  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.

Group Theory;Functional Analysis;Operator Algebras