Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A complete characterization of connected Lie groups with the
  Approximation Property   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.

21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
18
The categories of this paper are: econ
10
The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
20
The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas