Type-I permanence

Chirvasitu, Alexandru

doi:10.1090/ert/648

Type-I permanence
HTML articles powered by AMS MathViewer

by Alexandru Chirvasitu;

Represent. Theory 27 (2023), 574-607

DOI: https://doi.org/10.1090/ert/648

Published electronically: July 24, 2023

HTML | PDF | Request permission

Abstract:

We prove a number of results on the survival of the type-I property under extensions of locally compact groups: (a) that given a closed normal embedding $\mathbb {N}\trianglelefteq \mathbb {E}$ of locally compact groups and a twisted action $(\alpha ,\tau )$ thereof on a (post)liminal $C^*$-algebra $A$ the twisted crossed product $A\rtimes _{\alpha ,\tau }\mathbb {E}$ is again (post)liminal and (b) a number of converses to the effect that under various conditions a normal, closed, cocompact subgroup $\mathbb {N}\trianglelefteq \mathbb {E}$ is type-I as soon as $\mathbb {E}$ is. This happens for instance if $\mathbb {N}$ is discrete and $\mathbb {E}$ is Lie, or if $\mathbb {N}$ is finitely-generated discrete (with no further restrictions except cocompactness). Examples show that there is not much scope for dropping these conditions.

In the same spirit, call a locally compact group $\mathbb {G}$ type-I-preserving if all semidirect products $\mathbb {N}\rtimes \mathbb {G}$ are type-I as soon as $\mathbb {N}$ is, and linearly type-I-preserving if the same conclusion holds for semidirect products $V\rtimes \mathbb {G}$ arising from finite-dimensional $\mathbb {G}$-representations. We characterize the (linearly) type-I-preserving groups that are (1) discrete-by-compact-Lie, (2) nilpotent, or (3) solvable Lie.

