ON THE KK-THEORY AND THE E-THEORY OF
AMALGAMATED FREE PRODUCTS OF C ∗ -ALGEBRAS
KLAUS THOMSEN
Abstract. We establish six terms exact sequences relating the KK-theory groups
and the E-theory groups of an amalgamated free product C ∗ -algebra, A1 ∗B A2 ,
to the respective groups for the three constituents, A1 , A2 and B. In the KKtheory case we assume the existence of conditional expectations from Ak onto B
or that A1 , A2 and B are all nuclear, and in the E-theory case that there exist
sequences Rnk : Ak → B, n ∈ N, of completely positive contractions such that
limn→∞ Rnk (b) = b for all b ∈ B, k = 1, 2. This condition is fullfilled e.g. when B
is nuclear or sits as a hereditary C ∗ -subalgebra of the Ak ’s.
1. Introduction
Cuntz and Germain have conjectured the existence of two short exact sequences
which should relate the KK-groups of an amalgamated free product A1 ∗B A2 to
the KK-groups of A1 , A2 , B. See Remark 2 of [C1], Conjecture 0.1 of [G2] and
Conjecture 3.11 of [G3] where the conjecture is formulated in increasing generality.
In [C1] Cuntz proved the conjecture when there are retractions from the Ak ’s onto
B, in [G1] Germain proved it when B = C sits unitally inside the Ak ’s which
were assumed to be ’K-pointed’, cf. Definition 5.1 of [G1], a condition which he
subsequently, in [G2], weakened to K-nuclearity (in the sense of Skandalis, [S]).
Finally, in [G3] he announced a proof of the conjecture under certain technical
assumptions (’relative K-nuclearity’) which among other things require the existence
of conditional expectations Pk : Ak → B. In another direction the conjecture was
established in increasing general! ity for examples coming from groups or actions
by groups in [C2], [N], [L]. In this direction the ultimate result seems to be that
of Pimsner, [Pi], who obtained results which, among others, verify the conjecture
when G1 and G2 are countable discrete groups containing a common subgroup H,
Ak = A n Gk , k = 1, 2, and B = A n H for some actions of G1 and G2 on A which
agree on H. However, in the general case the conjecture remained open even when
B = C.
In this paper we prove the conjecture when there are conditional expectations
Ak → B, k = 1, 2, or A1 , A2 and B are all nuclear. In principle the methods that
we use for this is the same as that of Germain. In [G2] and [G3] Germain wrote
down a ∗-homomorphism ϕ : C → S(A1 ∗B A2 ) between the mapping cone C for
the inclusion B → A1 ⊕ A2 and the suspension S(A1 ∗B A2 ) of A1 ∗B A2 , and made
the observation that the conjecture is equivalent to the KK-invertability of ϕ. He
was then able to invert ϕ in KK-theory when B = C under the assumption on A1
and A2 mentioned above. The method of proof that we shall use is in principle the
same, but the goal - to invert ϕ in KK-theory - is achieved by completely different
means. In fact, we shall obtain the proof by working with extension groups in much
Version: March 26, 2001.
2
KLAUS THOMSEN
the same way as in the work of L. Brown, [Br], who obtained partial results which
inspi! red Cuntz in the formulation of the conjecture, cf. Remark 2 of [C1]. By
working with extensions we shall establish enough of the desired exact sequences
to deduce that Germains homomorphism is invertible in KK-theory. To do this we
use two important ingredients which were not available when Brown did his work,
namely Boca’s result on free products of completely positive unital maps, [Bo], and
the automatic existence of absorbing trivial extensions together with the related
duality results for KK-theory obtained in full generality by the author in [Th1].
A major part of the paper is an attack on the analogous conjecture in the E-theory
of Connes and Higson, [CH], and we obtain the desired six terms exact sequences
under even weaker conditions in this setting, as described in the abstract. The
approach we take for this is new: Provided B is properly embedded in both Ak ’s,
meaning that an approximate unit in B is also an approximate unit in Ak , there is
an exact sequence
0
/
S(A1 ∗B A2 )
/
cone(A1 ) ∗SB cone(A2 )
/
A1 ∗ A 2
/
0,
(1.1)
where A1 ∗ A2 is the unrestricted free product. As shown by Cuntz,[C3], A1 ∗ A2
is KK-equivalent to A1 ⊕ A2 . Based on methods and results from [DE] and [Th2]
we show here that cone(A1 ) ∗SB cone(A2 ) is equivalent to B in E-theory provided
there are sequences of completely positive contractions Rkn : Ak → B such that
limn→∞ Rkn (b) = b for all b ∈ B, k = 1, 2. The desired exact sequences then come up
as the E-theory exact sequences arising from (1.1). Note that the extension (1.1) is
actually semi-split (this follows from Boca’s result, [Bo]), so it is not inconceivable
that this extension can be used to obtain the result in KK-theory rather than Etheory. However, the methods that we use here to show that cone(A1 ) ∗SB cone(A2 )
is equivalent to B works only in E-theory. In a final section we point a serious
limitation of our methods which prevents them from taking us all the wa! y to a
proof of the general conjecture.
2. On absorbing extensions and asymptotic homomorphisms
In this section we gather a series of lemmas. Only the first two are needed for our
results in KK-theory. Let A and D be separable C ∗ -algebras, D stable. Let M (D)
denote the multiplier algebra of D. Since D is stable there are isometries V1 , V2 ∈
M (D) such that V1 V1∗ + V2 V2∗ = 1 and V1∗ V2 = 0 and we can define the orthogonal
sum a ⊕ b of two elements a, b ∈ M (D) to be V1 aV1∗ + V2 bV2∗ . Similarly, we can add
maps, ϕ, ψ : A → M (D), orthogonally; viz. (ϕ ⊕ ψ)(a) = V1 ϕ(a)V1∗ + V2 ψ(a)V2∗ .
We call a ∗-homomorphism ϕ : A → M (D) absorbing when the following holds:
When π : A → M (D) is a ∗-homomorphism, there is a sequence of unitaries
{Un } ⊆ M (D) such that limn→∞ Un (ϕ ⊕ π)(a)Un∗ − ϕ(a) = 0 for all a ∈ A.
See Theorem 2.5 of [Th1] for alternative characterizations of absorbing ∗-homomorphisms
which we shall use quite freely. By Theorem 2.7 of [Th1] there always exists an absorbing ∗-homomorphism.
Lemma 2.1. Let A, D be separable C ∗ -algebras, D stable and B ⊆ A a C ∗ -subalgebra
of A. Assume that that there is a sequence of completely positive contractions R n :
A → B such that limn Rn (b) = b, b ∈ B. If π : A → M (D) is an absorbing extension,
then π|B : B → M (D) is an absorbing extension.
KK-THEORY AND E-THEORY OF AMALGAMATED FREE PRODUCTS
3
Proof. By Theorem 2.5 of [Th1] we must show that the unitization (π|B )+ : B + →
M (D) of π|B is unitally absorbing. We check that condition 1) of Theorem 2.1 of
[Th1] is satisfied. Consider therefore a completely positive contraction ϕ : B + → D.
Then ϕ◦Rk+ : A+ → D is also a completely positive contraction and since π + : A+ →
M (D) is unitally absorbing, we know that there is a sequence {Wnk } ⊆ M (D) such
∗
∗
that limn→∞ kϕ◦Rk+ (a)−Wnk π + (a)Wnk k = 0 for all a ∈ A+ and limn→∞ kWnk dk = 0
for all d ∈ D. Since limk→∞ Rk+ (b) = b for all b ∈ B + ⊆ A+ , it follows that also
(π|B )+ = π + |B + satisfies condition 1).
Lemma 2.2. Let A, D be separable C ∗ -algebras, D stable and B ⊆ A a C ∗ -subalgebra
of A. Assume that B is nuclear. If π : A → M (D) is an absorbing extension, then
π|B : B → M (D) is an absorbing extension.
Proof. Since B is nuclear there are sequences Sn : B → Fn , Tn : Fn → B, n ∈ N, of
completely positive contractions, where the Fn ’s are finite dimensional C ∗ -algebras,
such that limn→∞ Tn ◦ Sn (b) = b for all b ∈ B. By Arvesons extension theorem, [A1],
there is for each n a completely positive contraction, Vn : A → Fn , extending Sn .
Set Rn = Tn ◦ Vn and apply Lemma 2.1.
