1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

9.2. Tensor product properties 287

Proof. Let C C A be a C* -subalgebra. We only prove the case of property

C as the other two are similar.

Since C** c A** and (C ® B)** c (A® B)**, we have a canonical

commutative diagram

C** 8 B** -----+ (C ® B)**

1 1

A** 8 B** -----+ (A® B)**

for every C* -algebra B. Since C ® B C A ® B, min-continuity of the
bottom arrow implies that of the top. D

Proposition 9.2.4. If I <J A is an ideal and A has property C {resp. C"),
then A/ I has property C (resp. C").

Proof. First assume that A has property C. Let 7r: A ---+ A/ I be the
quotient map, n be its normal extension to double duals and p E A be
the central support projection of n (i.e., (A/I)= pA**). Recall that if
{ ei} is an approximate unit I, then ei / 1 - p ultraweakly.

Let e be the (continuous) *-homomorphism defined via the following
diagram:

(A/I)® B e .,,.. ((A/I)® B)**

l £o! I ( 1f@idB) **

pA** ® B**'-----o--A** ® B** __ _,,..(A® B)**,

where i: A® B ---+ (A® B) is the inclusion coming from property C.
We claim that e restricted to (A/ I) 8 B coincides with the canonical
inclusion (A/ I) 8 B ---+ ((A/ I)® B) (which evidently implies the result
we are after). Indeed, for every a E A and b E B, we have

B(n(a)@ b) = (n@ idB)** o i(pa@ b)

= lim (n ® idB)** o i((l - ei)a ® b)

i

= li~ n((l - ei)a) ® b

i

=n(a)®b

and hence e[(A/I)0B coincides with the canonical inclusion. But e is clearly
bi-normal and this completes the proof.

We leave the case of property C" to the reader as it is virtually identical.

D