9.2. Tensor product properties 285Suppose now that both I and B are locally reflexive and the extension
is locally split. Let a finite-dimensional operator system E c A be given.
Again writing A = I EB B, we may regard E c Er EB EB c I EB B,
where Er and EB are finite-dimensional. Since I and Bare locally reflexive,
there exist nets of c.c.p. maps 'l/J{: Er-----> I and 'l/Jf: EB -----> B which converge
to the relevant identities. Let ,(/;f: EB -----> A be a c.c.p. lifting of 'l/Jf. We
denote the units of I and B in A** by er and eB, respectively. If (ej) is an
approximate unit for I, then e}f^2 /er and (1-ej)^112 "'..,, eB ultrastrongly.
Now define a net of c.c.p. maps 'Pi,j: E-----> A by
'Pi,j(x) = e}^12 '1/J{(xer)e}^12 + (1-ej)^1 l^2 ,(/;f(xeB)(l - ej)^112.
Then, for every x E E, we have
li:i;nli:i;n<pi,j(x) = li:i;ner'l/J{ (xer)er + eB,(/;f (xeB)eB = xer + xeB = x
i J isince e B,(/;f ( xe B) e B = 'l/Jf ( xe B) under the identification A = I EBB**. D
Corollary 9.1.5. Assume I <I A is an ideal in a locally reflexive C-algebra
A. For every C -algebra B the sequence
0-----> I® B-----> A® B-----> (A/I)® B-----> 0
is exact.Proof. We may assume A is unital, whence the result follows from Propo-
sition 3.7.6. D
Corollary 9.1.6. The C*-algebra JEB('H) is not locally reflexive. Neither is
C*(lF2).Proof. The previous corollary together with Exercise 3.9.7 implies that if
A = JEB('H) is locally reflexive, then every C -algebra is exact; hence JEB('H)
is not locally reflexive. The free group case follows from Proposition 3.7.11,
taking A= B = C(lF2). D
Exercise
Exercise 9.1.1. Use the definition to show local reflexivity passes to hered-
itary subalgebras.
9.2. Tensor product properties
To give tensor-product characterizations of local reflexivity, we must spend
time with C* -tensor products of double duals; there are subtleties around
every corner.
If Mand N are von Neumann algebras, a *-homomorphism M 8 N----->
JEB('H) is said to be bi-normal if both of the restriction maps M = M ®ClN ----->