Throughout the rest of the paper A1 , A2 , B, D are separable C ∗ -algebras with D
stable, and ik : B → Ak , k = 1, 2, are embeddings.
Lemma 2.3. Assume that there are sequences Rnk : Ak → B, n = 1, 2, 3, · · · , of
completely positive contractions such that limn→∞ Rnk (ik (b)) = b for all b ∈ B, k =
1, 2. Let πk : Ak → M (D), k = 1, 2, be saturated and absorbing ∗-homomorphisms.
It follows that there is a norm-continuous path {ut }t∈[1,∞) of unitaries in M (D) such
that ut π1 ◦i1 (b)u∗t −π2 ◦i2 (b) ∈ D for all t ∈ [1, ∞), b ∈ B, and limt→∞ ut π1 ◦i1 (b)u∗t −
π2 ◦ i2 (b) = 0 for all b ∈ B.
Proof. Recall from [Th2] that πk being saturated means that the infinite direct sum
0 ⊕ πk ⊕ πk ⊕ πk ⊕ · · · is unitarily equivalent to πk . It follows from Lemma 2.1
that πk ◦ ik : B → M (D), k = 1, 2, are both absorbing (and saturated). From the
uniqueness of absorbing ∗-homomorphisms it follows that π1 ◦ i1 ⊕ (π2 ◦ i2 )∞ ∼
π1 ◦ i1 and (π1 ◦ i1 )∞ ⊕ π2 ◦ i2 ∼ π2 ◦ i2 , in the notation of [DE]. It follows
therefore from Lemma 2.4 of [DE] that there is a norm-continuous path {wt }t∈[1,∞)
of unitaries in M (D) such that wt (π1 ◦ i1 )∞ (b)wt∗ − (π2 ◦ i2 )∞ (b) ∈ D for all t, b, and
limt→∞ wt (π1 ◦ i1 )∞ (b)wt∗ − (π2 ◦ i2 )∞ (b) = 0 for all b ∈ B. Since πk is saturated, πk
is unitarily equivalent to ! (πk )∞ , so the conclusion follows.
In the following we shall consider the suspensions SA1 , SA2 , SB and the cones
cone(A1 ), cone(A2 ), cone(B). The embeddings ik : B → Ak , k = 1, 2, induce embeddings between some of these algebras (e.g. SB → cone(Ak )) in a natural way, and
in order to avoid too heavy notation we shall denote a map induced by ik : B → Ak
by ik again. It will always be clear from the context which domain and target is
meant.
Lemma 2.4. Assume that there are sequences Rnk : Ak → B, n = 1, 2, 3, · · · , of completely positive contractions such that lim n→∞ Rnk (ik (b)) = b for all b ∈ B, k = 1, 2.
There exist absorbing and saturated ∗-homomorphisms αk : cone(Ak ) → M (D), k =
1, 2, such that αk ◦ ik : cone(B) → M (D), k = 1, 2, are both absorbing and saturated,
and there exist normcontinuous paths, {pt }t∈[0,∞) , {wt }t∈[0,∞) , of elements in M (D)
such that the wt ’s are unitaries, and
4
KLAUS THOMSEN
1)
2)
3)
4)
5)
6)
7)
8)
0 ≤ pt ≤ 1, t ∈ [0, ∞),
pt αk (cone(Ak )) ⊆ D , t ∈ [0, ∞), k = 1, 2,
(p2t − pt )αk (cone(Ak )) = {0} , t ∈ [0, ∞),
limt→∞ pt d = d , d ∈ D,
limt→∞ kpt αk (a) − αk (a)pt k = 0 , a ∈ cone(Ak ), k = 1, 2,
p0 = 0, p2n = pn , n = 1, 2, 3, · · · ,
limt→∞ wt α1 ◦ i1 (b)wt∗ − α2 ◦ i2 (b) = 0 for all b ∈ cone(B),
limt→∞ pt wt − wt pt = 0.
The proof is an elaboration of the proof of Theorem 3.7 of [Th2], so we need first
the appropriate versions of Lemma 3.5 and Lemma 3.6 in [Th2].
Lemma 2.5. Let D be a separable C ∗ -algebra. Let K1 ⊆ K2 ⊆ M (D) and F ⊆ D
be compact subsets. Let δ > 0 and assume that p ∈ M (D) a projection such that
[p, m] ∈ D, m ∈ K2 , and
kmp − pmk < δ , a ∈ K1 .
Let 0 ≤ z ≤ 1 be a strictly positive element in (1 − p)D(1 − p) and let 1 , 2 ∈ ]0, 1[
be given. There is then a continuous function h : [0, 1] → [0, 1] such that h is zero
in a neighbourhood of 0, h(t) = 1, t ≥ 1 ,
sup k[m, p + h(tz)]k < 5δ ,
t∈[0,1]
and
k[m, p + h(z)]k < 2 ,
m ∈ K1 ,
m ∈ K2 ,
kpb + h(z)b − bk < 2 , b ∈ F .
Proof. The proof is the same as the proof of Lemma 3.5 in [Th2]
Let H be an infinite-dimensional separable Hilbert space. We can then define
g : [0, ∞[→ [0, 2] by
√
g(s) = sup{k[a, x]k : a, x ∈ B(H), kak ≤ 1, 0 ≤ x ≤ 1, k[a, x]k ≤ s} .
By the lemma on page 332 of [A2], g is continuous at 0, i.e. lims→0 g(s) = 0. g will
feature in the next lemma.
Lemma 2.6. Let A and D be separable C ∗ -algebras with A contractible. Let ϕt :
A → A, t ∈ [0, 1], be a homotopy of endomorphisms of A such that ϕ0 = id and
ϕ1 = 0. Let F0 ⊆ F1 ⊆ D and G1 ⊆ B, G2 ⊆ G3 ⊆ M (D) be compact subsets.
Let π : A → M (D) be a ∗-homomorphism and p ∈ M (D) a projection such that
pπ(A) ⊆ D, [p, m] ∈ D, m ∈ G3 , kpπ(ϕt (a)) − π(ϕt (a))pk < κ, a ∈ F0 , t ∈ [0, 1],
and kpm − mpk < κ, m ∈ G2 , for some κ > 0.
For any > 0 there is then an n ∈ N, a ∗-homomorphism π1 : A → M (Mn (D))
of the form
π1 (a) = diag(π(ϕs1 (a)), π(ϕs2 (a)), · · · , π(ϕsn (a)))
for some s1 , s2 , · · · , sn ∈ [0, 1], s1 = 0, sn = 1, and a normcontinuous path pt , t ∈
[0, 1], of elements pt ∈ M (Mn+1 (D)) such that
1) 0 ≤ pt ≤ 1, t ∈ [0, 1],
2) (p2t − pt ) π(a) π1 (a) = 0 , a ∈ D, t ∈ [0, 1],
3) pt π(a) π1 (a) ∈ Mn+1 (B) , a ∈ D, t ∈ [0, 1],
KK-THEORY AND E-THEORY OF AMALGAMATED FREE PRODUCTS
4) kpt
7)
8)
9)
10)
11)
π(a)
π1 (a)
−
π(a)
0n
) ≤ pt , t ∈ [0,1],
kp1 π(ϕt (a)) π1 (ϕt (a)) −
5) (
6)
p
π1 (a)
5
pt k ≤ 6g(20κ) + 3κ , a ∈ F0 , t ∈ [0, 1],
π(ϕt (a))
π1 (ϕt (a))
p1 k ≤ , a ∈ F1 , t ∈ [0, 1],
kp1 b 0n − b 0n k < , b ∈ G1 ,
p
p1 = p21 , p0 = ( 0n ),
[(1Mn+1 (C) ⊗ m), pt ] ∈ Mn+1 (D), m ∈ G3 , t ∈ [0, 1],
k[(1Mn+1 (C) ⊗ m), pt ]k ≤ 6g(20κ) + 3κ , m ∈ G2 , t ∈ [0, 1],
k[p1 , (1Mn+1 (C) ⊗ m)]k ≤ , m ∈ G3 .
Proof. The proof is basically the same as the proof of Lemma 3.6 in [Th2]; to obtain
10) and 11) it suffices to ensure, in the notation of that proof, that kxtj m−mxtj k < 5κ
for all t, j and all m ∈ G2 , while kxj m − mxj k ≤ δ for all j and all m ∈ G3 . This is
possible by Lemma 2.5. 9) follows from the construction, and the assumption that
[p, m] ∈ D, m ∈ G3 .
