11.3. Inner quasidiagonality 323and use the t.pn,k, n > k, for connecting maps. Since we have defined Ck+i+l
to be the algebra generated by the image of the previous algebra, we always
have *-homomorphisms Ck+i+l -+ Ck+i which reverse the complete order
embedding connecting maps. (This fact will be important below.)
The generalized inductive limit Ak of this sequence is obviously a strong
NF (hence nuclear) subalgebra of A for each k E N. To prove residual
finite-dimensionality of Ak, we note that for each n there is a commutative
diagram
Ck+n ----+ Ck+n+l ----+ Ck+n+2 ----+... ----+ Ak
1 1 1 1
Ck+n ----+ Ck+n ----+ Ck+n ----+ · · · ----+ Ck+n
where the bottom arrows are all the identity and the vertical arrows are
-homomorphisms obtained by composing the -homomorphisms Ck+n+i-+
Ck+n+i-1 -+ · · · -+ Ck+n· This is enough to imply that each Ak is residually
finite-dimensional, and it is clear that their union is dense since <Pk(Bk) C
Ak. D
11.3. Inner quasidiagonality
This section is devoted to a natural subclass of QD algebras, first introduced
by Blackadar and Kirchberg ([19]). Though interesting in their own right,
we have a specific goal in mind and hence do not present everything that is
known about inner QD C* -algebras.
Definition 11.3.1. A separable^5 C*-algebra A is inner QD if there exist
projections Pn EA** such that
(1) ll[pn, aJll-+ 0 for all a EA CA**,
(2) llall = lim llPnaPnll for all a EA and
(3) Pn is in the socle^6 of A**, for every n EN.Note that a nonunital algebra is inner QD if and only if its unitization
is inner QD (since (A) =A EB C).
Remark 11.3.2. Every inner QD algebra is evidently QD, simply define
'Pn: A-+ PnA**Pn by compression.
Remark 11.3.3. Every RFD C*-algebra A is inner QD, since the central
covers c( 1f) E Z (A) of finite-dimensional representations always live in the
socle of A.
5 As usual, this is an assumption of convenience. The interested reader can work out the
nonseparable case.
(^6) This means PnA**Pn is finite-dimensional.