The structure of homomorphisms of connected locally compact groups into compact groups

A. I. Shtern

Izvestiya. Mathematics , Tome 75 (2011) no. 6, pp. 1279-1304.

Résumé

We obtain consequences of the theorem concerning the automatic continuity of locally bounded finite-dimensional representations of connected Lie groups on the commutator subgroup of the group and also of an analogue of Lie's theorem for (not necessarily continuous) finite-dimensional representations of soluble Lie groups. In particular, we prove that an almost connected locally compact group admitting a (not necessarily continuous) injective homomorphism into a compact group also admits a continuous injective homomorphism into a compact group, and thus the given group is a finite extension of the direct product of a compact group and a vector group. We solve the related problem of describing the images of (not necessarily continuous) homomorphisms of connected locally compact groups into compact groups. Moreover, we refine the description of the von Neumann kernel of a connected locally compact group and describe the intersection of the kernels of all (not necessarily continuous) finite-dimensional unitary representations of a given connected locally compact group. Some applications are mentioned. We also show that every almost connected locally compact group admitting a (not necessarily continuous) locally bounded injective homomorphism into an amenable almost connected locally compact group is amenable.

Détail

Keywords: locally compact group, almost connected locally compact group, Freudenthal–Weil theorem, $[\mathrm{MAP}]$-group, semisimple locally compact group, locally bounded map.

Bibliographie

[1] E. Cartan, “Sur les représentations linéaires des groupes clos”, Comment. Math. Helv., 2:1 (1930), 269–283 | DOI | MR | Zbl

[2] B. L. van der Waerden, “Stetigkeitssätze für halbeinfache Liesche Gruppen”, Math. Z., 36:1 (1933), 780–786 | DOI | MR | Zbl

[3] J. D. Lawson, “Intrinsic topologies in topological lattices and semilattices”, Pacific J. Math., 44 (1973), 593–602 | MR | Zbl

[4] W. W. Comfort, “Topological groups”, Handbook of set-theoretic topology, North-Holland, Amsterdam–New York, 1984, 1143–1263 | MR | Zbl

[5] A. Pillay, “An application of model theory to real and $p$-adic algebraic groups”, J. Algebra, 126:1 (1989), 139–146 | DOI | MR | Zbl

[6] K. H. Hofmann, S. A. Morris, The structure of compact groups, de Gruyter Stud. Math., 25, de Gruyter, Berlin, 1998 | MR | Zbl

[7] W. W. Comfort, D. Remus, H. Szambien, “Extending ring topologies”, J. Algebra, 232:1 (2000), 21–47 | DOI | MR | Zbl

[8] J. E. Hart, K. Kunen, “Bohr compactifications of non-abelian groups”, Topology Proc., 26:2 (2001–2002), 593–626 | MR | Zbl

[9] A. I. Shtern, “A version of van der Waerden's theorem and a proof of Mishchenko's conjecture for homomorphisms of locally compact groups”, Izv. Math., 72:1 (2008), 169–205 | DOI | MR | Zbl

[10] W. W. Comfort, L. C. Robertson, “Images and quotients of $\operatorname{SO}(3,\mathbf{R})$: remarks on a theorem of Van der Waerden”, Rocky Mountain J. Math., 17:1 (1987), 1–14 | DOI | MR | Zbl

[11] A. Borel, Essays in the history of Lie groups and algebraic groups, Hist. Math., 21, Amer. Math. Soc., Providence, RI, 2001 | MR | Zbl

[12] O. Schreier, B. L. van der Waerden, “Die Automorphismen der projektiven Gruppen”, Abh. Math. Semin. Univ. Hambg., 6:1 (1928), 303–322 | DOI | Zbl

[13] H. Freudenthal, “Die Topologie der Lieschen Gruppen als algebraisches Phänomen. I”, Ann. of Math. (2), 42 (1941), 1051–1074 | DOI | MR

[14] W. T. van Est, “Dense imbeddings of Lie groups”, Indagationes Math., 13 (1951), 321–328 | MR | Zbl

[15] W. T. van Est, “Dense imbeddings of Lie groups. II (I, II)”, Indagationes Math., 14 (1952), 255–266, 267–274 | MR | Zbl