Proof. (of Lemma 2.4) We apply first Lemma 2.3 to obtain saturated and absorbing
∗-homomorphisms Θk : cone(Ak ) → M (D), k = 1, 2, and a norm-continuous path
{ut }t∈[1,∞) of unitaries in M (D) such that limt→∞ ut Θ1 ◦ i1 (b)u∗t − Θ2 ◦ i2 (b) = 0
for all b ∈ cone(B). Define Θ : cone(A1 ) ⊕ cone(A2 ) → M (D) by Θ(a1 , a2 ) =
Θ1 (a1 ) ⊕ Θ2 (a2 ). There is then a norm-continuous path {vt }t∈[0,∞) of unitaries in
M (D) such that vt Θ(i1 (b), 0)vt∗ − Θ(0, i2 (b)) ∈ D for all t ∈ [0, ∞), b ∈ cone(B),
and limt→∞ vt Θ(i1 (b), 0)vt∗ − Θ(0, i2 (b)) = 0 for all b ∈ cone(B). Let F1 ⊆ F2 ⊆
F3 ⊆ · · · and G1 ⊆ G2 ⊆ G3 ⊆ · · · be sequences of finite sets with dense union in
cone(A1 ) ⊕ cone(A2 ) and D, respectively. ! By using Lemma 2.6 we can construct
a sequence 1 = n0 < n1 < n2 < · · · of natural numbers, paths pi (t), t ∈ [i, i + 1],
in Mni (M (D)), i = 0, 1, 2, · · · , and ∗-homomorphisms πei : cone(A1 ) ⊕ cone(A2 ) →
Mni −ni−1 (M (D)), i = 1, 2, · · · , such that π0 = Θ and πi = πi−1 ⊕ πei : cone(A1 ) ⊕
cone(A2 ) → Mni (M (D)), i = 1, 2, · · · , satisfy
1) 0 ≤ pi (t) ≤ 1, t ∈ [i, i + 1], i = 0, 1, 2, · · · ,
2) kpi (t)πi (a) − πi (a)pi (t)k ≤ 1i , a ∈ Fi , t ∈ [i, i + 1], i = 0, 1, 2, · · · ,
3) pi (t)πi (cone(A
cone(A2 )) ⊆ Mni (D),
t ∈ [i, i + 1], i = 0, 1, 2, · · · ,
1) ⊕
b
b
k ≤ 1i when all the entries of b ∈
−
4) kpi+1 (t)
0ni −ni−1
0ni −ni−1
Mni−1 (D) come from Gi , t ∈ [i, i + 1], i = 1, 2, 3, · · · ,
5) (pi (t)2 − pi (t))πi (cone(A1 ) ⊕ cone(A
2 )) = {0}
, t ∈ [i, i + 1], i = 0, 1, 2, · · · ,
pi−1 (i)
0
2
6) pi (i + 1) = pi (i + 1) , pi (i) =
, i = 1, 2, 3, · · · ,
0
0ni −ni−1
7) kpi (t)(1Mni (C) ⊗ vs ) − (1Mni (C) ⊗ vs )pi (t)k ≤ 1i , t ∈ [i, i + 1], s ∈ [0, i + 1], i =
0, 1, 2, · · · ,
and p0 = 0. Note that thanks to the way π1 is constructed in Lemma 2.6 we find
that πei has the form πei = πi−1 ⊕ ϕi ⊕ 0 for some ∗-homomorphism ϕi : cone(A1 ) ⊕
cone(A2 ) → Mni −2ni−1 −1 (M (D)), and that
lim sup k(1Mni (C) ⊗ vt )πi (i1 (b), 0)(1Mni (C) ⊗ vt∗ ) − πi (0, i2 (b))k = 0
t→∞
i
for all b ∈ cone(B). Now define ϕ0 : cone(A1 ) ⊕ cone(A2 ) → LD (l2 (D)) by ϕ0 (a) =
diag(Θ(a), πe1 (a), πe2 (a), πe3 (a), · · · ), and set
pi (t)
0
pt =
, t ∈ [i, i + 1] , i = 0, 1, 2, · · · ,
0∞
6
KLAUS THOMSEN
and wt0 = diag(vt , vt , vt , · · · ), t ∈ [0, ∞). ϕ0 is unitarily equivalent to a ∗-homomorphism
π : cone(A1 ) ⊕ cone(A2 ) → M (D) since l2 (D) ' D as Hilbert D-modules. Via
the isomorphism l2 (D) ' D, p0 and w 0 become paths in M (D) which satisfy 1)-8)
relative to α1 (a) = π(a, 0) and α2 (a) = π(0, a), in the statement of the lemma. It is
easy to see that αk and αk ◦ ik , k = 1, 2, are all absorbing and saturated.
Lemma 2.7. Assume that there are sequences Rnk : Ak → B, n = 1, 2, 3, · · · , k =
1, 2, of completely positive contractions such that lim n→∞ Rnk (ik (b)) = b for all
x ∈ B, k = 1, 2. For any pair of asymptotic homomorphisms ϕ, ψ : cone(B) → D,
there are asymptotic homomorphisms µk : cone(Ak ) → D, k = 1, 2, such that
limt→∞ µ1t ◦ i1 (x) − µ2t ◦ i2 (x) = 0, x ∈ cone(B), and a normcontinuous path of
unitaries {Wt }t∈[1,∞) in M2 (D)+ such that
ψt (x)
ϕt (x)
∗
=0
Wt −
lim Wt
µ1 ◦i1 (x)
µ1 ◦i1 (x)
t→∞
t
t
for all x ∈ cone(B).
Proof. Let ψ̃, ϕ̃ : cone(B) → Cb ([1, ∞), D)/C0([1, ∞), D) be the ∗-homomorphisms
arising from ψ and ϕ, respectively. By Lemma 2.6 of [Th2] there is a stable separable
C ∗ -algebra D0 ⊆ Cb ([1, ∞), D)/C0([1, ∞), D) such that ψ̃(cone(B)) ∪ ϕ̃(cone(B)) ⊆
D0 . For any C ∗ -algebra X, define p : cone(X) → cone(X) and s : cone(X) → SX ⊆
cone(X) by p(f )(t) = f ( 2t ) and
(
f (2t),
t ∈ [0, 12 ]
s(f )(t) =
.
f (2 − 2t), t ∈ [ 21 , 1]
Note that ψ̃ ◦ p|SB and ϕ̃ ◦ p|SB both represent zero in [[SB, D0 ]]. Let pt , wt , α1
and α2 be as in Lemma 2.4, relative to D0 . By Theorem 4.1 in [Th2] there is an
increasing continuous function r : [1, ∞) → [1, ∞) with limt→∞ r(t) = ∞ and a
normcontinuous path {St }t∈[1,∞) of unitaries in M2 (D0 )+ such that
ϕ̃◦p(x)
ψ̃◦p(x)
∗
lim St
=0
pr(t) α1 ◦i1 (x)pr(t) St −
pr(t) α1 ◦i1 (x)pr(t)
t→∞
for all x ∈ SB. Set
Tt =
1
wr(t)
St
1
∗
wr(t)
∗
and lt1 (·) = wr(t) pr(t) α1 (·)pr(t) wr(t)
, lt2 (·) = pr(t) α2 (·)pr(t) . Then lk : cone(Ak ) →
D0 , k = 1, 2, are asymptotic homomorphisms such that limt→∞ lt1 ◦i1 (b)−lt2 ◦i2 (b) = 0
for all b ∈ cone(B) and
ϕ̃◦p(b)
ψ̃◦p(b)
∗
lim Tt
=0
l1 ◦i1 (b) Tt −
l1 ◦i1 (b)
t→∞
t
t
for all b ∈ SB. Note that {Tt }t∈[1,∞) is a normcontinuous path of unitaries in
M2 (D0 )+ . Let χ : Cb ([1, ∞), D + )/C0 ([1, ∞), D +) → Cb ([1, ∞), D + ) be a continuous
right-inverse for the quotient map q : Cb ([1, ∞), D + ) → Cb ([1, ∞), D +)/C0 ([1, ∞), D + ).
