3.3. The spatial and maximal C*-norms 73It requires a little work, but it's a fact that C*-norms on algebraic tensor
products always exist. Here are the two most natural candidates.
Definition 3.3.3. (Maximal norm) Given A and B, we define the maximal
C*-norm on A 8 B to be
[[x[[max = sup{[[7r(x)JJ : ?T: A 8 B-+ IIB(H) a (cyclic) *-homomorphism}for x E AGE. We let A®maxB denote the completion of A8B with respect
to [[ · [[max·
Definition 3.3.4. (Spatial norm) Let 1T: A -+ IIB(H) and <J: B -+ IIB(JC) be
faithful representations. Then the spatial (or minimal) C* -norm on A 8 B is
[[Lai® bi [[min = [[ L 7r( ai) ® <Y(bi) [[JIB(7-l@J().
The completion of A 8 B with respect to [[·[[min is denoted A® B.^10
Remark 3.3.5 (Von Neumann algebra tensor products). If 1\11 c IIB(H) and
NC IIB(JC) are von Neumann algebras, then there are a number of C-norms
that one can put on M 8 N. However, the norm completions won't be von
Neumann algebras and researchers have virtually forgotten about the subject
of C-tensor products of von Neumann algebras. On the other hand, the
von Neumann algebraic tensor product is still very important and is denoted
by M ® N. By definition, this is the von Neumann algebra generated by
M ® CClJ( c IIB(H ®JC) and CC17-l ® N c IIB(H ®JC) -i.e., the weak closure of
M ® N c IIB(H ®JC).
Remark 3.3.6 (Operator space tensor products). For completeness we also
mention that one defines the spatial tensor product norm on operator sys-
tems (or spaces) in exactly the same way. Given X and Y, we take em-
beddings X c IIB(H) and Y C IIB(JC) which induce the given operator space
structures and then define X ® Y to be the norm closure of the span of
{ x ® y E IIB(H ® JC) : x E X, y E Y}. As we'll soon see for C*-algebras,
X ® Y is independent of the embeddings (so long as they induce the proper
operator space structures, of course).
There are numerous technical points which one should worry about. The
first is whether or not [[ · [[max is even finite. This is the case thanks to the
existence of restrictions (Theorem 3.2.6). Indeed, if 1T : A 8 B -+ IIB(H) is a
*-representation with restrictions 1TA and 1TB, then
[[7r( a® b) II :::; [[7rA( a) [[ J[7rB(b) II :::; JJa[[ J[b[[for all elementary tensors. This implies that JJx[[max < oo for all x E AGE.
lOYou will also see A ®min B in the literature.