1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

Chapter 10

Summary and Open

Problems

Since a number of important characterizations and permanence properties
are scattered throughout this text, we thought it may be useful to collect
them in one place - hence this chapter. At the end, there are a few open
problems too.

10.1. Nuclear C*-algebras

Here are the main characterizations of nuclear C* -algebras:

Theorem 10.1.1. For a C* -algebra A, the following are equivalent:

(1) the identity map idA: A -----+ A is nuclear (Definitions 2.1.1 and

2.3.li see also Proposition 2.3.8 and Exercise 2.3.13)i

(2) for every C*-algebra B, A ®max B =A® B (Theorem 3.8. 'l)i

(3) A** is injective (or semidiscretei see Proposition 2.3.8 and Theorem

9.3.3).

Subalgebras. In Remark 4.4.4 we saw that subalgebras of a nuclear C* -
algebra need not be nuclear. However, there are two natural conditions
which ensure that nuclearity is preserved.

Proposition 10.1.2. A hereditary subalgebra of a nuclear C* -algebra is

nuclear. If A is nuclear, B c A and there exists a conditional expectation

qi: A-----+ B, then B is also nuclear (Exercises 2.3.4 and 2.3.3).

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