Standard arguments give us a normcontinuous path of unitaries {Vt } in M2 (D)+
such that (idM2 ⊗q)(Vt ) = Tt . Set νtk (x) = χ(lsk−1 (t) (x))(t) for a sufficiently rapidly
increasing continuous bijection s : [1, ∞) → [1, ∞). If s increases fast enough, this
will give us asymptotic homomorphisms ν k : cone(Ak ) → D, k = 1, 2, such that
KK-THEORY AND E-THEORY OF AMALGAMATED FREE PRODUCTS
7
limt→∞ νt1 ◦ i1 (x) − νt2 ◦ i2 (x) = 0, x ∈ cone(B), and a normcontinuous path of
unitaries Wt = Vs−1 (t) (t) in M2 (D)+ such that
ψt ◦p(x)
ϕ ◦p(x)
∗
=0
W
−
lim Wt t
t
ν 1 ◦i1 (x)
ν 1 ◦i1 (x)
t→∞
t
t
k
for all x ∈ SB. Set µ = ν ◦ s : cone(Ak ) → D, k = 1, 2. Then µk ◦ ik = ν k ◦ s ◦ ik =
ν k ◦ ik ◦ s and hence limt→∞ µ1t ◦ i1 (x) − µ2t ◦ i2 (x) = 0, x ∈ cone(B), and
ϕt (x)
ψt (x)
∗
lim Wt
=0
µ1 ◦i1 (x) Wt −
µ1 ◦i1 (x)
t→∞
k
t
t
for all x ∈ cone(B).
3. Results in KK-theory
Assume now that A1 , A2 , B and ik : B → Ak , k = 1, 2, are all unital. Let jk :
Ak → A1 ∗B A2 , k = 1, 2, be the natural maps. The basic assumption in this section
is that there is an absorbing ∗-homomorphism α : A1 ∗B A2 → M (D) such that α ◦ jk
and α ◦ jk ◦ ik , k = 1, 2, are all absorbing. (Of course, α ◦ j1 ◦ i1 = α ◦ j2 ◦ i2 .) This
is the case when either,
a) there are conditional expectations Pk : Ak → B, k = 1, 2,
or
b) A1 , A2 and B are all nuclear.
Indeed, in case a) it follows from [Bo] that there are also conditional expectations
idA1 ∗B P2 : A1 ∗B A2 → A1 and P1 ∗B idA2 : A1 ∗B A2 → A2 . Hence by Lemma
2.1 any absorbing ∗-homomorphism α : A1 ∗B A2 → M (D) will have the desired
property. And α exists by Theorem 2.7 of [Th1]. In case b) it suffices to use Lemma
2.2 instead, plus the non-trivial fact that jk : Ak → A1 ∗B A2 , k = 1, 2, are injective,
see Theorem 3.1 of [Bl] or Theorem 4.2 of [P2].
This α will be fixed throughout this section. To simplify notation we set αk =
α ◦ jk , k = 1, 2. Set
Ak = {x ∈ M (D) : xαk (a) − αk (a)x ∈ D, a ∈ Ak },
Bk = {x ∈ Ak : xαk (a) ∈ D, a ∈ Ak },
A = {x ∈ M (D) : xα1 ◦ i1 (b) − α1 ◦ i1 (b)x ∈ D, b ∈ B},
B = {x ∈ A : xα1 ◦ i1 (b) ∈ D, b ∈ B}.
Obviously, Ak ⊆ A, k = 1, 2, and since A1 , A2 , B share the same unit, we see that
B1 = B2 = B. Hence Ak /Bk ⊆ A/B, k = 1, 2. By Theorem 3.2 of [Th1] we can make
the following identifications
KK(Ak , D) = K1 (Ak /Bk ), k = 1, 2,
KK(B, D) = K1 (A/B).
In the following, when given a ∗-homomorphism ϕ : E → F between C ∗ -algebras,
we will denote the ∗-homomorphism E → Mn (F ) given by
E 3 e 7→ diag(ϕ(e), ϕ(e), · · · , ϕ(e))
by [1n ⊗ ϕ]. Let qD : M (D) → Q(D) = M (D)/D be the quotient map. We
define a map ρ : KK(B, D) → Ext−1 (A1 ∗B A2 , D) in the following way. Let
8
KLAUS THOMSEN
u be a unitary Mn (A/B) for some n, and let ũ ∈ Mn (A) be a lift of u. Then
ũ[1n ⊗ α1 ◦ i1 ](b)ũ∗ − [1n ⊗ α2 ◦ i2 ](b) ∈ Mn (D) for all b ∈ B and ũũ∗ [1 ⊗ α1 ](a) =
ũ∗ ũ[1n ⊗ α1 ](a) = [1 ⊗ α1 ](a) modulo Mn (D) for all a ∈ A1 , so
ρ(u) = (qMn (D) ◦ Ad ũ ◦ [1n ⊗ α1 ]) ∗B (qMn (D) ◦ [1 ⊗ α2 ])
is a well-defined extension, ρ(u) : A1 ∗B A2 → Q(Mn (D)) ' Q(D).
Lemma 3.1. ρ(u) is an invertible extension.
Proof. We assert that ρ(u) ρ(u∗ ) is a split extension. To see this choose first a
unitary lift w ∈ M2n (A) of ( u u∗ ). Then
ρ(u)
= (qM2n (D) ◦ Ad w ◦ [12n ⊗ α1 ]) ∗B (qM2n (D) ◦ [12n ⊗ α2 ]).
∗
ρ(u )
As is wellknown, there is a continuous path of unitaries in M2n (A) connecting w to
a unitary w0 ∈ B + . Set w1 = ww0∗ ∈ M2n (A), and observe that
ρ(u) ⊕ ρ(u∗ ) = (qM2n (D) ◦ Ad w1 ◦ [12n ⊗ α1 ]) ∗B (qM2n (D) ◦ [12n ⊗ α2 ]).
Note that Ad w1 leaves [12n ⊗ α1 ◦ i1 ]+ (B + ) + M2n (D) globally invariant and that
the path of unitaries in M2n (A) connecting w1 to 1 shows that the automorphism
of [12n ⊗ α1 ◦ i1 ]+ (B + ) + M2n (D) given by Ad w1 is homotopic to the identity in the
uniform normtopology. Consequently this automorphism is inner by Corollary 8.7.8
of [P1], i.e. there is a unitary T ∈ [12n ⊗ α1 ◦ i1 ]+ (B + ) + M2n (D) such that w1 xw1∗ =
T xT ∗ for all x ∈ [12n ⊗ α1 ◦ i1 ]+ (B + ) + M2n (D). Write T = [12n ⊗ α1 ◦ i1 ]+ (S) + d,
where S ∈ B + and d ∈ M2n (D). Since α1 ◦ i1 is absorbing, qM2n (D) ◦ [12n ⊗ α1 ◦ i1 ]+
is injective, so we conclude that S is a unitary. Furthermore, since
qM2n (D) ◦ [12n ⊗ α1 ◦ i1 ](b)
= qM2n (D) ◦ Ad w1 ◦ [12n ⊗ α1 ◦ i1 ](b) = qM2n (D) ◦ [12n ⊗ α1 ◦ i1 ](SbS ∗ )
for all b ∈ B, we conclude that SbS ∗ = b for all b ∈ B. We can therefore define an
∗ +
automorphism Φ of A1 ∗B A2 such that Φ ◦ j1 (x) = j1 (i+
1 (S) xi1 (S)), x ∈ A1 , and
Φ ◦ j2 (y) = j2 (y), y ∈ A2 . Then
ρ(u)
∗
ρ(u ) ◦ Φ
= (qM2n (D) ◦ Ad(w1 [12n ⊗ α1 ◦ i1 ]+ (S)∗ ) ◦ [12n ⊗ α1 ]) ∗B (qM2n (D) ◦ [12n ⊗ α2 ])
= (qM2n (D) ◦ Ad(w1 T ∗ ) ◦ [12n ⊗ α1 ]) ∗B (qM2n (D) ◦ [12n ⊗ α2 ])
∗
which admits the
1 T ) ◦ [12n ⊗ α1 ]) ∗B [12n ⊗ α2 ]) : A1 ∗B A2 → M2n (M (D)).
lift (Ad(w
It follows that ρ(u) ρ(u∗ ) admits the lift (Ad(w1 T ∗ ) ◦ [12n ⊗ α1 ]) ∗B [12n ⊗ α2 ]) ◦ Φ−1 .
