4.9. References 165where T(<C) is the Toeplitz algebra and w is the vacuum state. (Really long
hint, which some consider a proof: It is not hard to see that the nondegen-
eracy of Ef> and Ert'P A implies that of Ef> o E1i<p. A Since T(HA'P) is generated
by A and T(H'b) ~ T(<C) Q9 D, it suffices to show that A and T(HiJ) are
free over D in (T(H~), Ef> o Ertj,_), by Theorem 4.7.2. Since dT = Tr.p(d)
for every d E D, we have
ker E1i'b =span{ dTm(rr : d E D, (m, n) =/= (0, O)} c T(HiJ).
Thus, it suffices to show that
(Ef> o Ert'P A )(a0Tm^1 (Tr^1 a1 · · · Tmk(Trkak) = 0
fork 2: 1, (mj,nj) =/= (0,0) and aj E kerE-f> (except that a 0 and ak are
possibly 1). But this easily follows from the fact that TaT = Ef>(a) = 0
for a E ker Ef> and Theorem 4.6.6.)
Exercise 4.8.2. Let 1 E D C A be C -algebras with a nondegenerate
conditional expectation Ef> from A onto D. Prove that
(A, Ef>) D (Mn(CC) Q9 D, w Q9 idv)
is nuclear if A is nuclear and w is a pure state on Mn(<C). (Hint: The
previous exercise says free products with the Toeplitz algebra - using the
vacuum state - preserve nuclearity, by Theorem 4.6.25. Thus, given Mn(<C)
and a vector state, it would suffice to find a really nice embedding into the
Toeplitz algebra, since Theorem 4.8.5 would provide some c.p. maps to work
with.)
- References
Amenable actions were introduced by Zimmer in the measurable context.( cf.
[200]). They were transported into topological/C-terms by Anantharaman-
Delaroche [3], where Theorem 4.4.3 was proved. This naturally led to
amenable groupoids; see [6] for a detailed study. Pimsner's influential class
of algebras was introduced in [146]; our treatment is highly influenced by
Katsura's work [99], where he dealt with C -correspondences with noninjec-
tive left action. The gauge-invariant uniqueness theorem is due to Fowler,
Muhly and Raeburn [66]. Theorem 4.6.25 is due to Dykema and Shlyakht-
enko [58], where they gave a new proof of Dykema's theorem on exactness
of amalgamated free products: Corollary 4.8.3 ([57]). The existence of free
product maps was established, in the present generality, by Blanchard and
Dykema [20]. For more on free products and free probability, see the mono-
graph [193].