References

Michèle Audin, The topology of torus actions on symplectic manifolds, Progress in Mathematics, vol. 93, Birkhäuser Verlag, Basel, 1991. Translated from the French by the author. MR 1106194, DOI 10.1007/978-3-0348-7221-8
Louis Auslander and Calvin C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. 62 (1966), 199. MR 207910
Bachir Bekka and Siegfried Echterhoff, On unitary representations of algebraic groups over local fields, Represent. Theory 25 (2021), 508–526. MR 4273170, DOI 10.1090/ert/574
B. Blackadar, Operator algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Theory of $C^*$-algebras and von Neumann algebras; Operator Algebras and Non-commutative Geometry, III. MR 2188261, DOI 10.1007/3-540-28517-2
Armand Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122. MR 146301, DOI 10.1016/0040-9383(63)90026-0
Joachim Cuntz, Siegfried Echterhoff, Xin Li, and Guoliang Yu, $K$-theory for group $C^*$-algebras and semigroup $C^*$-algebras, Oberwolfach Seminars, vol. 47, Birkhäuser/Springer, Cham, 2017. MR 3618901, DOI 10.1007/978-3-319-59915-1
Anton Deitmar and Siegfried Echterhoff, Principles of harmonic analysis, 2nd ed., Universitext, Springer, Cham, 2014. MR 3289059, DOI 10.1007/978-3-319-05792-7
J. Dixmier, Sur les représentations unitaires des groupes de Lie algébriques, Ann. Inst. Fourier (Grenoble) 7 (1957), 315–328 (French). MR 99380, DOI 10.5802/aif.73
Jacques Dixmier, $C^*$-algebras, North-Holland Mathematical Library, Vol. 15, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French by Francis Jellett. MR 458185
Gerald B. Folland, A course in abstract harmonic analysis, 2nd ed., Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2016. MR 3444405, DOI 10.1201/b19172
A. M. Gleason, The structure of locally compact groups, Duke Math. J. 18 (1951), 85–104. MR 39730, DOI 10.1215/S0012-7094-51-01808-X
James Glimm, Locally compact transformation groups, Trans. Amer. Math. Soc. 101 (1961), 124–138. MR 136681, DOI 10.1090/S0002-9947-1961-0136681-X
Elliot C. Gootman and Robert R. Kallman, The left regular representation of a $p$-adic algebraic group is type $\textrm {I}$, Studies in algebra and number theory, Adv. Math. Suppl. Stud., vol. 6, Academic Press, New York-London, 1979, pp. 273–284. MR 535768
Johannes Grassberger and Günther Hörmann, A note on representations of the finite Heisenberg group and sums of greatest common divisors, Discrete Math. Theor. Comput. Sci. 4 (2001), no. 2, 91–100. MR 1835586
Philip Green, The local structure of twisted covariance algebras, Acta Math. 140 (1978), no. 3-4, 191–250. MR 493349, DOI 10.1007/BF02392308
Harish-Chandra, Representations of a semisimple Lie group on a Banach space. I, Trans. Amer. Math. Soc. 75 (1953), 185–243. MR 56610, DOI 10.1090/S0002-9947-1953-0056610-2
G. Hochschild, The automorphism group of Lie group, Trans. Amer. Math. Soc. 72 (1952), 209–216. MR 45735, DOI 10.1090/S0002-9947-1952-0045735-2
Karl H. Hofmann and Sidney A. Morris, The Lie theory of connected pro-Lie groups, EMS Tracts in Mathematics, vol. 2, European Mathematical Society (EMS), Zürich, 2007. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. MR 2337107, DOI 10.4171/032
Karl H. Hofmann and Sidney A. Morris, The structure of almost connected pro-Lie groups, J. Lie Theory 21 (2011), no. 2, 347–383. MR 2828721
Karl H. Hofmann and Sidney A. Morris, The structure of compact groups, De Gruyter Studies in Mathematics, vol. 25, De Gruyter, Berlin, 2013. A primer for the student—a handbook for the expert; Third edition, revised and augmented. MR 3114697, DOI 10.1515/9783110296792
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 323842, DOI 10.1007/978-1-4612-6398-2
Kenkichi Iwasawa, On some types of topological groups, Ann. of Math. (2) 50 (1949), 507–558. MR 29911, DOI 10.2307/1969548
Robert R. Kallman, Certain topological groups are type I, Bull. Amer. Math. Soc. 76 (1970), 404–406. MR 255725, DOI 10.1090/S0002-9904-1970-12491-0
Robert R. Kallman, Certain topological groups are type I. II, Advances in Math. 10 (1973), 221–255. MR 338256, DOI 10.1016/0001-8708(73)90109-6
Eberhard Kaniuth and Keith F. Taylor, Induced representations of locally compact groups, Cambridge Tracts in Mathematics, vol. 197, Cambridge University Press, Cambridge, 2013. MR 3012851
Irving Kaplansky, Infinite abelian groups, University of Michigan Press, Ann Arbor, MI, 1954. MR 65561
M. I. Kargapolov and Ju. I. Merzljakov, Fundamentals of the theory of groups, Graduate Texts in Mathematics, vol. 62, Springer-Verlag, New York-Berlin, 1979. Translated from the second Russian edition by Robert G. Burns. MR 551207, DOI 10.1007/978-1-4612-9964-6