Given Lemma 3.1 it is clear that the construction gives us a homomorphism
ρ : KK(B, D) → Ext−1 (A1 ∗B A2 , D).
Lemma 3.2.
KK(A1 , B) ⊕ KK(A2 , B)
i∗1 −i∗2
/
ρ
KK(B, D)
/
Ext−1 (A1 ∗B A2 , D)
(j1∗ ,j2∗ )
Ext−1 (B, D) o i∗ −i∗ Ext−1 (A1 , D) ⊕ Ext−1 (A2 , D)
1
2
KK-THEORY AND E-THEORY OF AMALGAMATED FREE PRODUCTS
9
is exact.
Proof. Exactness at KK(B, D): Consider elements vk ∈ Mn (Ak ) that are unitaries
modulo Mn (Bk ). Then
(qMn (D) ◦ Ad v2∗ v1 ◦ [1n ⊗ α1 ]) ∗B (qMn (D) ◦ [1 ⊗ α2 ])
is unitarily equivalent to
(qMn (D) ◦ Ad v1 ◦ [1n ⊗ α1 ]) ∗B (qMn (D) ◦ Ad v2 ◦ [1 ⊗ α2 ]) =
(qMn (D) ◦ [1n ⊗ α1 ]) ∗B (qMn (D) ◦ [1 ⊗ α2 ]) = qMn (D) ◦ [1n ⊗ α],
which is a split extension. This shows that ρ ◦ (i∗1 − i∗2 ) = 0. Consider then a
unitary u ∈ Mn (A/B) and assume that [ρ(u)] = 0 in Ext−1 (A1 ∗B A2 , D). Since α
is absorbing this implies that
q
ρ(u)
Mn (D) ◦[1n ⊗α]
Ad qMn+1 (D) (W ) ◦
qD ◦α =
qD ◦α
for some unitary W ∈ Mn+1 (M (D)). Alternatively,
qMn (D) ◦[1⊗α1 ]
qMn (D) ◦Ad ũ◦[1n ⊗α1 ]
=
Ad qMn+1 (D) (W ) ◦
qD ◦α1
qD ◦α1
and
Ad qMn+1 (D) (W ) ◦
qMn (D) ◦[1n ⊗α2 ]
qD ◦α2
=
qMn (D) ◦[1⊗α2 ]
qD ◦α2
Hence W and W ( ũ 1 ) represent unitaries in Mn+1 (A2 /B2 ) and Mn+1 (A1 /B1 ), respectively, and since the product of their images in Mn+1 (A/B) is ( u 1 ), we conclude
that [u] is in the range of i∗1 − i∗2 .
Exactness at Ext−1 (A1 ∗B A2 , D): It is obvious that the composition (j1∗ , j2∗ ) ◦ ρ is
zero, so consider an extension - a priori not necessarily invertible - ϕ : A1 ∗B A2 →
Q(D) with the property that ϕ ◦ jk , k = 1, 2, are both split. Since αk is absorbing
there are unitaries Sk ∈ M2 (M (D)) such that
Ad qM2 (D) (Sk ) ◦ ϕ◦jk qD ◦αk = ( qD ◦αk qD ◦αk ) ,
∗
k = 1, 2. It follows that S2 S1∗ ∈ M2 (A) and that ( ϕ qD ◦α ) is unitarily equivalent
to (Ad qM2 (D) (S2 S1∗ ) ◦ [12 ⊗ α1 ]) ∗B [12 ⊗ α2 ] which is clearly in the range of ρ. In
particular, ϕ is invertible afterall, and we have exactness at Ext −1 (A1 ∗B A2 , D).
Exactness at Ext−1 (A1 , D) ⊕ Ext−1 (A2 , D) : It is trivial that (i∗1 − i∗2 ) ◦ (j1∗ , j2∗ ) = 0,
so consider a pair of invertible extensions ϕk : Ak → Q(B), k = 1, 2, with the
property that i∗1 [ϕ1 ] = i∗2 [ϕ2 ]. There is then a unitary S ∈ M2 (M (D)) such that
Ad qM2 (D) (S) ◦ ϕ1 ◦i1 qD ◦α1 ◦i1 = ϕ2 ◦i2 qD ◦α2 ◦i2 .
After adding qD ◦ αk to ϕk we may assume that ϕ1 ◦ i1 = ϕ2 ◦ i2 . Similarly, if
ψk : Ak → Q(D) represents the inverse of ϕk in Ext−1 (Ak , D), k = 1, 2, we may
assume that ψ1 ◦ i1 = ψ2 ◦ i2 . We can then consider the two extensions ϕ1 ∗B
ϕ2 , ψ1 ∗B ψ2 : A1 ∗B A2 → Q(B) whose sum µ = (ϕ1 ∗B ϕ2 ) ⊕ (ψ1 ∗B ψ2 ) has the
property that µ ◦ jk : Ak → Q(D), k = 1, 2, both split. By the arguments in the last
paragraph we conclude that µ and hence also ϕ1 ∗B ϕ2 is an invertible extension.
Since ϕk = (ϕ1 ∗B ϕ2 ) ◦ jk , k = 1, 2, the proof is complete.
Theorem 3.3. Let A1 , A2 , B be separable C ∗ -algebras. Assume that ik : B →
Ak , k = 1, 2, are embeddings, and that there are conditional expectations P k : Ak →
ik (B), k = 1, 2, or that A1 , A2 and B are all nuclear. Let jk : Ak → A1 ∗B A2 , k = 1, 2,
10
KLAUS THOMSEN
be the natural maps. For any separable C ∗ -algebra D there are six terms exact
sequences
KK(D, B)
(i1 ∗ ,i2∗ )
/
O
KK(SD, A1 ∗B A2 )jo
1 ∗ −j2 ∗
KK(D, A1 ) ⊕ KK(D, A2 )
j1 ∗ −j2 ∗
KK(SD, A1 ) ⊕ KK(SD, A2 ) o
/
KK(D, A1 ∗B A2 )
(i1 ∗ ,i2∗ )
KK(SD, B)
and
KK(B, D) o
i1 ∗ −i2 ∗
KK(A1 ∗B A2 , SD)
KK(A1 , D) ⊕ KK(A2 , D) o
/
(j1∗ ,j2∗ )
(j1∗ ,j2∗ )
KK(A1 , SD) ⊕ KK(A2 , SD)
KK(A1 ∗B A2 , D)
O
i1 ∗ −i2 ∗
/
KK(B, SD)
Proof. Consider first the case where A1 , A2 and B share the same unit, and let
ϕ : C → S(A1 ∗B A2 ) be Germain’s ∗-homomorphism, cf. [G3], where C is the
mapping cone for the embedding B → A1 ⊕ A2 . ϕ relates the Puppe exact sequence
of Theorem 1 in [CS] to the exact sequence in Lemma 3.2 in such a way that we can
conclude from the five lemma that ϕ∗ : KK(S(A1 ∗B A2 ), D) → KK(C, D) is an
isomorphism. Since D is arbitrary here, standard KK-theory arguments show that
+
[ϕ] ∈ KK(C, S(A1 ∗B A2 )) is invertible. By using that (A1 ∗B A2 )+ = A+
1 ∗B + A2 it
follows straightforwardly that [ϕ] is also invertible in the general (non-unital) case.
As pointed out by Germain in [G3], this completes the proof.
4. An appropriate picture of the E-theory groups
Let A, D be separable C ∗ -algebras, D stable. In this section an E-pair for (A, D)
will be a pair (W, ϕ), where ϕ : cone(A) → D is an asymptotic homomorphism and
W = {Wt }t∈[1,∞) is a strictly continuous path of unitaries in M (D) such that
lim kWt ϕt (a) − ϕt (a)Wt k = 0
t→∞
(4.1)
for all a ∈ SA. The pair (W, ϕ) is degenerate when (4.1) holds for all a ∈ cone(A).
