A version of van der Waerden's theorem and a proof of Mishchenko's

A. I. Shtern

Izvestiya. Mathematics , Tome 72 (2008) no. 1, pp. 169-205.

Résumé

van der Waerden proved in 1933 that every finite-dimensional locally bounded representation of a semisimple compact Lie group is automatically continuous. This theorem evoked an extensive literature, which related the assertion of the theorem (and its converse) to properties of Bohr compactifications of topological groups and led to the introduction and study of classes of so-called van der Waerden groups and algebras. In the present paper we study properties of (not necessarily continuous) locally relatively compact homomorphisms of topological groups (in particular, connected locally compact groups) from the point of view of this theorem and obtain a classification of homomorphisms of this kind from the point of view of their continuity or discontinuity properties (this classification is especially simple in the case of Lie groups because it turns out that every locally bounded finite-dimensional representation of a connected Lie group is continuous on the commutator subgroup). Our main results are obtained by studying new objects, namely, the discontinuity group and the final discontinuity group of a locally bounded homomorphism, and the new notion of a finally continuous homomorphism from one locally compact group into another. The notion of local relative compactness of a homomorphism is naturally related to the notion of point oscillation (at the identity element of the group) introduced by the author in 2002. According to a conjecture of A. S. Mishchenko, the (reasonably defined) oscillation at a point of any finite-dimensional representation of a ‘good’ topological group can take one of only three values: $0$, $2$ and $\infty$. We shall prove this for all connected locally compact groups.

Bibliographie

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

[2] L. W. Anderson, R. P. Hunter, “On the continuity of certain homomorphisms of compact semigroups”, Duke Math. J., 38:2 (1971), 409–414 | DOI | MR | Zbl

[3] E. Khyuitt, K. Ross, Abstraktnyi garmonicheskii analiz. T. II. Struktura i analiz kompaktnykh grupp. Analiz na lokalno kompaktnykh abelevykh gruppakh, Nauka, M., 1975 ; E. Hewitt, K. A. Ross, Abstract harmonic analysis. V. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Grundlehren Math. Wiss., 152, Springer-Verlag, New York–Berlin, 1970 | MR | MR | Zbl

[4] J. E. Hart, K. Kunen, “Bohr compactifications of discrete structures”, Fund. Math., 160:2 (1999), 101–151 | MR | Zbl

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

[6] J. E. Hart, K. Kunen, “Bohr topologies and compact function spaces”, Topology Appl., 125:2 (2002), 183–198 | DOI | MR | Zbl

[7] G. Lukács, Lifted closure operators, arXiv: math/0502410

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

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

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

[11] K. H. Hofmann, S. A. Morris, The structure of compact groups. A primer for the student—a handbook for the expert, de Gruyter Stud. Math., 25, de Gruyter, Berlin, 1998 | MR | Zbl

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

[13] W. W. Comfort, L. C. Robertson, “Images and quotients of $\operatorname{SO}(3,R)$: remarks on a theorem of Van der Waerden”, Rocky Mountain J. Math., 17:1 (1987), 1–13 | MR | Zbl

[14] K. de Leeuw, I. Glicksberg, “The decomposition of certain group representations”, J. Analyse Math., 15 (1965), 135–192 | DOI | MR | Zbl

[15] R. T. Moore, Measurable, continuous and smooth vectors for semi-groups and group representations, Mem. Amer. Math. Soc., no. 78, Amer. Math. Soc., Providence, RI, 1968 | MR | Zbl

[16] L. G. Brown, “Continuity of actions of groups and semigroups on Banach spaces”, J. London Math. Soc. (2), 62:1 (2000), 107–116 | DOI | MR | Zbl

[17] A. I. Shtern, “Criteria for weak and strong continuity of representations of topological groups in Banach spaces”, Sb. Math., 193:9 (2002), 1381–1396 | DOI | MR | Zbl

[18] A. I. Shtern, “Continuity of Banach representations in terms of point variations”, Russ. J. Math. Phys., 9:2 (2002), 250–252 | MR | Zbl

[19] A. I. Shtern, “Representations of topological groups in locally convex spaces: continuity properties and weak almost periodicity”, Russ. J. Math. Phys., 11:1 (2004), 81–108 | MR | Zbl

[20] A. I. Shtern, “Continuity criteria for finite-dimensional representations of compact connected groups”, Adv. Stud. Contemp. Math. (Kyungshang), 6:2 (2003), 141–156 | MR | Zbl

[21] A. I. Shtern, “Kriterii nepreryvnosti konechnomernykh predstavlenii grupp”, Funktsionalnye prostranstva. Differentsialnye operatory. Problemy matematicheskogo obrazovaniya, Nauka, M., 2003, 122–124

[22] A. I. Shtern, “Continuity conditions for finite-dimensional representations of some compact totally disconnected groups”, Adv. Stud. Contemp. Math. (Kyungshang), 8:1 (2004), 13–22 | MR | Zbl

[23] A. I. Shtern, “Criteria for the continuity of finite-dimensional representations of connected locally compact groups”, Sb. Math., 195:9 (2004), 1377–1391 | DOI | MR | Zbl

[24] A. I. Shtern, “Continuity criterion for finite-dimensional representations of locally compact groups”, Math. Notes, 75:5–6 (2004), 890–892 | DOI | MR | Zbl

[25] A. I. Shtern, “Weak and strong continuity of representations of topologically pseudocomplete groups in locally convex spaces”, Sb. Math., 197:3 (2006), 453–473 | DOI | MR

[26] J.-P. Serre, “Exemples de variétés projectives conjuguées non homéomorphes”, C. R. Acad. Sci. Paris, 258 (1964), 4194–4196 | MR | Zbl

