1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.9. Exactness and tensor products 107

The converse follows from the previous lemma since the top row of the

commutative diagram

is an isometric isomorphism whenever A is @-exact. D

It follows that we may always assume separability.
The next result is quasi-obvious, there are just a bunch of identifications
to check. For the record, EBn An denotes the co-direct sum, i.e., the set of
sequences (an) such that limn llanll = 0.
Lemma 3.9.4. If E C A is a finite-dimensional operator system and En
are unital C* -algebras, then there is a u.c.p. isometric isomorphism

n n

defined on elementary tensors by

e Q9 (bn)n ~ ( e @ bn)n·

This map also gives an identification of E Q9 ( EBn En) and EBn (E Q9 En).

Proof. Fix some concrete representations En C JE(Hn) and AC JE(JC). The

natural "diagonal" embedding

n n

induces an inclusion

A Q9 (fJ En) C lBl (/( Q9 (EB Hn)).

n n

Similarly, we have

II A Q9 En C II JE(JC @ Hn) C lBl ( E9(JC @ Hn)).

n n n

The canonical isomorphism JC Q9 ( EBn Hn) ----+ EBn (JC@ Hn) gives an identifi-

cation