We let X0 (A, D) denote the set of homotopy classes of E-pairs, where a homotopy
is given by an E-pair for (A, C[0, 1] ⊗ D). The direct sum of E-pairs, performed
with the aid of any pair V1 , V2 of isometries in M (D) such that V1∗ V2 = 0 and
V1 V1∗ + V2 V2∗ = 1, makes obviously X0 (A, D) into an abelian semigroup. The subsemigroup of X0 (A, D) consisting of the elements of X0 (A, D) that can be represented by a degenerate E-pair will be denoted by X00 (A, D). The quotient semigroup X(A, D) = X0 (A, D)/X00 (A, D) is then an abelian group; a standard rotation
argument shows that (W, ϕ) ⊕ (W ∗ , ϕ) is homotopic to a degenerate E-pair. Define
a map κ : X(A, D) → [[S 2 A, D]] in the following way. Given an E-pair (W, ϕ) we
can define an asymptotic homomorphism W ⊗ ϕ : C(T) ⊗ SA → D suc! h that
lim (W ⊗ ϕ)t (g ⊗ f ) − g(Wt )ϕt (f ) = 0
t→∞
for all g ∈ C(T), f ∈ SA. We set κ[W, ϕ] = i∗ [W ⊗ϕ], where i : S 2 A → C(T)⊗SA is
the canonical embedding. To show that κ is an isomorphism, we need two lemmas.
KK-THEORY AND E-THEORY OF AMALGAMATED FREE PRODUCTS
11
Lemma 4.1. Let λ : C(T) ⊗ A → D be an asymptotic homomorphism. There is
then a strictly continuous path W = {Wt }t∈[1,∞) of unitaries in M2 (M (D)) such that
lim g(Wt ) ϕt (1C(T) ⊗a) 0 − ϕt (g⊗a) 0 = 0
t→∞
for all g ∈ C(T), a ∈ A.
Proof. Let ϕ1 : C(T) ⊗ A → Cb ([1, ∞), D)/C0 ([1, ∞), D) be the ∗-homomorphism
defined from ϕ in the usual way. Set H = ϕ1 (C(T) ⊗ A), and H0 = q −1 (H) ⊆
Cb ([1, ∞), D), where q : Cb ([1, ∞), D) → Cb ([1, ∞), D)/C0([1, ∞), D) is the quotient
map. Since ϕ1 and q extend to surjections ϕ1 : M (C(T) ⊗ A) → M
(H) and q:
M (H0 ) → M (H), we can find a unitary lift W ∈ M2 (M (H0 )) of ϕ1 (z) ϕ1 (z ∗ ) ,
where z ∈ M (C(T) ⊗ A) is the element given by the identity function on T. Since
M (M2 (H0 )) ⊆ M (M2 (C0 ([1, ∞), D))), W is given by a strictly continuous path of
unitaries in M2 (M (D)) with the desired property.
Let c : SA → C(T) ⊗ SA be the ∗-homomorphism c(f ) = 1C(T) ⊗ f .
Lemma 4.2. Let ψ : C(T) ⊗ SA → D be an asymptotic homomorphism such
that c∗ [ψ] = 0 in [[SA, D]]. It follows that there are asymptotic homomorphisms
ψ 00 : C(T) ⊗ cone(A) → D, ψ 0 : cone(A) → D and a strictly continuous path
{Wt }t∈[1,∞) of unitaries in M (D) such that
[ψt ⊕ ψt00 ](g ⊗ f ) − g(Wt )ψt0 (f ) = 0
for all g ∈ C(T), f ∈ SA.
Proof. Since c∗ [ψ] = 0 it follows from Theorem 4.2 of [Th2] that there is an asymptotic homomorphism ν : cone(A) → D and a normcontinuous path of unitaries
{Ut }t∈[1,∞) ⊆ M2 (D)+ such that
ψ (1
⊗f )
lim Ut t C(T)
Ut∗ − 0 νt (f ) = 0
νt (f )
t→∞
for all f ∈ SA. Let ev : C(T) ⊗ SA → SA be the ∗-homomorphism obtained by
evaluation at some point in T. It follows from Lemma 4.1 that there is a strictly
continuous path {Wt }t∈[1,∞) of unitaries in M (D) such that
lim g(Wt )(ψt ⊕ νt ◦ ev ⊕ 0)(1 ⊗ f ) − (ψt ⊕ νt ◦ ev ⊕ 0)(g ⊗ f ) = 0.
t→∞
Set ψ 00 = ν ◦ ev ⊕ 0 and ψ 0 = Ad(U ⊕ 1)∗ ◦ (0 ⊕ ν ⊕ 0).
To use these two lemmas to define a map δ : [[S 2 A, D]] → X(A, D), we remind
the reader that [[S−, D]] = E(S−, D), cf. [DL]. In particular, the contravariant
functor [[S−, D]] is split-exact, and this will be used now. Let ψ : S 2 A → D
be an asymptotic homomorphism. There is then an asymptotic homomorphism
ϕ : C(T) ⊗ SA → D such that c∗ [ϕ] = 0 and [ψ] = [ϕ ◦ i] in [[S 2 A, B]]. Since
c∗ [ϕ] = 0, Lemma 4.2 gives us asymptotic homomorphisms ϕ00 : C(T) ⊗ cone(A) →
D, ϕ0 : cone(A) → D and a strictly continuous path {Wt }t∈[1,∞) of unitaries in
M (D) such that
[ϕt ⊕ ϕ00t ](g ⊗ f ) − g(Wt )ϕ0t (f ) = 0
for all g ∈ C(T), f ∈ SA. Then (W, ϕ0 ) is an E-pair and we claim that we can
define δ such that δ[ψ] = [W, ϕ0 ]. To see this, the only non-trivial point is to
show that the class of (W, ϕ0 ) is independent of the choices made. So assume that
12
KLAUS THOMSEN
λ00 : C(T) ⊗ cone(A) → D, λ0 : cone(A) → D are asymptotic homomorphisms and
{St }t∈[1,∞) a strictly continuous path of unitaries in M (D) such that
[ϕt ⊕ λ00t ](g ⊗ f ) − g(St )λ0t (f ) = 0
for all g ∈ C(T), f ∈ SA. By Lemma 4.1 there are strictly continuous pathes,
{Yt }t∈[1,∞) , {Xt }t∈[1,∞) of unitaries in M (D) such that
lim g(Yt )ϕ00t (1 ⊗ f ) − ϕ00t (g ⊗ f ) = 0,
t→∞
lim g(Xt )λ00t (1 ⊗ f ) − λ00t (g ⊗ f ) = 0
t→∞
for all g ∈ C(T), f ∈ cone(A). Since (Y, ϕ00 ◦c) and (X, λ00 ◦c) are degenerate E-pairs,
, and
[W, ϕ0 ] = [W, ϕ0 ] + [X, λ00 ◦ c] = [W ⊕ X, ϕ0 ⊕ λ00 ◦ c]
[S, λ0 ] = [S, λ0 ] + [Y, ϕ00 ◦ c] = [S ⊕ Y, λ0 ⊕ ϕ00 ◦ c]
in X(A, D). Conjugating the pair (S ⊕ Y, λ0 ⊕ ϕ00 ◦ c) by a unitary, we see that
[W, ϕ0 ] = [W 1 , ϕ1 ] and [S, λ0 ] = [W 2 , ϕ2 ], where the E-pairs (W 1 , ϕ1 ) and (W 2 , ϕ2 )
are related such that
lim g(Wt1 )ϕ1t (f ) − g(Wt2 )ϕ2t (f ) = 0
t→∞
for all g ∈ C(T), f ∈ SA. In particular, limt→∞ ϕ1t (f ) − ϕ2t (f ) = 0 for all f ∈ SA,
∗
so a standard rotation argument shows that (W 1 ⊕ W 2 , ϕ1 ⊕ ϕ2 ) is homotopic to
∗
(W 2 W 1 ⊕ 1, ϕ1 ⊕ ϕ2 ). This shows that [W 1 , ϕ1 ] − [W 2 , ϕ2 ] is represented by an
E-pair (V, µ) where limt→∞ g(Vt )µt (f ) − g(1)µt (f ) = 0 for all g ∈ C(T), f ∈ SA.
By rotation we find that [V, µ] + [1, 0] = [V, 0] + [1, µ] = 0 in X(A, D). Hence
[W, ϕ0 ] = [S, λ0 ] and we conclude that δ is well-defined. Since δ is clearly an inverse
for κ, we have obtained the following proposition.
Proposition 4.3. κ : X(A, D) → [[S 2 A, D]] is an isomorphism with inverse δ.
