9.5. References 299Hence we can find pure states <p on A and 'ljJ on B such that ( <p@'lj;) Iker er =
0 and ( <p 0 'ljJ) ( z) =/= 0. Let ai E A excise <p and bi E B excise 'ljJ (Theorem
1.4.10). Since (<p 0 'l/;)(ai 0 bi) = 1, one has JJai 0 bi+ JJJ = 1 for all i. We
claim that llai 0 bi+ Ill = 1 as well (in particular, liar 0 br +Ill is bounded
away from zero). Once demonstrated, the proof will be complete because
(<p 0 'l/;)(z)(ar 0 br +I) ~ (ai 0 bi)z(ai 0 bi)+ I, by the excision property,
and hence z +I=/= 0.
Suppose by contradiction that Jlai 0 bi + Ill = 8 < 1. Consider the
*-homomorphism C[O, 1] 0 C[O, 1] -----+ A 0 B defined by L:k fk 0 9k f--+
L:k Jk(ai) 0 9k(bi)· Let Jo E C[O, 1] be such that 0 :S Jo :S 1, Jo = 0
on [O, 8112 ] and Jo = 1 on [8^114 , 1]. Then, fo(ai) 0 J 0 (bi) E I and hence
fo(ai) 0 fo(bi) E J, by definition. This implies that JJai 0 bi+ Jll :S (jl/^4 < 1
in contradiction to our observations above, so the proof is complete. D
9.5. References
Local reflexivity was imported to C -algebras by Effros and Haagerup [59].
Propositions 9.1.4 and 9.2.5 appeared in that paper. Property C was in-
troduced by Archbold and Batty in [8], where they showed property C
implies exactness, it passes to subquotients, and it is enjoyed by nuclear C -
algebras. Theorem 9.3.1 is due to Kirchberg [103], [106]. That injectivity
implies semidiscreteness (and even hyperfiniteness) for factors is Connes's
remarkable discovery [41] (the general case is found in [39]). The proof
given here is inspired by Wassermann's work in [194].