1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

Nuclear and Exact

C* -Algebras:

Definitions, Basic Facts

and Examples

Chapter 2

Nuclearity and exactness have dominated the C*-scene for quite a while. We

will define these classes in terms of the nuclearity of certain maps; historically

they were defined via tensor products, as we'll see in Sections 3.8 and 3.9.

For the most part, this chapter consists of easy propositions and exam-

ples. We have attempted to lay out, as simply as possible, the main themes,

subtleties and techniques. Sections 2.1 and 2.3 contain numerous exercises

which newcomers are highly encouraged to work through. Most are quite

easy, but we aren't trying to insult your intelligence. These exercises tend

to get used without explanation in the literature (and this book), so we

thought it might be helpful to isolate them from the get-go.

2.1. Nuclear maps

The following definition is the cornerstone of nuclearity and exactness;.
Definition 2.1.1. A map e: A ---+ B is called nuclear if there exist c.c.p.
maps 1Pn: A---+ Mk(n:) (C) and 'I/Jn: Mk(n) (C) ---+ B such that 'I/Jn o 1Pn ---+ e in
the point-norm topology:

ll'l/Jn 01Pn(a) - e(a)ll---+ 0,

for all a EA.

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