5. cone(A1 ) ∗SB cone(A2 ) is equivalent to B in E-theory
Assume now that ik : B → Ak , k = 1, 2, are proper embeddings, i.e. that ik (B)Ak
spans a dense subspace in Ak , and that there are sequences Rnk : Ak → B, n =
1, 2, 3, · · · , k = 1, 2, of completely positive contractions such that lim n→∞ Rnk (ik (x)) =
x for all x ∈ B, k = 1, 2. By Theorem 5.5 of [P2] we have natural isomorphisms
S(A1 ∗B A2 ) = SA1 ∗SB SA2 , cone(A1 ∗B A2 ) = cone(A1 ) ∗cone(B) cone(A2 ).
We can then define a map X(B, D) → [[cone(A1 ) ∗SB cone(A2 ), D]] in the following
way. Let (W, ϕ) be an E-pair. It follows from Lemma 2.7 that there are asymptotic
homomorphisms µk : cone(Ak ) → D and ν k : cone(Ak ) → D, k = 1, 2, such that
limt→∞ µ1t ◦ i1 (x) − µ2t ◦ i2 (x) = limt→∞ νt1 ◦ i1 (x) − νt2 ◦ i2 (x) = 0 and
lim ϕt (x) ⊕ µ1t ◦ i1 (x) − νt1 ◦ i1 (x) = 0
t→∞
for all x ∈ cone(B). Since limt→∞ Ad(Wt ⊕ 1) ◦ νt1 ◦ i1 (x) − νt2 ◦ i2 (x) = 0 for all
x ∈ SB, there is an asymptotic homomorphism
(Ad(W ⊕ 1) ◦ ν 1 ) ∗SB ν 2 : cone(A1 ) ∗SB cone(A2 ) → D
such that limt→∞ ((Ad(W ⊕ 1) ◦ ν 1 ) ∗SB ν 2 )t ◦ j1 (x) − Ad(Wt ⊕ 1) ◦ νt1 ◦ j1 (x) = 0 for
all x ∈ cone(A1 ) and limt→∞ ((Ad(W ⊕ 1) ◦ ν 1 ) ∗SB ν 2 )t ◦ j2 (x) − νt2 ◦ j2 (x) = 0 for
KK-THEORY AND E-THEORY OF AMALGAMATED FREE PRODUCTS
13
all x ∈ cone(A2 ). We claim that we can define a map ρ : X(B, D) → [[cone(A1 ) ∗SB
cone(A2 ), D]] such that ρ[W, ϕ] = [(Ad(W ⊕ 1) ◦ ν 1 ) ∗SB ν 2 ]. If κk : cone(Ak ) → D
and lk : cone(Ak ) → D, k = 1, 2, are other asymptotic homomorphisms such that
limt→∞ lt1 ◦ i1 (x) − lt2 ◦ i2 (x) = limt→∞ κ1t ◦ i1 (x) − κ2t ◦ i2 (x) = 0 and
lim ϕt (x) ⊕ κ1t ◦ i1 (x) − lt1 ◦ i1 (x) = 0
t→∞
for all x ∈ cone(B), observe first that κ1 ∗SB κ2 , l1 ∗SB l2 and µ1 ∗SB µ2 are all restrictions of asymptotic homomorphisms defined on cone(A1 ∗B A2 ) = cone(A1 ) ∗cone(B)
cone(A2 ), and hence null-homotopic. Consequently
[(Ad(W ⊕1)◦ν 1 )∗SB ν 2 ] = [((Ad(W ⊕1)◦ν 1 )∗SB ν 2 )⊕(κ1 ∗SB κ2 )⊕(l1 ∗SB l2 )⊕(µ1 ∗SB µ2 )].
Note that there is a unitary S ∈ M (D) which commutes both with W ⊕1⊕1⊕1⊕1⊕1
and 1 ⊕ 1 ⊕ 1 ⊕ W ⊕ 1 ⊕ 1, and has the property that
lim Ad S ◦ (((Ad(W ⊕ 1) ◦ ν 1 ) ∗SB ν 2 ) ⊕ (κ1 ∗SB κ2 ) ⊕ (l1 ∗SB l2 ) ⊕ (µ1 ∗SB µ2 ))t ◦ jk ◦ ik (x)
t→∞
− (ϕ ⊕ κ1 ⊕ µ1 ⊕ ϕ ⊕ κ1 ⊕ µ1 )t ◦ ik (x) = 0
for all x ∈ SB, k = 1, 2. By first ’rotating’ W ⊕1⊕1⊕1⊕1⊕1 to 1⊕1⊕1⊕W ⊕1⊕1
and then connecting S to 1 through a strictly continuous path of unitaries in M (D),
we get a homotopy connecting
((Ad(W ⊕ 1) ◦ ν 1 ) ∗SB ν 2 ) ⊕ (κ1 ∗SB κ2 ) ⊕ (l1 ∗SB l2 ) ⊕ (µ1 ∗SB µ2 )
to
(ν 1 ∗SB ν 2 ) ⊕ (κ1 ∗SB κ2 ) ⊕ ((Ad(W ⊕ 1) ◦ l1 ) ∗SB l2 ) ⊕ (µ1 ∗SB µ2 ).
Hence [(Ad(W ⊕ 1) ◦ ν 1 ) ∗SB ν 2 ] = [(Ad(W ⊕ 1) ◦ l1 ) ∗SB l2 ], and we conclude that ρ
is a well-defined homomorphism. Note, however, that at this point we do not know
that [[cone(A1 ) ∗SB cone(A2 ), D]] is a group.
To obtain an inverse to ρ, consider an asymptotic homomorphism ψ : cone(A1 )∗SB
cone(A2 ) → D. By Lemma 2.7 there are asymptotic homomorphisms ν k : cone(Ak ) →
D, k = 1, 2, such that limt→∞ νt1 ◦ i1 (x) − νt2 ◦ i2 (x) = 0 for all x ∈ cone(B) and a
normcontinuous path of unitaries {Wt } in D + such that
lim Wt (ψt ◦ j1 ◦ i1 (x) ⊕ νt1 ◦ i1 (x))Wt∗ − ψt ◦ j2 ◦ i2 (x) ⊕ νt2 ◦ i2 (x) = 0
t→∞
for all x ∈ cone(B). Then (W ∗ , ψ ◦ j2 ◦ i2 ⊕ ν 2 ◦ i2 ) is an E-pair, and we claim that
δ[ψ] = [W ∗ , ψ ◦j2 ◦i2 ⊕ν 2 ◦i2 ] is a welldefined map δ : [[cone(A1 )∗SB cone(A2 ), D]] →
X(B, D). To see this, let µk : cone(Ak ) → D, k = 1, 2, be another pair of asymptotic
homomorphisms and {St } another normcontinuous path of unitaries in D + such that
lim St (ψt ◦ j1 ◦ i1 (x) ⊕ µ1t ◦ i1 (x))St∗ − ψt ◦ j2 ◦ i2 (x) ⊕ µ2t ◦ i2 (x) = 0
t→∞
for all x ∈ cone(B). There is a unitary in M (D) conjugating (S, ψ ◦ j2 ◦ i2 ⊕ µ2 ◦
i2 ) ⊕ (1, ν 2 ◦ i2 ) to (T, ψ ◦ j2 ◦ i2 ⊕ ν 2 ◦ i2 ⊕ µ2 ◦ i2 ), where T ∈ D + is a normcontinuous
path of unitaries in D + such that
lim Tt (ψt ◦j1 ◦i1 (x)⊕νt1 ◦i1 (x)⊕µ1t ◦i1 (x))Tt∗ −ψt ◦j2 ◦i2 (x)⊕νt2 ◦i2 (x)⊕µ2t ◦i2 (x) = 0
t→∞
14
KLAUS THOMSEN
for all x ∈ cone(B). Then a standard rotation argument shows that
[W ∗ , ψ ◦ j2 ◦ i2 ⊕ ν 2 ◦ i2 ] + [S, ψ ◦ j2 ◦ i2 ⊕ µ2 ◦ i2 ]
= [W ∗ , ψ ◦ j2 ◦ i2 ⊕ ν 2 ◦ i2 ] + [1, µ2 ◦ i2 ] + [S, ψ ◦ j2 ◦ i2 ⊕ µ2 ◦ i2 ] + [1, ν 2 ◦ i2 ]
= [T (W ∗ ⊕ 1), ψ ◦ j2 ◦ i2 ⊕ ν 2 ◦ i2 ⊕ µ2 ◦ i2 ] + [1, ψ ◦ j2 ◦ i2 ⊕ ν 2 ◦ i2 ⊕ µ2 ◦ i2 ]
=0
in X(B, D). Since X(A, B) is a group we deduce that
[W ∗ , ψ ◦ j2 ◦ i2 ⊕ ν 2 ◦ i2 ] = [S ∗ , ψ ◦ j2 ◦ i2 ⊕ µ2 ◦ i2 ],
proving that δ is well-defined. It is straightforward to see that δ is an inverse to ρ
so we have proved the following
Lemma 5.1. [[cone(A1 )∗SB cone(A2 ), D]] is a group, and ρ : X(B, D) → [[cone(A1 )∗SB
cone(A2 ), D]] is an isomorphism.
