Let be a locally compact group with an open subgroup with the Kunze–Stein property, and let be a unitary representation of . We show that the representation of induced from is an -representation if and only if is an -representation. We deduce the following consequence for a large natural class of almost automorphism groups of trees: For every , the group has a unitary -representation that is not an -representation for any . This in particular applies to the Neretin groups.
Soit un groupe localement compact avec un sous-groupe ouvert ayant la propriété de Kunze–Stein, et soit une représentation unitaire de . Nous montrons que la représentation de induite par est une représentation si et seulement si est une représentation . Nous en déduisons la conséquence suivante pour une grande classe naturelle de groupes de presqu’automorphismes d’un arbre : pour tout , le groupe a une représentation unitaire qui n’est pas une représentation pour tout . Ceci s’applique en particulier aux groupes de Neretin.
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Antje Dabeler 1; Emilie Mai Elkiær 2; Maria Gerasimova 1; Tim de Laat 1

@article{CRMATH_2024__362_G3_245_0, author = {Antje Dabeler and Emilie Mai Elki{\ae}r and Maria Gerasimova and Tim de Laat}, title = {Unitary $L^{p+}$-representations of almost automorphism groups}, journal = {Comptes Rendus. Math\'ematique}, pages = {245--249}, publisher = {Acad\'emie des sciences, Paris}, volume = {362}, year = {2024}, doi = {10.5802/crmath.549}, language = {en}, }
TY - JOUR AU - Antje Dabeler AU - Emilie Mai Elkiær AU - Maria Gerasimova AU - Tim de Laat TI - Unitary $L^{p+}$-representations of almost automorphism groups JO - Comptes Rendus. Mathématique PY - 2024 SP - 245 EP - 249 VL - 362 PB - Académie des sciences, Paris DO - 10.5802/crmath.549 LA - en ID - CRMATH_2024__362_G3_245_0 ER -
%0 Journal Article %A Antje Dabeler %A Emilie Mai Elkiær %A Maria Gerasimova %A Tim de Laat %T Unitary $L^{p+}$-representations of almost automorphism groups %J Comptes Rendus. Mathématique %D 2024 %P 245-249 %V 362 %I Académie des sciences, Paris %R 10.5802/crmath.549 %G en %F CRMATH_2024__362_G3_245_0
Antje Dabeler; Emilie Mai Elkiær; Maria Gerasimova; Tim de Laat. Unitary $L^{p+}$-representations of almost automorphism groups. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 245-249. doi : 10.5802/crmath.549. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.549/
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