Amenable Traces and
Kirchberg's
Factorization Property
Chapter 6
In this chapter we explore another duality between finite-dimensional ap-
proximation properties and tensor products. This time the context is tra-
cial states, but the end results have the same flavor as those characterizing
nuclear or exact C* -algebras.
The first section contains some classical work of Murray and von Neu-
mann. Section 6.2 contains the main result, Theorem 6.2.7, which connects
tensor products with approximation properties of traces. The third section,
perhaps out of place, discusses motivation and examples related to Theorem
6.2.7. In the final section we discuss Kirchberg's factorization property and
see what this means for groups with Kazhdan's property (T).
6.1. Traces and the right regular representation
A classical theorem of Murray and von Neumann asserts that the commutant
of the left regular representation of a group is just the weak closure of the
right regular representation. Let's mimic their proof in the more general
context of tracial GNS representations.
Definition 6.1.1. The opposite algebra of a C* -algebra A, denoted A op,
is simply A with reversed multiplication. That is, as a normed, involutive
linear space A^0 P =A, but we define a new multiplication by a· b = ba.
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