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Transference for Banach space representations of nilpotent Lie groups. I: Irreducible representations. (English) Zbl 1439.22017

Summary: We establish a general CCR (liminarity) property for uniformly bounded irreducible representations of nilpotent Lie groups on reflexive Banach spaces, extending the well-known property of unitary irreducible representations of these groups on Hilbert spaces. We also prove that this conclusion fails for many representations on non-reflexive Banach spaces. Our approach to these results blends the method of transference from abstract harmonic analysis and a systematic use of spaces of smooth vectors with respect to Lie group representations.

MSC:

22E25 Nilpotent and solvable Lie groups
22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
17B30 Solvable, nilpotent (super)algebras
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[1] Auslander, L.; Kostant, B., Polarization and unitary representations of solvable Lie groups, Invent. Math., 14, 255-354 (1971) · Zbl 0233.22005 · doi:10.1007/BF01389744
[2] Auslander, Louis; Moore, Calvin C., Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc. No., 62, 199 pp. (1966) · Zbl 0204.14202
[3] Belti\c t\u a, Ingrid; Belti\c t\u a, Daniel, Smooth vectors and Weyl-Pedersen calculus for representations of nilpotent Lie groups, Ann. Univ. Buchar. Math. Ser., 1(LIX), 1, 17-46 (2010) · Zbl 1224.22009
[4] Corwin, Lawrence J.; Greenleaf, Frederick P., Representations of nilpotent Lie groups and their applications. Part I, Cambridge Studies in Advanced Mathematics 18, viii+269 pp. (1990), Cambridge University Press, Cambridge · Zbl 0704.22007
[5] Dixmier, Jacques, Sur les repr\'esentations unitaires des groupes de Lie nilpotents. V, Bull. Soc. Math. France, 87, 65-79 (1959) · Zbl 0152.01302
[6] DkJW18 S. Dirksen, M. de Jeu, and M. Wortel, Crossed products of Banach algebras. I. Dissertationes Math. (to appear).
[7] ter Elst, A. F. M.; Robinson, Derek W., Reduced heat kernels on nilpotent Lie groups, Comm. Math. Phys., 173, 3, 475-511 (1995) · Zbl 0838.22002
[8] Fell, J. M. G., A new proof that nilpotent groups are CCR, Proc. Amer. Math. Soc., 13, 93-99 (1962) · Zbl 0105.09602 · doi:10.2307/2033779
[9] Fujiwara, Hidenori; Ludwig, Jean, Harmonic analysis on exponential solvable Lie groups, Springer Monographs in Mathematics, xii+465 pp. (2015), Springer, Tokyo · Zbl 1311.22001 · doi:10.1007/978-4-431-55288-8
[10] Gardella, Eusebio; Thiel, Hannes, Group algebras acting on \(L^p\)-spaces, J. Fourier Anal. Appl., 21, 6, 1310-1343 (2015) · Zbl 1334.22007 · doi:10.1007/s00041-015-9406-1
[11] Hille, Einar; Phillips, Ralph S., Functional analysis and semi-groups, xii+808 pp. (1974), American Mathematical Society, Providence, R. I. · Zbl 0392.46001
[12] Howe, Roger E., On a connection between nilpotent groups and oscillatory integrals associated to singularities, Pacific J. Math., 73, 2, 329-363 (1977) · Zbl 0383.22009
[13] Ludwig, Jean, Topologically irreducible representations of the Schwartz-algebra of a nilpotent Lie group, Arch. Math. (Basel), 54, 3, 284-292 (1990) · Zbl 0664.43002 · doi:10.1007/BF01188525
[14] Ludwig, Jean; Elhacen, Salma Mint; Molitor-Braun, Carine, Characterization of the simple \(L^1(G)\)-modules for exponential Lie groups, Pacific J. Math., 212, 1, 133-156 (2003) · Zbl 1046.22001 · doi:10.2140/pjm.2003.212.133
[15] Ludwig, J.; Molitor-Braun, C., Repr\'esentations irr\'eductibles born\'ees des groupes de Lie exponentiels, Canad. J. Math., 53, 5, 944-978 (2001) · Zbl 0990.43004 · doi:10.4153/CJM-2001-038-0
[16] Neeb, Karl-Hermann, On differentiable vectors for representations of infinite dimensional Lie groups, J. Funct. Anal., 259, 11, 2814-2855 (2010) · Zbl 1204.22016 · doi:10.1016/j.jfa.2010.07.020
[17] Poulsen, Neils Skovhus, On \(C^{\infty} \)-vectors and intertwining bilinear forms for representations of Lie groups, J. Functional Analysis, 9, 87-120 (1972) · Zbl 0237.22013
[18] Puk\'anszky, L., On the unitary representations of exponential groups, J. Functional Analysis, 2, 73-113 (1968) · Zbl 0172.18502
[19] Radjavi, Heydar; Rosenthal, Peter, Simultaneous triangularization, Universitext, xii+318 pp. (2000), Springer-Verlag, New York · Zbl 0981.15007 · doi:10.1007/978-1-4612-1200-3
[20] Tr\`eves, Fran\c{c}ois, Topological vector spaces, distributions and kernels, xvi+624 pp. (1967), Academic Press, New York-London · Zbl 1111.46001
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