[16] M. Goto, “Dense imbedding of topological groups”, Proc. Amer. Math. Soc., 4 (1953), 653–655 | DOI | MR | Zbl

[17] M. Goto, “Dense imbeddings of locally compact connected groups”, Ann. of Math. (2), 61 (1955), 154–169 | DOI | MR | Zbl

[18] A. Borel, J. Tits, “Homomorphismes “abstraits” de groupes algebriques simples”, Ann. of Math. (2), 97 (1973), 499–571 | DOI | MR | Zbl

[19] J. Tits, “Homorphismes “abstraits” de groupes de Lie”, Symposia Mathematica (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), v. XIII, Academic Press, London, 1974, 479–499 | MR | Zbl

[20] D. James, W. Waterhouse, B. Weisfeiler, “Abstract homomorphisms of algebraic groups: problems and bibliography”, Comm. Algebra, 9:1 (1981), 95–114 | DOI | MR | Zbl

[21] Y. Chen, “Homomorphisms from linear groups over division rings to algebraic groups”, Group theory (Beijing, 1984), Lecture Notes in Math., 1185, Springer-Verlag, Berlin–New York, 1984, 231–265 | DOI | MR | Zbl

[22] G. M. Seitz, “Abstract homomorphisms of algebraic groups”, J. London Math. Soc. (2), 56:1 (1997), 104–124 | DOI | MR | Zbl

[23] H. Freudenthal, “Topologische Gruppen mit genügend vielen fastperiodischen Funktionen”, Ann. of Math. (2), 37:1 (1936), 57–77 | DOI | MR | Zbl

[24] A. Weil, L'intégration dans les groupes topologiques et ses applications, Hermann, Paris, 1940 | MR | Zbl

[25] A. I. Shtern, “Applications of automatic continuity results to analogs of the Freudenthal–Weil and Hochschild theorems”, Adv. Stud. Contemp. Math. (Kyungshang), 20:2 (2010), 203–212 | MR | Zbl

[26] A. I. Shtern, “Hochschild kernel for locally bounded finite-dimensional representations of a connected reductive Lie group”, Proc. Jangjeon Math. Soc., 13:2 (2010), 127–132 | MR | Zbl

[27] A. I. Shtern, “Von Neumann kernel of a connected locally compact group, revisited”, Adv. Stud. Contemp. Math. (Kyungshang), 20:3 (2010), 313–318 | MR | Zbl

[28] A. I. Shtern, “Homomorphic images of connected Lie groups in compact groups”, Adv. Stud. Contemp. Math. (Kyungshang), 20:1 (2010), 1–6 | MR | Zbl

[29] A. I. Shtern, “Connected Lie groups having faithful locally bounded (not necessarily continuous) finite-dimensional representations”, Russ. J. Math. Phys., 16:4 (2009), 566–567 | DOI | MR | Zbl

[30] A. I. Shtern, “Freudenthal–Weil theorem for arbitrary embeddings of connected Lie groups in compact groups”, Adv. Stud. Contemp. Math. (Kyungshang), 19:2 (2009), 157–164 | MR | Zbl

[31] W. Banaszczyk, Additive subgroups of topological vector spaces, Lecture Notes in Math., 1466, Springer-Verlag, Berlin, 1991 | DOI | MR | Zbl

[32] C. C. Moore, “Groups with finite dimensional irreducible representations”, Trans. Amer. Math. Soc., 166 (1972), 401–410 | DOI | MR | Zbl

[33] L. C. Robertson, “A note on the structure of Moore groups”, Bull. Amer. Math. Soc., 75 (1969), 594–599 | DOI | MR | Zbl

[34] A. I. Štern, “Locally bicompact groups with finite-dimensional irreducible representations”, Math. USSR-Sb., 19:1 (1973), 85–94 | DOI | MR | Zbl | Zbl

[35] A. I. Shtern, “Topological groups with finite von Neumann group algebras of type I”, Sb. Math., 196:3 (2005), 447–463 | DOI | MR | Zbl

