Title: A complete characterization of connected Lie groups with the Approximation Property
Authors: Haagerup, Uffe * #
Knudby, Søren * #
de Laat, Tim * # ×
Issue Date: 2016
Publisher: Gauthier-Villars
Series Title: Annales Scientifiques de l'École Normale Supérieure Paris vol:49 issue:4 pages:927-946
Abstract: We give a complete characterization of connected Lie groups with the Approximation Property for groups (AP). To this end, we introduce a strengthening of property (T), that we call property (T*), which is a natural obstruction to the AP. In order to define property (T*), we first prove that for every locally compact group G, there exists a unique left invariant mean on the space of completely bounded Fourier multipliers of G. A locally compact group G is said to have property (T*) if this mean is a weak* continuous functional. After proving that the groups SL(3,R), Sp(2,R), and the universal
covering of Sp(2,R) have property (T*), we address the question which connected Lie groups have the AP. A technical problem that arises when considering this question from the point of view of the AP is that the semisimple part of the global Levi decomposition of a connected Lie group need not be closed. Because of an important permanence property of property (T*), this problem vanishes. It follows that a connected Lie group has the AP if and only if all simple factors in the semisimple part of its Levi decomposition have real rank 0 or 1. Finally, we are able to establish property (T*) for all connected simple higher rank Lie groups with finite center.
ISSN: 0012-9593
Publication status: published
KU Leuven publication type: IT
Appears in Collections:Analysis Section
* (joint) first author
× corresponding author
# (joint) last author

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