1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

2.1. Nuclear maps 27

Proof. Let { PihEI be a net of finite-rank projections which increases to

the identity (i.e., i ~ j :::}-Pi ~ Pj and llPiv - vii --+ 0 for all v EH). If Pi

has rank k(i), then we define c.c.p. maps <fJi: M--+ Mk(i)(C) ~ P/B(H)Pi

by compression (i.e., <fJi(T) = PiTPi) and we let 'l/Ji: Mk(i)(<C) --+ JIB(H) be

the natural inclusion maps. Since the predual of JIB(H) is the trace class

operators, a routine exercise shows that these maps converge to the identity

(on all of JIB(H), in fact) in the point-ultraweak topology and hence M <--+

JIB(H) is weakly nuclear. D

Here are some simple, but very useful, observations to get you warmed

up. As mentioned above, we will use these exercises - often without reference

• throughout this book.

Exercises

Exercise 2.1.1 (Finite sets and i::'s). Show that(): A--+ Bis nuclear if and

only if for each finite set~ CA and c > 0 there exist n EN and c.c.p. maps

<p: A--+ Mn(<C), 'l/J: Mn(<C) --+ B such that ll()(a) - 'ljJ o <p(a)ll < c for all

a E ~.^1

Exercise 2.1.2 (Matrices versus finite-dimensional algebras). Prove that
(): A --+ B is nuclear if and only if there exist c.c.p. maps <fJn: A --+ Cn and
'I/Jn: Cn --+ B, where the Cn 's are finite-dimensional C* -algebras, such that
'I/Jn o <fJn --+ e in the point-norm topology. (Hint: Find integers k(n) such
that there is a unital embedding Cn C Mk(n) (<C) and construct a conditional
expectation Mk(n)(<C)--+ Cn.)

Exercise 2.1.3 (Restriction to subalgebras). If e: A --+ B is nuclear and
C c A is a C -subalgebra, then e Io: C --+ B is also nuclear. (It need not
be true that ()Io : C --+ e ( C) is nuclear. But don't try to construct examples
until we prove that subalgebras of nuclear C-algebras need not be nuclear.)

Exercise 2.1.4 (Compositions). If c.c.p. maps(): A--+ B and er: B--+ C
are given and either e or (J is nuclear, then so is the composition (J 0 e.

Exercise 2.1.5 (Special case of compositions). If A is a C*-algebra such
that the identity map on A is nuclear, then any other c.c.p. map (): A--+ B
is also nuclear.

Exercise 2.1.6 (Special case of compositions). Assume A c JIB(H) is a
concretely represented C* -algebra such that the inclusion map A <--+ JIB(H)
is nuclear. Show that if (): A --+ JIB(JC) is any c.c.p. map, then () is nuclear.
(Hint: You will need Arveson's Extension Theorem.)

lyou may have noticed that we haven't specified in Definition 2.1.1 whether one should work
with sequences or nets. Of course, one should use nets in general and show that sequences suffice
in the separable setting. However, the main point of this exercise is that nuclearity is really a
local property and hence there is little difference between the separable and nonseparable worlds.
This exercise also tests whether or not you know how to construct a net.