Note that it follows from Lemma 5.1 and [DL] that E(cone(A1 )∗SB cone(A2 ), D) =
[[cone(A1 ) ∗SB cone(A2 ), D]].
Theorem 5.2. Assume that ik : B → Ak , k = 1, 2, are proper embeddings, and that
there are sequences Rnk : Ak → B, n = 1, 2, 3, · · · , k = 1, 2, of completely positive
contractions such that limn→∞ Rnk (ik (x)) = x for all x ∈ B, k = 1, 2. There is an
asymptotic homomorphism Φ : cone(A1 ) ∗SB cone(A2 ) → B ⊗ K which is invertible
in E-theory.
Proof. By Proposition 4.3 and Lemma 5.1, ρ ◦ κ−1 : [[S 2 B, D]] → [[cone(A1 ) ∗SB
cone(A2 ), D]] is an isomorphism. Let ϕ : S 2 B → B ⊗ K be the asymptotic homomorphism obtained by applying the Connes-Higson construction to the Toeplitzextension tensored with B. (K denotes the C ∗ -algebra of compact operators on l2 .)
Let Φ be an asymptotic homomorphism such that [Φ] = ρ ◦ κ−1 [ϕ] in [[cone(A1 ) ∗SB
cone(A2 ), B ⊗ K]]. Standard KK- and E-theory arguments show that Φ must be
invertible in E-theory because of Lemma 5.1.
6. Results in E-theory
Theorem 6.1. Let A1 , A2 , B be separable C ∗ -algebras. Assume that ik : B →
Ak , k = 1, 2, are embeddings, and that there are sequences Rnk : Ak → B, n =
1, 2, 3, · · · , k = 1, 2, of completely positive contractions such that limn→∞ Rnk (ik (x)) =
x for all x ∈ B, k = 1, 2. Let jk : Ak → A1 ∗B A2 , k = 1, 2, be the natural maps. For
any separable C ∗ -algebra D there are six terms exact sequences
E(D, B)
(i1 ∗ ,i2∗ )
/
O
E(SD, A1 ∗B A2 )jo
1 ∗ −j2 ∗
E(D, A1 ) ⊕ E(D, A2 )
j1 ∗ −j2 ∗
E(SD, A1 ) ⊕ E(SD, A2 ) o
/
E(D, A1 ∗B A2 )
(i1 ∗ ,i2∗ )
E(SD, B)
and
E(B, D) o
i1 ∗ −i2 ∗
E(A1 ∗B A2 , SD)
/
E(A1 , D) ⊕ E(A2 , D) o
(j1∗ ,j2∗ )
(j1∗ ,j2∗ )
E(A1 , SD) ⊕ E(A2 , SD)
E(A1 ∗B A2 , D)
O
i1 ∗ −i2 ∗
/
E(B, SD)
KK-THEORY AND E-THEORY OF AMALGAMATED FREE PRODUCTS
15
+
+
∗
Proof. Let A+
1 , A2 , B denote the C -algebras obtained by adjoining units to A1 , A2
+
and B. Let X denote the kernel of the natural map cone(A+
1 ) ∗S(B + ) cone(A2 ) →
+
cone(C) ∗SC cone(C) and Y the kernel of the natural map A+
1 ∗ A2 → C ∗ C. There
is then a commuting diagram
0
0
/
0
0
0
/
S(A1 ∗B A2 )
S(A+
1
/
∗B +
SC
0
A+
2)
/
/
cone(A+
1)
/
0
/
X
∗S(B + ) cone(A+
2)
cone(C) ∗SC cone(C)
0
/
/
0
Y
+
A+
1 ∗ A2
/
C∗C
/
/
0
0
0
(6.1)
+
+
+
By [C2] A+
∗
A
and
C
∗
C
are
KK-equivalent
to
A
⊕
A
and
C
⊕
C,
respectively,
1
2
1
2
so we find that Y is equivalent to A1 ⊕ A2 in E-theory. In the same way it follows from Theorem 5.2 that X is equivalent to B in E-theory. The two six terms
exact sequences of the theorem now arise by writing down the two six terms exact
sequences of E-theory coming from the first row in (6.1), substituting A1 ⊕ A2 for
Y and B for X, and finally identifying the resulting maps in the diagram. We leave
this to the reader.
7. Conclusion
As pointed out by Germain in [G3] the six terms exact sequences of Theorem 3.3
imply that S(A1 ∗B A2 ) is KK-equivalent to the mapping cone C of the embedding
B → A1 ⊕ A2 (and vice versa, essentially). It follows therefore from Theorem 3.3
that S(A1 ∗B A2 ) is equivalent to C in E-theory under the assumptions of that
theorem, and the six terms exact sequences of Theorem 6.1 follow from this by
writing down the E-theory Puppe sequences for the inclusion B → A1 ⊕ A2 . In
other words, Theorem 6.1 is a consequence of Theorem 3.3 when there are conditional
expectations from the Ak ’s onto B, and when A1 , A2 and B are all nuclear. But the
assumptions of Theorem 6.1 are much weaker than this; it suffices for example that
B is nuclear, or that B sits as a hereditary C ∗ -subalgebra of the Ak ’s. Nonetheless
it would be nice to be able to remove the condition in Theorem 6.1 alltogether, and!
in Theorem 3.3 for that matter. Let us therefore conclude by pointing out that the
methods we have used can not, without some serious adjustments, give the six terms
exact sequences of Theorem 6.1 in full generality. It is clear that the assumption
of Theorem 6.1 was used above to guarantee that some absorbing ∗-homomorphism
cone(B) → M (D) can be extended to a ∗-homomorphism cone(Ak ) → M (B). The
existence of such an extension will not exist in general. To see this, observe that
if B ⊆ A are separable C ∗ -algebras and D is a stable separable C ∗ -algebra then
there can only be a ∗-homomorphism π : A → M (D) such that π|B : B → M (D) is
16
KLAUS THOMSEN
absorbing when
{ϕ|B : ϕ : A → D is a completely positive contraction}
is dense for the topology of pointwise normconvergence among all the completely
positive contractions B → D, see [Th1]. (In fact, this condition is also sufficient.)
Now consider a separable exact C ∗ -algebra B for which Ext(cone(B)) is not a group
- such C ∗ -algebras exist in abundance by [Ki1]. Then B ⊆ A for some nuclear
separable C ∗ -algebra A; in fact one can take A = O2 , cf. [Ki2]. That Ext(cone(B))
is not a group means that there is a ∗-homomorphism χ : cone(B) → Q ( = the
Calkin algebra) which does not lift to a completely positive map cone(B) → B(l2 ).
Consider D = χ(cone(B)) ⊗ K which is certainly a separable stable C ∗ -algebra. If
ϕn : cone(A) → D, n ∈ N, is a sequence of completely positive contractions such that
limn→∞ ϕn (b) = χ(b) ⊗ e11 , b ∈ cone(B), we would clearly also have a sequence of
completely positive con! tractions ψn : cone(A) → Q such that limn→∞ ψn (b) = χ(b)
for all b ∈ cone(B). Since cone(A) is nuclear each ψn would be liftable in the sense of
[A2] and hence this would force χ to be liftable by Theorem 6 in [A2], contradicting
the choice of it. It follows that for such an inclusion B ⊆ A the approach we have
taken to prove Theorem 6.1 does not suffice to prove the general conjecture in Etheory. The method we use to prove Theorem 3.3 require even more; namely that
we can extend some absorbing ∗-homomorphism out of B, not only to A1 and A2 ,
but all the way to A1 ∗B A2 . The obstacle we have just identified is therefore even
more serious in regards to Theorem 3.3.
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[Bl]
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E-mail address: matkt@imf.au.dk