1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

72 3. Tensor Products

is replaced by a C -algebra. Indeed, let A be a unital C -algebra and let

SA= Co(R, A) = {f : R .,A: f is continuous and lim Iii (t) II = O}
t-+±oo
be the suspension of A. Show that there is a *-homomorphism Co (R) 0 A .,
SA defined on elementary tensors by f ®a f----+ f(·)a. This map cannot be
restricted to A since SA has no projections and A is unital.

Exercise 3.2.6. Are the restrictions given by Theorem 3.2.6 unique? Let
7r1: A., lIB(H) and 7r2: B ., lIB(H) be (possibly degenerate) representations
with commuting ranges and let

7r := 7r1 x 7r2: A 0 B .___, lIB(H).

Find necessary and sufficient conditions which ensure that 7f A
1fB = 7f2.

3.3. The spatial and maximal C*-norms

'lfl and

When A and B are C -algebras, it can happen that numerous different norms
make A 0 B into a pre-C -algebra. In other words, A 0 B may carry more
than one C* -norm.

Definition 3.3.1. A C* -norm II · Ila on A 0 B is a norm such that llxylla :=::;

llxllallYlla, llx*lla = llxlla and llx*xlla = llxll; for all x, YE A 0 B. We will

let A ®a B denote the completion of A 0 B with respect to II· Ila·

The following example is both of fundamental importance and also illus-
trates the fact that even "trivial" examples in this subject can have subtleties
which require care.

Proposition 3.3.2. For each C* -algebra A there is a C* -norm on the alge-

braic tensor product Mn(<C) 0 A and it is unique.

Proof. We assume the reader knows how to make Mn(A) into a C-algebra
and hence the existence of a C -norm follows from the existence of an alge-
braic -isomorphism (Exercise 3.1.3)
Mn(<C) 0 A~ Mn(A).
Uniqueness is then a consequence of the fact that C-algebras have
unique norms since Mn(<C) 0 A is a C*-algebra with respect to the norm
it gets from Mn(A).^9 D

(^9) Depending on what the phrase "C-algebras have unique norms" means to you, there may or
may not be a subtlety here. If this statement only means, "Whenever an algebra B is a C -algebra
with respect to two norms JI· 11 and 11 ·II', then those norms agree," then the proof of uniqueness
has a gap. Luckily, the more general statement, "If (B, II· II) is a C-algebra and (B, II· II') is a
pre-C-algebra (i.e., not necessarily complete), then II · II = II · II'," is true and this is what we are
using above.