1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

17.2. Property (T) and Kazhdan projections 435

In [11] Arveson proved the previous result and simplified the proof of the

Choi-Effros Lifting Theorem (Theorem C.3). This latter result immediately

implies

Theorem 17.1.9. The semigroup Ext (A) is a group for every nuclear C* -

algebra A.

Exercises

Exercise 17.1.1. Prove that every isomorphism Cf?: IK(H) -+ IK(JC) is inner

• i.e., there exists a unitary U: 1t -+ JC such that <f?(T) = UTU* for all
  TE IK(H). (Hint: Cf? must rnap rank-one projections to rank-one projections
  and thus we can define U via the correspondence between orthonormal bases
  and orthogonal rank-one projections.)

Exercise 17.1.2. Let IK(H) C £1 C JIB(H) and IK(JC) C £2 C JIB(JC) be given
and assume there is an isomorphism Cf?: £1-+ E2 such that <f?(IK(H)) = IK(JC).
Let U : 1t -+ JC be the unitary from the previous exercise and show that
<f?(T) = UTU for all T E £1. Use this fact to show that there is a one-
to-one correspondence between Busby invariants and equivalence classes of
essential extensions. (Hint: If {Pn} E IK(H) is an approximate unit, then
{ <f?(pn)} must be an approximate unit for IK(JC) and hence <f?(T) is the strong
limit of <f?(pn)<r?(T) = (UpnU)(UTU*), for all TE £1.)

Exercise 17.1.3. Prove that addition in Ext( A) is well-defined and abelian.

17.2. Property (T) and Kazhdan projections

This section contains an important structure theorem for the universal C -
algebras of discrete groups with property (T) (Definition 6.4.4). Essentially
the result states that every finite-dimensional unitary representation of r
arises from compression by a projection in C(r). In particular, C(r) has
at least one nontrivial projection, corning from the trivial representation.
(Compare with C(lFn), which contains no nontrivial projections, [53, The-
orem VII.6.6]).

We'll need a well-known observation of Schur.

Lemma 17.2.1 (Schur). Let 7r: A -+ JIB(H) and O": A -+ JIB(JC) be two *-
representations which have a nontrivial intertwiner - i.e., assume there ex-
ists a bounded linear operator T: 1t-+ JC such that T7r(a) = O"(a)T for all
a E A. If 7r is irreducible, then 7r is unitarily equivalent to a subrepresenta-
tion of O".^3

3Recall that this means there is a subspace.CC JC which is invariant for o-(A) and such that
7r is unitarily equivalent to the restriction o-(A) le·