14.3. References 403is a normal u.c.p. map such that (7/J o c,D)IMi®Mz = idMi®Mz· Hence the
restriction <p of cp to M1 ®M2 C (M1 ®M2)** satisfies 1/Jo<p = idMi®Mz· D
Remark 14.2.6. The separability assumption in Corollary 14.2.5 can be
dropped if one employs the following fact (which is proved using Takesaki's
conditional expectation theorem - see Theorem IX.4.2 in [184]): For any
von Neumann algebra M, there exists a net Ei of normal conditional expec-
tations such that Ei(M) are all von Neumann subalgebras with separable
predual and Ei --t idM in the point-ultraweak topology.
Exercises
Exercise 14.2.1. Let J be an ideal in a C*-algebra Band Q: B --t B/J
be the quotient map. Let S C B be an exact (and hence locally reflexive)
operator subsystem such that Jc S. Prove that S = Q(S) c B / J is exact.
Exercise 14.2.2. Let SC A and TC B be operator systems. Prove that
there exists a bi-normal embedding S 0 T '---t (S ® T) which extends
the canonical inclusion S 0 Tc (S ® T).
Exercise 14.2.3. Let Sand T be operator systems and assume that Sis
exact (and hence locally reflexive). Prove that the bi-normal embedding
S 0 T '---t (S ® T)** is contractive with respect to the minimal norm.
14.3. References
The notion of weak exactness was introduced by Kirchberg in [104], where
he proved all the results in Section 14.1, except for Theorem 14.1.6 and
Proposition 14.1.8. Proposition 14.1.8 comes from [63] and [80], while The-
orem 14.1.6 and Section 14.2 are adapted from [132].