# Enveloping algebras by Jacques Dixmier

By Jacques Dixmier

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2 Property (T) and Fell’s topology 35 and a + b = 1. We then have ϕ = aψ + bψ . We claim that either b = 0 or ψ = 0. Indeed, assume by contradiction that b = 0 and ψ (e) = 0. Then ψ (e) = 1, since ϕ(e) = 1 and ψ(e) ≤ 1, ψ (e) ≤ 1. 6 that limi ψi = ψ uniformly on compact subsets of G. 8 shows that ψ is a sum of functions of positive type associated to π0 . 1) and this is a contradiction. Therefore, we have limi ai ψi = ϕ and limi ai = 1. Thus, limi ψi = ϕ in the weak* topology; by Raikov’s Theorem, this holds also uniformly on compact subsets of G.

12). 12 again. Since π0 ⊗ π 0 = π1 ⊕ · · · ⊕ πn , it follows that σi is unitarily equivalent to one of the πk ’s. This is a contradiction to the choice of (σi )i∈I . P. 1]. 3 Compact generation and other consequences The ﬁrst spectacular application of Property (T) is the following result, due to Kazhdan. 1 Let G be a locally compact group with Property (T). Then G is compactly generated. In particular, a discrete group with Property (T) is ﬁnitely generated. 3 Compact generation and other consequences 37 Proof Let C be the set of all open and compactly generated subgroups of G.

Indeed, SLn (Q) is not ﬁnitely generated, since every ﬁnite subset {x1 , . . , xm } of SLn (Q) is contained in SLn (Z[1/N ]), where N is a common multiple of the denominators of the matrix coefﬁcients of x1 , . . , xm . 1. 4 Property (T) for SLn (K), n ≥ 3 Let K be a local ﬁeld . 4, for more details). 6). 4. Some general facts We collect the common ingredients used in the proofs of Property (T) for SLn (K) and Sp2n (K). 1 Let G be a topological group, and let (π , H) be a unitary representation of G with 1G ≺ π.