Property (T) for locally compact groups and C*-algebras
Bachir Bekka, Chi-Keung Ng

TL;DR
This paper explores the connection between Property (T) for locally compact groups and their associated C*-algebras, establishing equivalences and conditions under which these properties coincide or differ.
Contribution
It proves that Property (T) for a group is equivalent to Property (T) for its full C*-algebra and characterizes Property (T) for IN-groups via the reduced C*-algebra, also linking strong Property (T) to nuclearity.
Findings
G has Property (T) iff C*(G) has Property (T)
G has Property (T) iff C*_r(G) has strong Property (T) for IN-groups
C*_r(G) has strong Property (T) for non-amenable, nuclear C*-algebras
Abstract
Let be a locally compact group and let and be the full group -algebra and the reduced group -algebra of . We investigate the relationship between Property for and Property as well as its strong version for and . We show that has Property if (and only if) has Property . In the case where is a locally compact IN-group, we prove that has Property if and only if has strong Property . We also show that has strong Property for every non-amenable locally compact group for which is nuclear. Some of these groups (as for instance ) do not have Property .
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Banach Space Theory · Holomorphic and Operator Theory
