A pproxirnation
Properties for Groups
Chapter 12
This chapter contains a number of other approximation properties, as well
as a fundamental notion of Kazhdan which typically prevents nice approx-
imations from existing. Truth be told, we barely scratch the surface of
any of these topics, but we hope our survey will help the reader tackle the
literature.12.1. Kazhdan's property (T)Kazhdan's property (T) is extremely useful for constructing counterexam-
ples and proving certain isomorphism theorems (among other things). Our
goal in this section is quite modest: Discuss the basics and prove the very
first application of this concept to operator algebras (Theorem 12.1.19).Introduction to (relative) property (T).Definition 12.1.1. Let r be a group and (7r, 1-i) be a unitary representation
of r. A vector e E 1i is I'-invariant if 7r(s)e = e for alls Er. A net (en) of
unit vectors is almost I'-invariant if lim JJ7r(s)en - enJJ = 0 for every s EI'.
If E c rands> 0, we say a (nonzero) vector e E 1i is (E,s)-invariant if
sup JJ7r(s)e - e11 < sJJeJJ.
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