann. Inst. Fourier (Grenoble), 11 (1961), 83–87 | DOI | MR | Zbl

[11] J. Lindenstrauss, G. Olsen, Y. Sternfeld, “The Poulsen simplex”, Ann. Inst. Fourier (Grenoble), 28:1 (1978), 91–114 | DOI | MR | Zbl

[12] E. B. Dynkin, “Nachalnoe i finalnoe povedenie traektorii markovskikh protsessov”, UMN, 26:4(160) (1971), 153–172 | MR

[13] W. Lusky, “The Gurarij spaces are unique”, Arch. Math. (Basel), 27:1 (1976), 627–635 | DOI | MR | Zbl

[14] A. M. Vershik, “Opiisanie invariantykh mer dlya deistvii nekotrykh beskonechnomernykh grupp”, Dokl. AN SSSR, 218:4 (1974), 749–752 | Zbl

[15] E. B. Vinberg, “Invariantnye normy v kompaktnykh algebrakh Li”, Funkts. analiz i ego pril., 2:2 (1968), 89–90 | MR | Zbl

[16] D. Ornstein, Ergodic Theory, Randomness, and Dynamical Systems, Yale Mathematical Monographs, 5, Yale Univ. Press, New Haven–London, 1974 | MR | Zbl

[17] J. Melleray, F. V. Petrov, A. M. Vershik, “Linearly rigid metric spaces and the embedding problem”, Fund. Math., 199:2 (2008), 177–194 | DOI | MR | Zbl

[18] A. M. Vershik, Invariantnye mery, graduirovannye grafy, filtratsii, Kurs lektsii v Laboratorii im. P. L. Chebysheva, 2014