[27] H. A. Keller, “On valued, complete fields and their automorphisms”, Pacific J. Math., 121:2 (1986), 397–406 | MR | Zbl

[28] J. Tits, “Sur le groupe des automorphismes d'un arbre”, Essays on topology and related topics, Mémoires dédiés á Georges de Rham, Springer, New York, 1970, 188–211 | MR | Zbl

[29] J. Tits, “Homomorphismes et automorphismes “abstraits” de groupes algébriques et arithmétiques”, Actes du Congrès International des Mathématiciens (Nice, 1970), vol. 2, Gauthier-Villars, Paris, 1971, 349–355 | MR | Zbl

[30] S. A. Adeleke, “Discontinuous automorphisms of the proper Galilei and Euclidean groups”, Internat. J. Theoret. Phys., 28:4 (1989), 469–479 | DOI | MR | Zbl

[31] M. Kuczma, An introduction to the theory of functional equations and inequalities, Paǹstwowe Wydawnictwo Naukowe, Warsaw, 1985 | MR | Zbl

[32] D. Kazhdan, “On $\varepsilon$-representations”, Israel J. Math., 43:4 (1982), 315–323 | DOI | MR | Zbl

[33] A. I. Shtern, “Kazhdan–Milman problem for semisimple compact Lie groups”, Russian Math. Surveys, 62:1 (2007), 113–174 | DOI | MR

[34] K. Grove, H. Karcher, E. A. Ruh, “Jacobi fields and Finsler metrics on compact Lie groups with an application to differential pinching problems”, Math. Ann., 211:1 (1974), 7–21 | DOI | MR | Zbl

[35] A. I. Shtern, “Roughness and approximation of quasi-representations of amenable groups”, Math. Notes, 65:6 (1999), 760–769 | DOI | MR | Zbl

[36] A. I. Shtern, “Van der Waerden continuity theorem for semisimple Lie groups”, Russ. J. Math. Phys., 13:2 (2006), 210–223 | DOI | MR | Zbl

[37] A. I. Shtern, “Van der Waerden continuity theorem for the Poincaré group and for some other group extensions”, Adv. Theor. Appl. Math., 1:1 (2006), 79–90 | MR

[38] A. I. Shtern, “Van der Waerden's continuity theorem for the commutator subgroups of connected Lie groups and Mishchenko's conjecture”, Adv. Stud. Contemp. Math. (Kyungshang), 13:2 (2006), 143–158 | MR | Zbl

[39] M. A. Naimark, A. I. Štern [Shtern], Theory of group representations, Grundlehren Math. Wiss., 246, Springer-Verlag, New York, 1982 | MR | Zbl

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

[41] Kh. Shefer, Topologicheskie vektornye prostranstva, Mir, M., 1971 ; H. Schaefer, Topological vector spaces, The Macmillan Co., New York; Collier–Macmillan Ltd., London, 1966 | MR | Zbl | MR | Zbl

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

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

[44] D. Montgomery, L. Zippin, Topological transformation groups, Interscience Publishers, New York–London, 1955 | MR | Zbl

[45] E. Breuillard, T. Gelander, “On dense free subgroups of Lie groups”, J. Algebra, 261:2 (2003), 448–467 | DOI | MR | Zbl

[46] R. W. Bagley, T. S. Wu, J. S. Yang, “Pro-Lie groups”, Trans. Amer. Math. Soc., 287:2 (1985), 829–838 | DOI | MR | Zbl

[47] M. Rajagopalan, “Characters of locally compact abelian groups”, Math. Z., 86:4 (1964), 268–272 | DOI | MR | Zbl

[48] H. Rindler, “Unitary representations and compact groups”, Arch. Math. (Basel), 58:5 (1992), 492–499 | MR | Zbl

[49] A. I. Shtern, “Quasirepresentations and pseudorepresentations”, Funct. Anal. Appl., 25:2 (1991), 140–143 | DOI | MR | Zbl

[50] A. I. Shtern, “Quasisymmetry. I”, Russian J. Math. Phys., 2:3 (1994), 353–382 | MR | Zbl

[51] A. I. Shtern, Ustoichivost predstavlenii i psevdokharaktery, Lomonosovskie chteniya, MGU, M., 1983

[52] A. I. Shtern, “Remarks on pseudocharacters and the real continuous bounded cohomology of connected locally compact groups”, Ann. Global Anal. Geom., 20:3 (2001), 199–221 | DOI | MR | Zbl

[53] A. I. Shtern, “Bounded continuous real 2-cocycles on simply connected simple Lie groups and their applications”, Russ. J. Math. Phys., 8:1 (2001), 122–133 | MR | Zbl

[54] A. I. Shtern, “Structure properties and real continuous bounded 2-cohomology of locally compact groups”, Funct. Anal. Appl., 35:4 (2001), 294–304 | DOI | MR | Zbl

[55] A. I. Shtern, “A criterion for the second real continuous bounded cohomology of a locally compact group to be finite-dimensional”, Acta Appl. Math., 68:1–3 (2001), 105–121 | DOI | MR | Zbl

[56] A. I. Shtern, “Automatic continuity of pseudocharacters on Hermitian symmetric semisimple Lie groups and some applications”, Adv. Stud. Contemp. Math. (Kyungshang), 12:1 (2006), 1–8 | MR | Zbl

[57] A. I. Shtern, “Automatic continuity of pseudocharacters on semisimple Lie groups”, Math. Notes, 80:3–4 (2006), 435–441 | DOI | MR | Zbl

[58] A. I. Shtern, “Continuity conditions for finite-dimensional representations of some locally bounded groups”, Russ. J. Math. Phys., 13:4 (2006), 438–457 | DOI | MR | Zbl