Polyhomomorphisms of locally compact groups

Yu. A. Neretin

Sbornik. Mathematics, Tome 212 (2021) no. 2, pp. 185-210

Résumé

Let $G$ and $H$ be locally compact groups with fixed two-sided invariant Haar measures. A polyhomomorphism $G\rightarrowtail H$ is a closed subgroup $R\subset G\times H$ with fixed Haar measure, whose marginals on $G$ and $H$ are dominated by the Haar measures on $G$ and $H$. A polyhomomorphism can be regarded as a multi-valued map sending points to sets equipped with ‘uniform’ measures. For two polyhomomorphisms $G\rightarrowtail H$ and $H\rightarrowtail K$ there is a well-defined product $G\rightarrowtail K$. The set of polyhomomorphisms $G\rightarrowtail H$ is a metrizable compact space with respect to the Chabauty-Bourbaki topology and the product is separately continuous. A polyhomomorphism $G\rightarrowtail H$ determines a canonical operator $L^2(H)\to L^2(G)$, which is a partial isometry up to a scalar factor. For example, we consider locally compact linear spaces over finite fields and examine the closures of groups of linear operators in semigroups of polyhomomorphisms. Bibliography: 40 titles.

Détail

Keywords: Haar measure, partial isometries, Chabauty-Bourbaki topology.
Mots-clés : polymorphism, multiplicative relation

Bibliographie

[1] E. Hewitt, K. A. Ross, Abstract harmonic analysis, v. I, Grundlehren Math. Wiss., 115, Structure of topological groups. Integration theory. Group representations, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin–Göttingen–Heidelberg, 1963, viii+519 pp. | DOI | MR | MR | Zbl

[2] N. Bourbaki, Éléments de mathématique. Fasc. III. Première partie. Livre III. Topologie générale, Chap. 3: Groupes topologiques. Chap. 4: Nombres réels, Actualités Sci. Indust., 1143, 3ème éd., rev. et augm., Hermann, Paris, 1960, 236 pp. | MR | MR | Zbl | Zbl

[3] N. Bourbaki, Éléments de mathématique. Fasc. XXIX. Livre VI. Intégration, Chap. 7: Mesure de Haar. Chap. 8: Convolution et représentations, Actualités Sci. Indust., 1306, Hermann, Paris, 1963, 222 pp. | MR | MR | Zbl | Zbl

[4] A. Weil, L'intégration dans les groupes topologiques et ses applications, Actualités Sci. Indust., 869, Hermann, Paris, 1940, 158 pp. | MR | Zbl

[5] D. Zhelobenko, Principal structures and methods of representation theory, Transl. Math. Monogr., 228, Amer. Math. Soc., Providence, RI, 2006, xii+430 pp. | DOI | MR | MR | Zbl | Zbl

[6] Yu. A. Neretin, Topologicheskie gruppy i invariantnye mery, arXiv: 1510.03082

[7] A. S. Kechris, Classical descriptive set theory, Grad. Texts in Math., 156, Springer-Verlag, New York, 1995, xviii+402 pp. | DOI | MR | Zbl

[8] V. I. Bogachev, Measure theory, v. 2, Springer-Verlag, Berlin, 2007, xiv+575 pp. | DOI | MR | Zbl

[9] S. Mac Lane, “An algebra of additive relations”, Proc. Nat. Acad. Sci. U.S.A., 47:7 (1961), 1043-1051 | DOI | MR | Zbl

[10] Yu. A. Neretin, Lectures on Gaussian integral operators and classical groups, EMS Ser. Lect. Math., Eur. Math. Soc. (EMS), Zürich, 2011, xii+559 pp. | DOI | MR | Zbl

[11] H. Schubert, Kategorien, v. I, Heidelberger Taschenbücher, 65, Springer-Verlag, Berlin–New York, 1970, ix+160 pp. ; v. II, 66, viii+148 pp. ; engl. transl.: H. Schubert, Categories, Springer-Verlag, Berlin–Heidelberg, 1972, xi+385 pp. | DOI | MR | Zbl | DOI | DOI | MR | Zbl

[12] Yu. A. Neretin, Categories of symmetries and infinite-dimensional groups, London Math. Soc. Monogr. (N.S.), 16, The Clarendon Press, Oxford Univ. Press, New York, 1996, xiv+417 pp. | MR | Zbl

[13] E. Michael, “Topologies on spaces of subsets”, Trans. Amer. Math. Soc., 71 (1951), 152–182 | DOI | MR | Zbl

[14] I. Biringer, “Metrizing the Chabauty topology”, Geom. Dedicata, 195 (2018), 19–22 | DOI | MR | Zbl

[15] Yu. A. Neretin, “Categories of bistochastic measures, and representations of some infinite-dimensional groups”, Russian Acad. Sci. Sb. Math., 75:1 (1993), 197–219 | DOI | MR | Zbl

[16] G. E. Shilov, B. L. Gurevich, Integral, measure and derivative: a unified approach, Dover Books Adv. Math., Dover Publications, Inc., New York, 1977, xiv+233 pp. | MR | MR | Zbl | Zbl