[36] A. I. Shtern, “Compact semitopological semigroups and reflexive representability of topological groups”, Russian J. Math. Phys., 2:1 (1994), 131–132 | MR | Zbl

[37] A. I. Shtern, “Nepreryvnye vlozheniya topologicheskikh grupp v lokalno kompaktnye gruppy”, Sovremennye problemy matematiki i mekhaniki, v. 2, vyp. 1, Izd-vo MGU, M., 2009, 89–98

[38] K. Iwasawa, “On some types of topological groups”, Ann. of Math. (2), 50 (1949), 507–558 | DOI | MR | Zbl

[39] A. L. T. Paterson, Amenability, Math. Surveys Monogr., 29, Amer. Math. Soc., Providence, RI, 1988 | MR | Zbl

[40] K. H. Hofmann, S. A. Morris, The Lie theory of connected pro-Lie groups, EMS Tracts Math., 2, Europ. Math. Soc., Zürich, 2007 | MR | Zbl

[41] S. Ahdout, C. Hurwitz, G. Itzkowitz, S. Rothman, H. Strassberg, “Maximal protori in compact topological groups”, Papers on general topology and applications (New York, USA, 1992), Ann. New York Acad. Sci., 728, New York Acad. Sci., New York, 1994, 227–236 | MR | Zbl

[42] N. Bourbaki, Éléments de mathématique. Fasc. XXIX. Livre VI: Intégration. Chap. 7: Mesure de Haar. Chap. 8: Convolution et représentations, Hermann, Paris, 1963 | MR | MR | Zbl | Zbl

[43] D. Montgomery, L. Zippin, Topological transformation groups, Interscience Publ., New York, 1955 | MR | Zbl

[44] M. Stroppel, “Lie theory for non-Lie groups”, J. Lie Theory, 4:2 (1994), 257–284 | MR | Zbl

[45] M. M. Day, Normed linear spaces, Ergeb. Math. Grenzgeb., v. 21, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1958 | MR | MR | Zbl | Zbl

[46] L. S. Pontryagin, Topological groups, Gordon and Breach, New York–London–Paris, 1966 | MR | MR | Zbl | Zbl

[47] A. I. Shtern, “Almost periodic functions and representations in locally convex spaces”, Russ. Math. Surv., 60:3 (2005), 489–557 | DOI | MR | Zbl

[48] A. I. Shtern, “Finite-dimensional quasirepresentations of connected Lie groups and Mishchenko's conjecture”, J. Math. Sci., 159:5 (2009), 653–751 | DOI | MR | Zbl

[49] V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice-Hall, Englewood Cliffs, NJ, 1974 | MR | Zbl

[50] J. von Neumann, “Almost periodic functions in a group. I”, Trans. Amer. Math. Soc., 36:3 (1934), 445–492 | DOI | MR | Zbl

[51] M. Kuranishi, “On non-connected maximally almost periodic groups”, Tôhoku Math. J. (2), 2 (1950), 40–46 | DOI | MR | Zbl

[52] S. Murakami, “Remarks on the structure of maximally almost periodic groups”, Osaka Math. J., 2 (1950), 119–129 | MR | Zbl

[53] S. Grosser, M. Moskowitz, “Compactness conditions in topological groups”, J. Reine Angew. Math., 246 (1971), 1–40 | DOI | MR | Zbl

[54] S. Rothman, “The von Neumann kernel and minimally almost periodic groups”, Trans. Amer. Math. Soc., 259:2 (1980), 401–421 | DOI | MR | Zbl

[55] S. Rothman, H. Strassberg, “The von Neumann kernel of a locally compact group”, J. Austral. Math. Soc. Ser. A, 36:2 (1984), 279–286 | DOI | MR | Zbl

[56] A. I. Shtern, “Von Neumann kernels of connected Lie groups, revisited and refined”, Russ. J. Math. Phys., 17:2 (2010), 262–266 | DOI | MR

[57] E. Hewitt, K. A. Ross, Abstract harmonic analysis, v. I, Grundlehren Math. Wiss., 115, Springer-Verlag, Berlin–New York, 1979 | MR | MR | Zbl

[58] F. P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand, London, 1969 | MR | Zbl | Zbl