1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.3. The spatial and maximal C*-norms 75

Rearranging terms, we get

((LSi0Ti)v0e,w0'1J) = L:\si0Ti(v0e),w0'1J)

i i

= L:\Siv,w)(Tie,'IJ)

i

= ((L(Tie, 'IJ)Si)v, w).

i

Since this holds for all v,w E 1-l, it follows that the operator L,i(Tie,'IJ)Si E

JE(J-l) is zero and hence, by linear independence, that each of the coefficients

(Tie, '17) is zero. Since this holds for all e, 'IJ E JC, it follows that 0 =Ti E JE(JC)

for all i, and the proof is complete. D

Corollary 3.3.10. For each x EA 0 B, if llxllmin = 0, then x = 0.

Proof. If 7r: A ---t JE(J-l) and O": B ---t JE(JC) are faithful representations,
then, by Proposition 3.1.12, the tensor product map

7r 0 O": A 0 B ---t JE(J-l) 0 lE(JC)

is also injective. Together with the previous lemma this implies the result.

D

We now resolve the second technical question.

Proposition 3.3.11. The spatial tensor product norm is independent of the
choices of faithful representations 7r: A ---t JE(J-l) and O": B ---t JE(JC).

Proof. For the moment we will let II· ll~i:) denote the minimal norm with
respect to 7r and O". Evidently it suffices to prove that if 0"^1 : B ---t JE(JC') is

another faithful representation, then II ·II~:) = II ·II~:').
For notational reasons it is slightly more convenient to give the proof
in the separable setting. It is a simple exercise to net-ify the argument and
deduce the general case. Let P1 :::; P2 :::; · · · be finite-rank projections in
JE(J-l) such that Pn has rank n and llPn(h) - hll ---t 0 for all h E 1-l. Then it
is not hard to show that for every X E JE(J-l 0 JC) we have
llXll = sup{ll(Pn 0 lJC)X(Pn 0 lJC)ll}.
n
Thus, if L, ai 0 bi E A 0 B is arbitrary, we have

II Lai 0 bill~i:) =sup{ II L(Pn7r(ai)Pn) 0 O"(bi)ll}

n

and