[17] E. Hopf, “The general temporally discrete Markoff process”, J. Rational Mech. Anal., 3 (1954), 13–45 | DOI | MR | Zbl

[18] Zh. Neve, Matematicheskie osnovy teorii veroyatnostei, Mir, M., 1969, 309 pp. ; J. Neveu, Bases mathématiques du calcul des probabilités, Masson et Cie, Éditeurs, Paris, 1964, xiii+203 pp. ; J. Neveu, Mathematical foundations of the calculus of probability, Holden-Day, Inc., San Francisco, CA–London–Amsterdam, 1965, xiii+223 с. | MR | Zbl | MR | Zbl | MR | Zbl

[19] A. M. Vershik, “Many-valued measure-preserving mappings (polymorphisms) and Markovian operators”, J. Soviet Math., 23:3 (1983), 2243–2266 | DOI | MR | Zbl

[20] U. Krengel, Ergodic theorems, De Gruyter Stud. Math., 6, Walter de Gruyter Co., Berlin, 1985, viii+357 pp. | DOI | MR | Zbl

[21] K. Schmidt, A. Vershik, “Algebraic polymorphisms”, Ergodic Theory Dynam. Systems, 28:2 (2008), 633-642 | DOI | MR | Zbl

[22] J. A. Wolf, “Elliptic spaces in Grassmann manifolds”, Illinois J. Math., 7:3 (1963), 447–462 | DOI | MR | Zbl

[23] T. Tsankov, “Unitary representations of oligomorphic groups”, Geom. Funct. Anal., 22:2 (2012), 528–555 | DOI | MR | Zbl

[24] G. I. Ol'shanskiĭ, “On semigroups related to infinite-dimensional groups”, Topics in representation theory, Adv. Soviet Math., 2, Amer. Math. Soc., Providence, RI, 1991, 67–101 | MR | Zbl

[25] Yu. A. Neretin, “The space $L^2$ on semi-infinite Grassmannian over finite field”, Adv. Math., 250 (2014), 320–350 | DOI | MR | Zbl

[26] Yu. A. Neretin, Groups $\mathrm{GL}(\infty)$ over finite fields and multiplications of double cosets, arXiv: 2002.09969

[27] Yu. A. Neretin, On the Weil representation of infinite-dimensional symplectic group over a finite field, arXiv: 1703.07238

[28] G. I. Olshanskiĭ, “Unitary representations of infinite-dimensional pairs $(G,K)$ and the formalism of R. Howe”, Representations of Lie groups and related topics, Adv. Stud. Contemp. Math., 7, Gordon and Breach, New York, 1990, 269–463 | MR | Zbl

[29] E. Nelson, “The free Markoff field”, J. Funct. Anal., 12:2 (1973), 211–227 | DOI | MR | Zbl

[30] J. Math. Sci. (N.Y.), 126:2 (2005), 1077–1094 | DOI | MR | Zbl

[31] Yu. Neretin, “Symmetries of Gaussian measures and operator colligations”, J. Funct. Anal., 263:3 (2012), 782–802 | DOI | MR | Zbl

[32] Yu. A. Neretin, “The Wishart–Pickrell distributions and closures of group actions”, J. Math. Sci. (N.Y.), 224:2 \year 2017, 328–334 | DOI | MR | Zbl

[33] R. E. Howe, C. C. Moore, “Asymptotic properties of unitary representations”, J. Funct. Anal., 32:1 (1979), 72–96 | DOI | MR | Zbl

[34] J. King, “The commutant is the weak closure of the powers, for rank-1 transformations”, Ergodic Theory Dynam. Systems, 6:3 (1986), 363–384 | DOI | MR | Zbl

[35] É. Janvresse, T. de la Rue, V. Ryzhikov, “Around King's rank-one theorems: flows and $\mathbb{Z}^n$-actions”, Dynamical systems and group actions, Contemp. Math., 567, Amer. Math. Soc., Providence, RI, 2012, 143–161 | DOI | MR | Zbl

[36] S. Solecki, “Closed subgroups generated by generic measure automorphisms”, Ergodic Theory Dynam. Systems, 34:3 (2014), 1011–1017 | DOI | MR | Zbl

[37] É. Janvresse, A. A. Prikhod'ko, T. de la Rue, V. V. Ryzhikov, “Weak limits of powers of Chacon's automorphism”, Ergodic Theory Dynam. Systems, 35:1 (2015), 128–141 | DOI | MR | Zbl

[38] A. Yu. Kushnir, V. V. Ryzhikov, “Weak closures of ergodic actions”, Math. Notes, 101:2 (2017), 277–283 | DOI | DOI | MR | Zbl

[39] V. V. Ryzhikov, “Weak closure of infinite actions of rank $1$, joinings, and spectrum”, Math. Notes, 106:6 (2019), 957–965 | DOI | DOI | MR | Zbl

[40] G. W. Mackey, “Borel structure in groups and their duals”, Trans. Amer. Math. Soc., 85 (1957), 134–165 | DOI | MR | Zbl