292 9. Local Reflexivityand N = JC** (canonically). Then, <T?!eB**<VfC** is a bi-normal and min-
continuous u.c.p. map on M 8 N. We claim that this u.c.p. map coincides
with <I>. Let (bi) be a bounded net in B such that bi --+ e ultraweakly in
B**. Then, bi --+ lM ultraweakly in M and hence
<f?(e Q91) = lii;n<f?(bi ® 1) = lii;n<I>(bi ® 1) = <I>(lM ® 1) = 1.
i i
Likewise one has <f?(l ® f) = 1. It follows that e ® f is in the multiplicative
domain of <J? and <f?(eb®fc) = <T?(b®c) = <I>(b®c) for every b EB and c EC.
Since both maps are bi-normal, this implies that <I> = <J? leB**0fC**. DWhen dealing with double duals and tensor products, one cannot be too
careful. As Pisier noted, "This subject is full of traps" ([152, page 309]).
Below are a few to avoid.
Exercises
Exercise 9.2.1. Find an example of unital C* -algebras Bo c B and an
ideal J <1 B such that the quotient map 7r: B --+ B / J is locally liftable but
7ro = 7r1Bo: Bo--+ 7r(Bo) is not. (Hint: Bo should not contain J.)
Exercise 9.2.2. Here's a "proof'' that 7ro (as in the previous exercise) is
always locally liftable, whenever 7r is locally liftable. Where is the gap in
the argument?Proof. First observe that if I <1 C, then there is a canonical embedding
C/I c (C/I)** c C** (since C** = (C/I)** EB I**). By Theorem C.4, local
liftability of 7r is equivalent to having a canonical identification
A ® BI J = (A ® B) I (A ® J)'
for every A. Since (A ® B) / (A ® J) c (A ® B) **, this in turn is equivalent
to knowing that the canonical embedding
B: A 8B/J c A8 (B/J)** c A GB**~ (A® B)**
is min-continuous.
Now consider the commutative diagramA®B/JC e > (A®B)**j eo j
A® 7ro(Bo)~ (A® Bo)**.Since the vertical inclusions are isometric, it follows that Bo is isometric
whenever B is isometric, and hence 7rO is locally liftable whenever 7r is. 0
Exercise 9.2.3. Here is a "proof'' of the false assertion that every C* -
algebra A is locally reflexive. Where's the gap?