1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

92 3. Tensor Products

Show that there is a natural isomorphism

Co(X) ©max A~ Co(X) 129 A~ Co(X, A)

such that h@a maps to the function x f---t h(x)a. (Hint: A partition of unity
argument will show density.)

3.7. Exact sequences

In this section we exhibit another peculiar feature of C -tensor products, one
related to exact sequences. Namely, we consider the analogues of Proposition
3.1.13 with our two C-norms 11 · llmin and 11 · llmax· In contrast to the previous
section, the spatial norm is now the problem child.

Proposition 3. 7 .1. If 0 ---+ J ---+ A ---+ (A/ J) ---+ 0 is an exact sequence,
then for every B, the natural sequence
0 ---+ J @max B ---+ A @max B ---+ (A/ J) @max B ---+ 0
is also exact.

Proof. Modulo one subtlety, the proof is fairly simple. First note that
injectivity of l@maxB---+ A@maxB follows from Corollary 3.6.4, while A@max
B---+ (A/ J) @max Bis surjective thanks to the fact that -homomorphisms
of C -algebras always have closed range. Moreover, it is clear that J @max B
is contained in the kernel of the map A@max B---+ (A/ J) @max B; hence the
only question is whether or not J @max B is the whole kernel.

To see that this is the case, we first observe that there is a C*-norm II· Ila
on (A/ J) 0 B such that

A@maxB ~ (AjJ)@aB,
l@maxB
since exactness of the sequence

O---+ J 0 B---+ A 0 B---+ (A/ J) 0 B---+ 0

guarantees that we have a dense embedding

(A/J)0B~ A0B "---+ A@maxB_1s

J0B l@maxB

On the other hand, we also have a quotient (hence contractive) mapping

(A/ J) @a B ~ ~@max; ---+ (A/ J) @max B.

@max

By maximality of 11 · llmax, it follows that II· Ila= 11 · llmax and thus J @max B
is precisely the kernel of A @max B---+ (A/ J) @max B.

(^18) The norm II · II°' is just the restriction of the quotient norm to this embedded copy of
(A/J) 8 B.