Johan Kustermans and Stefaan Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 6, 837–934 (English, with English and French summaries). MR 1832993, DOI 10.1016/S0012-9593(00)01055-7
Magnus B. Landstad, Algebras of spherical functions associated with covariant systems over a compact group, Math. Scand. 47 (1980), no. 1, 137–149. MR 600084, DOI 10.7146/math.scand.a-11880
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
George W. Mackey, Induced representations of locally compact groups. II. The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953), 193–221. MR 56611, DOI 10.2307/1969786
George W. Mackey, The theory of unitary group representations, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, Ill.-London, 1976. Based on notes by James M. G. Fell and David B. Lowdenslager of lectures given at the University of Chicago, Chicago, Ill., 1955. MR 396826
Paul Milnes, Enveloping semigroups, distal flows and groups of Heisenberg type, Houston J. Math. 15 (1989), no. 4, 563–572. MR 1045513
Deane Montgomery and Leo Zippin, Topological transformation groups, Robert E. Krieger Publishing Co., Huntington, NY, 1974. Reprint of the 1955 original. MR 379739
Calvin C. Moore, Extensions and low dimensional cohomology theory of locally compact groups. I, II, Trans. Amer. Math. Soc. 113 (1964), 40–63; ibid. 113 (1964), 64–86. MR 171880, DOI 10.1090/S0002-9947-1964-0171880-5
Sidney A. Morris, Pontryagin duality and the structure of locally compact abelian groups, London Mathematical Society Lecture Note Series, No. 29, Cambridge University Press, Cambridge-New York-Melbourne, 1977. MR 442141, DOI 10.1017/CBO9780511600722
Gert K. Pedersen, $C^*$-algebras and their automorphism groups, Pure and Applied Mathematics (Amsterdam), Academic Press, London, 2018. Second edition of [ MR0548006]; Edited and with a preface by Søren Eilers and Dorte Olesen. MR 3839621
John C. Quigg, Full $C^*$-crossed product duality, J. Austral. Math. Soc. Ser. A 50 (1991), no. 1, 34–52. MR 1094057, DOI 10.1017/S1446788700032535
Iain Raeburn, A duality theorem for crossed products by nonabelian groups, Miniconference on harmonic analysis and operator algebras (Canberra, 1987) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 15, Austral. Nat. Univ., Canberra, 1987, pp. 214–227. MR 935605
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 507234, DOI 10.1007/978-3-642-86426-1
Luis Ribes and Pavel Zalesskii, Profinite groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 40, 2nd ed., Springer-Verlag, Berlin, 2010.
Marc A. Rieffel, Induced representations of $C^{\ast }$-algebras, Advances in Math. 13 (1974), 176–257. MR 353003, DOI 10.1016/0001-8708(74)90068-1
Joseph J. Rotman, An introduction to the theory of groups, 4th ed., Graduate Texts in Mathematics, vol. 148, Springer-Verlag, New York, 1995. MR 1307623, DOI 10.1007/978-1-4612-4176-8
Joseph J. Rotman, An introduction to homological algebra, 2nd ed., Universitext, Springer, New York, 2009. MR 2455920, DOI 10.1007/b98977
Jörg Schulte, Harmonic analysis on finite Heisenberg groups, European J. Combin. 25 (2004), no. 3, 327–338. MR 2036470, DOI 10.1016/j.ejc.2003.10.003
Jean-Pierre Serre, Lie algebras and Lie groups, 2nd ed., Lecture Notes in Mathematics, vol. 1500, Springer-Verlag, Berlin, 1992. 1964 lectures given at Harvard University. MR 1176100, DOI 10.1007/978-3-540-70634-2
Hiroshi Takai, On a duality for crossed products of $C^{\ast }$-algebras, J. Functional Analysis 19 (1975), 25–39. MR 365160, DOI 10.1016/0022-1236(75)90004-x
M. Takesaki, Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, vol. 125, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 6. MR 1943006, DOI 10.1007/978-3-662-10451-4
M. Takesaki, Theory of operator algebras. III, Encyclopaedia of Mathematical Sciences, vol. 127, Springer-Verlag, Berlin, 2003. Operator Algebras and Non-commutative Geometry, 8. MR 1943007, DOI 10.1007/978-3-662-10453-8
Elmar Thoma, Über unitäre Darstellungen abzählbarer, diskreter Gruppen, Math. Ann. 153 (1964), 111–138 (German). MR 160118, DOI 10.1007/BF01361180
Fabio Elio Tonti and Asger Törnquist, A short proof of Thoma’s theorem on type I groups, arXiv:1904.08313, 2019
Stefaan Vaes, A new approach to induction and imprimitivity results, J. Funct. Anal. 229 (2005), no. 2, 317–374. MR 2182592, DOI 10.1016/j.jfa.2004.11.016
V. S. Varadarajan, Geometry of quantum theory, 2nd ed., Springer-Verlag, New York, 1985. MR 805158
G. Willis, Totally disconnected, nilpotent, locally compact groups, Bull. Austral. Math. Soc. 55 (1997), no. 1, 143–146. MR 1428537, DOI 10.1017/S0004972700030604

AMS Website Down

Representation Theory

Type-I permanence
HTML articles powered by AMS MathViewer

Abstract: