ON THE KK-THEORY OF STRONGLY SELF-ABSORBING
C∗-ALGEBRAS
MARIUS DADARLAT AND WILHELM WINTER
Abstract. Let D and A be unital and separable C∗-algebras; let D be strongly self-
absorbing. It is known that any two unital ∗-homomorphisms from D to A ⊗ D are
approximately unitarily equivalent. We show that, if D is also K1-injective, they are
even asymptotically unitarily equivalent. This in particular implies that any unital en-
domorphism of D is asymptotically inner. Moreover, the space of automorphisms of D is
compactly-contractible (in the point-norm topology) in the sense that for any compact
Hausdorff space X, the set of homotopy classes [X,Aut (D)] reduces to a point. The re-
spective statement holds for the space of unital endomorphisms of D. As an application,
we give a description of the Kasparov group KK(D, A⊗D) in terms of ∗-homomorphisms
and asymptotic unitary equivalence. Along the way, we show that the Kasparov group
KK(D, A⊗D) is isomorphic to K0(A⊗D).
0. Introduction
A unital and separable C∗-algebra D 6= C is strongly self-absorbing if there is an
isomorphism D
→ D ⊗ D which is approximately unitarily equivalent to the inclusion
map D → D ⊗ D, d 7→ d ⊗ 1D ([14]). Strongly self-absorbing C
∗-algebras are known to
be simple and nuclear; moreover, they are either purely infinite or stably finite. The only
known examples of strongly self-absorbing C∗-algebras are the UHF algebras of infinite
type (i.e., every prime number that occurs in the respective supernatural number occurs
with infinite multiplicity), the Cuntz algebras O2 and O∞, the Jiang–Su algebra Z and
tensor products of O∞ with UHF algebras of infinite type, see [14]. All these examples
are K1-injective, i.e., the canonical map U(D)/U0(D) → K1(D) is injective.
It was observed in [14] that any two unital ∗-homomorphisms σ, γ : D → A ⊗ D are
approximately unitarily equivalent, were A is another unital and separable C∗-algebra.
If D is K1-injective, the unitaries implementing the equivalence may even be chosen to
Date: August 3, 2021.
2000 Mathematics Subject Classification. 46L05, 47L40.
Key words and phrases. Strongly self-absorbing C∗-algebras, KK-theory, asymptotic unitary
equivalence, continuous fields of C∗-algebras.
Supported by: The first named author was partially supported by NSF grant #DMS-0500693.
The second named author was supported by the DFG (SFB 478).
http://arxiv.org/abs/0704.0583v1
2 MARIUS DADARLAT AND WILHELM WINTER
be homotopic to the unit. When D is O2, O∞, it was known that σ and γ are even
asymptotically unitarily equivalent – i.e., they can be intertwined by a continuous path of
unitaries, parametrized by a half-open interval. Up to this point, it was not clear whether
the respective statement holds for the Jiang–Su algebra Z. Theorem 2.2 below provides
an affirmative answer to this problem. Even more, we show that the path intertwining σ
and γ may be chosen in the component of the unit.
We believe this result, albeit technical, is interesting in its own right, and that it will
be a useful ingredient for the systematic further use of strongly self-absorbing C∗-algebras
in Elliott’s program to classify nuclear C∗-algebras by K-theory data. In fact, this point
of view is our main motivation for the study of strongly self-absorbing C∗-algebras; see
[8], [10], [16], [17], [18] and [15] for already existing results in this direction.
For the time being, we use Theorem 2.2 to derive some consequences for the Kasparov
groups of the form KK(D, A ⊗ D). More precisely, we show that all the elements of the
Kasparov group KK(D, A ⊗ D) are of the form [ϕ] − n[ι] where ϕ : D → K ⊗ A ⊗ D
is a ∗-homomorphism and ι : D → A ⊗ D is the inclusion ι(d) = 1A ⊗ d and n ∈ N.
Moreover, two non-zero ∗-homomorphisms ϕ,ψ : D → K⊗A⊗D with ϕ(1D) = ψ(1D) = e
have the same KK-theory class if and only if there is a unitary-valued continuous map
u : [0, 1) → e(K ⊗ A ⊗ D)e, t 7→ ut such that u0 = e and limt→1 ‖ut ϕ(d)u
t − ψ(d)‖ = 0
for all d ∈ D. In addition, we show that KKi(D,D ⊗A) ∼= Ki(D ⊗A), i = 0, 1.
One may note the similarity to the descriptions of KK(O∞,O∞ ⊗ A) ([8],[10]) and
KK(C,C ⊗ A). However, we do not require that D satisfies the universal coefficient
theorem (UCT) in KK-theory. In the same spirit, we characterize O2 and the universal
UHF algebra Q using K-theoretic conditions, but without involving the UCT.
As another application of Theorem 2.2 (and the results of [7]), we prove in [4] an
automatic trivialization result for continuous fields with strongly self-absorbing fibres over
finite dimensional spaces.
The second named author would like to thank Eberhard Kirchberg for an inspiring
conversation on the problem of proving Theorem 2.2.
1. Strongly self-absorbing C∗-algebras
In this section we recall the notion of strongly self-absorbing C∗-algebras and some facts
from [14].
1.1 Definition: Let A, B be C∗-algebras and σ, γ : A → B be ∗-homomorphisms.
Suppose that B is unital.
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 3
(i) We say that σ and γ are approximately unitarily equivalent, σ ≈u γ, if there is a
sequence (un)n∈N of unitaries in B such that
‖unσ(a)u
n − γ(a)‖
for every a ∈ A. If all un can be chosen to be in U0(B), the connected component of
1B of the unitary group U(B), then we say that σ and γ are strongly approximately
unitarily equivalent, written σ ≈su γ.
(ii) We say that σ and γ are asymptotically unitarily equivalent, σ ≈uh γ, if there is
a norm-continuous path (ut)t∈[0,∞) of unitaries in B such that
‖utσ(a)u
t − γ(a)‖
for every a ∈ A. If one can arrange that u0 = 1B and hence (ut ∈ U0(B) for
all t), then we say that σ and γ are strongly asymptotically unitarily equivalent,
written σ ≈suh γ.
1.2 The concept of strongly self-absorbing C∗-algebras was formally introduced in [14,
Definition 1.3]:
Definition: A separable unital C∗-algebra D is strongly self-absorbing, if D 6= C and
there is an isomorphism ϕ : D → D ⊗D such that ϕ ≈u idD ⊗ 1D.
1.3 Recall [14, Corollary 1.12]:
Proposition: Let A and D be unital C∗-algebras, with D strongly self-absorbing. Then,
any two unital ∗-homomorphisms σ, γ : D → A⊗D are approximately unitarily equivalent.
In particular, any two unital endomorphisms of D are approximately unitarily equivalent.
We note that the assumption that A is separable which appears in the original statement
of [14, Corollary 1.12] is not necessary and was not used in the proof.
1.4 Lemma: Let D be a strongly self-absorbing C∗-algebra. Then there is a sequence
of unitaries (wn)n∈N in the commutator subgroup of U(D ⊗ D) such that for all d ∈ D
‖wn(d⊗ 1D)w
n − 1D ⊗ d‖ → 0 as n→ ∞.
Proof: Let F ⊂ D be a finite normalized set and let ε > 0. By [14, Prop. 1.5] there is a
unitary u ∈ U(D⊗D) such that ‖u(d⊗1D)u
∗−1D⊗d‖ < ε for all d ∈ F . Let θ : D⊗D → D
be a ∗-isomorphism. Then ‖(θ(u∗) ⊗ 1D)u(d ⊗ 1D)u
∗(θ(u) ⊗ 1D) − 1D ⊗ d‖ < ε for all
d ∈ F . By Proposition 1.3 θ ⊗ 1D ≈u idD⊗D and so there is a unitary v ∈ U(D ⊗ D)
such that ‖θ(u∗) ⊗ 1D − vu
∗v∗‖ < ε and hence ‖(θ(u∗) ⊗ 1D)u − vu
∗v∗u‖ < ε. Setting
w = vu∗v∗u we deduce that ‖w(d ⊗ 1D)w
∗ − 1D ⊗ d‖ < 3ε for all d ∈ F .
1.5 Remark: In the situation of Proposition 1.3, suppose that the commutator subgroup
of U(D) is contained in U0(D). This will happen for instance if D is assumed to be K1-
injective. Then one may choose the unitaries (un)n∈N which implement the approximate
4 MARIUS DADARLAT AND WILHELM WINTER
unitary equivalence between σ and γ to lie in U0(A⊗D). This follows from [14, (the proof
of) Corollary 1.12], since the unitaries (un)n∈N are essentially images of the unitaries
(wn)n∈N of Lemma 1.4 under suitable unital
∗-homomorphisms.
2. Asymptotic vs. approximate unitary equivalence
It is the aim of this section to establish a continuous version of Proposition 1.3.
2.1 Lemma: Let D be separable unital strongly self-absorbing C∗-algebra. For any finite
subset F ⊂ D and ε > 0, there are a finite subset G ⊂ D and δ > 0 such that the following
holds:
If A is another unital C∗-algebra and σ : D → A⊗D is a unital ∗-homomorphism, and
if w ∈ U0(A⊗D) is a unitary satisfying
‖[w, σ(d)]‖ < δ
for all d ∈ G, then there is a continuous path (wt)t∈[0,1] of unitaries in U0(A ⊗ D) such
that w0 = w, w1 = 1A⊗D and
‖[wt, σ(d)]‖ < ε
for all d ∈ F , t ∈ [0, 1].
Proof: We may clearly assume that the elements of F are normalized and that ε < 1.
Let u ∈ D ⊗D be a unitary satisfying
(1) ‖u(d ⊗ 1D)u
∗ − 1D ⊗ d‖ <
for all d ∈ F . There exist k ∈ N and elements s1, . . . , sk, t1, . . . , tk ∈ D of norm at most
one such that
(2) ‖u−
si ⊗ ti‖ <
(3) δ :=
k · 10
(4) G := {s1, . . . , sk} ⊂ D.
Now let w ∈ U0(A⊗D) be a unitary as in the assertion of the lemma, i.e., w satisfies
(5) ‖[w, σ(si)]‖ < δ
for all i = 1, . . . , k. We proceed to construct the path (wt)t∈[0,1].
By [14, Remark 2.7] there is a unital ∗-homomorphism
ϕ : A⊗D ⊗D → A⊗D
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 5
such that
(6) ‖ϕ(a⊗ 1D)− a‖ <
for all a ∈ σ(F) ∪ {w}.
Since w ∈ U0(A⊗D), there is a path (w̄t)t∈[ 1
,1] of unitaries in A⊗D such that
(7) w̄ 1
= w and w̄1 = 1A⊗D.
For t ∈ [1
, 1] define
(8) wt := ϕ((σ ⊗ idD)(u)
∗(w̄t ⊗ 1D)(σ ⊗ idD)(u)) ∈ U(A⊗D);
then (wt)t∈[ 1
,1] is a continuous path of unitaries in A ⊗ D. For t ∈ [
, 1] and d ∈ F we
‖[wt, σ(d)]‖
= ‖wtσ(d)w
t − σ(d)‖
< ‖wtϕ(σ(d) ⊗ 1D)w
t − ϕ(σ(d) ⊗ 1D)‖+ 2 ·
≤ ‖((σ ⊗ idD)(u))
∗(w̄t ⊗ 1D)((σ ⊗ idD)(u(d ⊗ 1D)u
∗))(w̄∗t ⊗ 1D)
·((σ ⊗ idD)(u)) − ((σ ⊗ idD)(d⊗ 1D))‖ +
< ‖((σ ⊗ idD)(u))
∗(w̄t ⊗ 1D)((σ ⊗ idD)(1D ⊗ d))(w̄
t ⊗ 1D)
·((σ ⊗ idD)(u)) − ((σ ⊗ idD)(d⊗ 1D))‖ +
= ‖(σ ⊗ idD)(u
∗(1D ⊗ d)u− d⊗ 1D)‖+
6 MARIUS DADARLAT AND WILHELM WINTER
where for the last equality we have used that the w̄t are unitaries and that σ is a unital
∗-homomorphism. Furthermore, we have
(7),(8)
= ‖ϕ(((σ ⊗ idD)(u))
∗(w ⊗ 1D)((σ ⊗ idD)(u))) − w‖
< ‖ϕ(((σ ⊗ idD)(u))
∗(w ⊗ 1D)(
σ(si)⊗ ti))− w‖+
≤ ‖ϕ(((σ ⊗ idD)(u))
σ(si)⊗ ti)(w ⊗ 1D))− w‖
‖[w, σ(si)]‖ · ‖ti‖+
(5),(4),(2)
< ‖ϕ(w ⊗ 1D)− w‖+ k · δ + 2 ·
(6),(3)
+ 2 ·
The above estimate allows us to extend the path (wt)t∈[ 1
,1] to the whole interval [0, 1]
in the desired way: We have ‖w 1
w∗ − 1D‖ <
< 2, whence −1 is not in the spectrum
of w 1
w∗. By functional calculus, there is a = a∗ ∈ A ⊗ D with ‖a‖ < 1 such that
w∗ = exp(πia). For t ∈ [0, 1
) we may therefore define a continuous path of unitaries
wt := (exp(2πita))w ∈ U(A⊗D).
It is clear that w0 = w and wt → w 1
as t→ (1
)−, whence (wt)t∈[0,1] is a continuous path
of unitaries in A satisfying w0 = w and w1 = 1A ⊗D. Moreover, it is easy to see that
‖wt − w‖ ≤ ‖w 1
− w‖ <
for all t ∈ [0, 1
), whence
‖[wt, σ(d)]‖ < ‖[w 1
, σ(d)]‖ +
for t ∈ [0, 1
), d ∈ F .
We have now constructed a path (wt)t∈[0,1] ⊂ U(A) with the desired properties.
2.2 Theorem: Let A and D be unital C∗-algebras, with D separable, strongly self-
absorbing and K1-injective. Then, any two unital
∗-homomorphisms σ, γ : D → A⊗D are
strongly asymptotically unitarily equivalent. In particular, any two unital endomorphisms
of D are strongly asymptotically unitarily equivalent.
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 7
Proof: Note that the second statement follows from the first one with A = D, since
D ∼= D ⊗D by assumption.
Let A be a unital C∗-algebra such that A ∼= A ⊗ D and let σ, γ : D → A be unital
∗-homomorphisms. We shall prove that σ and γ are strongly asymptotically unitarily
equivalent. Choose an increasing sequence
F0 ⊂ F1 ⊂ . . .
of finite subsets of D such that
Fn is a dense subset of D. Let 1 > ε0 > ε1 > . . . be a
decreasing sequence of strictly positive numbers converging to 0.
For each n ∈ N, employ Lemma 2.1 (with Fn and εn in place of F and ε) to obtain a
finite subset Gn ⊂ D and δn > 0. We may clearly assume that
(10) Fn ⊂ Gn ⊂ Gn+1 and that δn+1 < δn < εn
for all n ∈ N.
Since σ and γ are strongly approximately unitarily equivalent by Proposition 1.3 and
Remark 1.5, there is a sequence of unitaries (un)n∈N ⊂ U0(A) such that
(11) ‖unσ(d)u
n − γ(d)‖ <
for all d ∈ Gn and n ∈ N. Let us set
wn := u
n+1un, n ∈ N.
Then wn ∈ U0(A) and
‖[wn, σ(d)]‖
= ‖wnσ(d)w
n − σ(d)‖
≤ ‖u∗n+1unσ(d)u
nun+1 − u
n+1γ(d)un+1‖
+‖u∗n+1γ(d)un+1 − σ(d)‖
for d ∈ Gn, n ∈ N. Now by Lemma 2.1 (and the choice of the Gn and δn), for each n there
is a continuous path (wn,t)t∈[0,1] of unitaries in U0(A) such that wn,0 = wn, wn,1 = 1A and
(12) ‖[wn,t, σ(d)]‖ < εn
for all d ∈ Fn, t ∈ [0, 1].
Next, define a path (ūt)t∈[0,∞) of unitaries in U0(A) by
ūt := un+1wn,t−n if t ∈ [n, n+ 1).
8 MARIUS DADARLAT AND WILHELM WINTER
We have that
(13) ūn = un+1wn = un
and that
ūt → un+1
as t → n + 1 from below, which implies that the path (ūt)t∈[0,∞) is continuous in U0(A).
Furthermore, for t ∈ [n, n+ 1) and d ∈ Fn we obtain
‖ūtσ(d)ū
t − γ(d)‖
= ‖un+1wn,t−nσ(d)w
n,t−nu
n+1 − γ(d)‖
< ‖un+1σ(d)u
n+1 − γ(d)‖ + εn
(11),(10)
< 2εn.
Since the Fn are nested and the εn converge to 0, we have
(14) ‖ūtσ(d)ū
t − γ(d)‖
for all d ∈
n=0Fn; by continuity and since
n=0Fn is dense in D, we have (14) for all
d ∈ D. Since ū0 ∈ U0(A) we may arrange that ū0 = 1A.
3. The group KK(D, A⊗D) and some applications
3.1 For a separable C∗-algebra D we endow the group of automorphisms Aut (D) with the
point-norm topology.
Corollary: Let D be a separable, unital, strongly self-absorbing and K1-injective C
algebra. Then [X,Aut(D)] reduces to a point for any compact Hausdorff space X.
Proof: Let ϕ,ψ : X → Aut (D) be continuous maps. We identify ϕ and ψ with unital
∗-homomorphisms ϕ,ψ : D → C(X) ⊗ D. By Theorem 2.2, ϕ is strongly asymptotically
unitarily equivalent to ψ. This gives a homotopy between the two maps ϕ,ψ : X →
Aut (D).
3.2 Remark: The conclusion of Corollary 3.1 was known before for D a UHF algebra of
infinite type and X a CW complex by [13], for D = O2 by [8] and [10], and for D = O∞
by [2]. It is new for the Jiang–Su algebra.
3.3 For unital C∗-algebras D and B we denote by [D, B] the set of homotopy classes of
unital ∗-homomorphisms from D to B. By a similar argument as above we also have the
following corollary.
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 9
Corollary: Let D and A be unital C∗-algebras. If D is separable, strongly self-absorbing
and K1-injective, then [D, A⊗D] reduces to a singleton.
3.4 For separable unital C∗-algebras D and B, let χi : KKi(D, B) → KKi(C, B) ∼= Ki(B),
i = 0, 1 be the morphism of groups induced by the unital inclusion ν : C → D.
Theorem: Let D be a unital, separable and strongly self-absorbing C∗-algebra. Then for
any separable C∗-algebra A, the map χi : KKi(D, A ⊗ D) → Ki(A ⊗ D) is bijective, for
i = 0, 1. In particular both groups KKi(D, A⊗D) are countable and discrete with respect
to their natural topology.
Proof: Since D is KK-equivalent to D ⊗ O∞, we may assume that D is purely infinite
and in particular K1-injective by [11, Prop. 4.1.4]. Let CνD denote the mapping cone C
algebra of ν. By [3, Cor. 3.10], there is a bijection [D, A⊗ D] → KK(CνD, SA⊗ D) and
hence KK(CνD, SA⊗D) = 0 for all separable and unital C
∗-algebras A as a consequence
of Corollary 3.3. Since KK(CνD, A ⊗ D) is isomorphic to KK(CνD, S
2A ⊗ D) by Bott
periodicity and the latter group injects in KK(CνD, SC(T) ⊗ A ⊗ D) = 0, we have that
KKi(CνD,D ⊗ A) = 0 for all unital and separable C
∗-algebras A and i = 0, 1. Since
KKi(CνD,D ⊗A) is a subgroup of KKi(CνD,D ⊗ Ã) = 0 (where Ã is the unitization of
A) we see that KKi(CνD,D ⊗ A) = 0 for all separable C
∗-algebras A. Using the Puppe
exact sequence, where χi = ν
KKi+1(CνD, A⊗D) // KKi(D, A⊗D)
// KKi(C, A⊗D) // KKi(CνD, A⊗D)
we conclude that χi is an isomorphism, i = 0, 1. The map χi = ν
∗ is continuous since it
is given by the Kasparov product with a fixed element (we refer the reader to [12], [9] or
[1] for a background on the topology of the Kasparov groups). Since the topology of Ki is
discrete and χi is injective, it follows that the topology of KKi(D, A⊗D) is also discrete.
The countability of KKi(D, A⊗D) follows from that of Ki(A⊗D), as A⊗D is separable.
3.5 Remark: In contrast to Theorem 3.4, if D is the universal UHF algebra, then
KK(D,C) ∼= Ext(Q,Z) ∼= QN has the power of the continuum [6, p. 221].
3.6 Let D and A be as in Theorem 3.4 and assume in addition that D is K1-injective and
A is unital. Let ι : D → A⊗D be defined by ι(d) = 1A ⊗ d.
Corollary: If e ∈ K ⊗ A⊗ D is a projection, and ϕ,ψ : D → e(K ⊗ A ⊗ D)e are two
unital ∗-homomorphisms, then ϕ ≈suh ψ and hence [ϕ] = [ψ] ∈ KK(D, A⊗D). Moreover:
KK(D, A⊗D) = {[ϕ]− n[ι] |ϕ : D → K⊗A⊗D is a ∗-homomorphism, n ∈ N}.
10 MARIUS DADARLAT AND WILHELM WINTER
Proof: Let ϕ, ψ and e be as in the first part of the statement. By [14, Cor. 3.1], the
unital C∗-algebra e(K⊗A⊗D)e is D-stable, being a hereditary subalgebra of a D-stable
C∗-algebra. Therefore ϕ ≈suh ψ by Theorem 2.2.
Now for the second part of the statement, let x ∈ KK(D, A ⊗ D) be an arbitrary
element. Then χ0(x) = [e]−n[1A⊗D] for some projection e ∈ K⊗A⊗D and n ∈ N. Since
e(K ⊗ A ⊗ D)e is D-stable, there is a unital ∗-homomorphism ϕ : D → e(K ⊗ A ⊗ D)e.
χ0([ϕ] − n[ι]) = [ϕ(1D)]− n[ι(1D)] = [e]− n[1A⊗D] = χ0(x),
and hence [ϕ]− n[ι] = x since χ0 is injective by Theorem 3.4.
In the remainder of the paper we give characterizations for the Cuntz algebra O2 and for
the universal UHF-algebra which do not require the UCT. The latter result is a variation
of a theorem of Effros and Rosenberg [5].
3.7 Proposition: Let D be a separable unital strongly self-absorbing C∗-algebra. If [1D] =
0 in K0(D), then D ∼= O2.
Proof: Since D must be nuclear (see [14]), D embeds unitally in O2 by Kirchberg’s
theorem. D is not stably finite since [1D] = 0. By the dichotomy of [14, Thm. 1.7] D must
be purely infinite. Since [1D] = 0 in K0(D), there is a unital embedding O2 → D, see [11,
Prop. 4.2.3]. We conclude that D is isomorphic to O2 by [14, Prop. 5.12].
3.8 Proposition: Let D, A be separable, unital, strongly self-absorbing C∗-algebras.
Suppose that for any finite subset F of D and any ε > 0 there is a u.c.p. map ϕ : D → A
such that ‖ϕ(cd) − ϕ(c)ϕ(d)‖ < ε for all c, d ∈ F . Then A ∼= A⊗D.
Proof: By [14, Thm. 2.2] it suffices to show that for any given finite subsets F of D, G of
A and any ε > 0 there is u.c.p. map Φ : D → A such that (i) ‖Φ(cd)− Φ(c)Φ(d)‖ < ε for
all c, d ∈ F and (ii) ‖[Φ(d), a]‖ < ε for all d ∈ F and a ∈ G. We may assume that ‖d‖ ≤ 1
for all d ∈ F . Since A is strongly self-absorbing, by [14, Prop. 1.10] there is a unital ∗-
homomorphism γ : A⊗A→ A such that ‖γ(a⊗1A)−a‖ < ε/2 for all a ∈ G. On the other
hand, by assumption there is a u.c.p. map ϕ : D → A such that ‖ϕ(cd) − ϕ(c)ϕ(d)‖ < ε
for all c, d ∈ F . Let us define a u.c.p. map Φ : D → A by Φ(d) = γ(1A ⊗ ϕ(d)). It is
clear that Φ satisfies (i) since γ is a ∗-homomorphism. To conclude the proof we check
now that Φ also satisfies (ii). Let d ∈ F and a ∈ G. Then
‖[Φ(d), a]‖
≤ ‖[Φ(d), a − γ(a⊗ 1A)]‖+ ‖[Φ(d), γ(a ⊗ 1A)]‖
≤ 2‖Φ(d)‖‖a − γ(a⊗ 1A)‖+ ‖[γ(1A ⊗ ϕ(d)), γ(a ⊗ 1A)]‖
< 2ε/2 + 0 = ε.
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 11
3.9 Proposition: Let D be a separable, unital, strongly self-absorbing C∗-algebra. Sup-
pose that D is quasidiagonal, it has cancellation of projections and that [1D] ∈ nK0(D)
for all n ≥ 1. Then D is isomorphic to the universal UHF algebra Q with K0(Q) ∼= Q.
Proof: Since D is separable unital and quasidiagonal, there is a unital ∗-representation
π : D → B(H) on a separable Hilbert space H and a sequence of nonzero projections
pn ∈ B(H) of finite rank k(n) such that limn→∞ ‖[pn, π(d)]‖ = 0 for all d ∈ D. Then
the sequence of u.c.p. maps ϕn : D → pnB(H)pn ∼= Mk(n)(C) ⊂ Q is asymptotically
multiplicative, i.e limn→∞ ‖ϕn(cd) − ϕn(c)ϕn(d))‖ = 0 for all c, d ∈ D. Therefore Q ∼=
Q⊗D by Proposition 3.8.
In the second part of the proof we show that D ∼= D ⊗Q. Let En : Q → Mn!(C) ⊂ Q
be a conditional expectation onto Mn!(C). Then limn→∞ ‖En(a)− a‖ = 0 for all a ∈ Q.
By assumption, for each n there is a projection e in D ⊗Mm(C) (for some m) such
that n![e] = [1D] in K0(D). Let ϕ : Mn!(C) → Mn!(C) ⊗ e(D ⊗ Mm(C))e be defined
by ϕ(b) = b ⊗ e. Since D has cancellation of projections and since n![e] = [1D], there
is a partial isometry v ∈ Mn!(C) ⊗ D ⊗Mm(C) such that v
∗v = 1Mn!(C) ⊗ e and vv
e11⊗1D⊗e11. Therefore b 7→ v ϕ(b) v
∗ gives a unital embedding ofMn!(C) into D. Finally,
ψn(a) = v (ϕ ◦ En(a)) v
∗ defines a sequence of asymptotically multiplicative u.c.p. maps
Q → D. Therefore D ∼= D ⊗Q by Proposition 3.8.
3.10 Remark: Let D be a separable, unital, strongly self-absorbing and quasidiagonal C∗-
algebra. Then D ⊗Q ∼= Q by the first part of the proof of Proposition 3.9. In particular
K1(D) ⊗ Q = 0 and K0(D) ⊗ Q ∼= Q by the Künneth formula (or by writing Q as an
inductive limit of matrices).
References
[1] M. Dadarlat. On the topology of the Kasparov groups and its applications., J. Funct. Anal. 228 (2005),
394–418.
[2] M. Dadarlat. Continuous fields of C∗-algebras over finite dimensional spaces , arXiv preprint
math.OA/0611405 (2006).
[3] M. Dadarlat. The homotopy groups of the automorphism group of Kirchberg algebras, J. Noncomm.
Geom. 1 (2007), 113–139.
[4] M. Dadarlat and W. Winter. Trivialization of C(X)-algebras with strongly self-absorbing fibres, preprint
(2007).
[5] E. G. Effros and J. Rosenberg. C∗-algebras with approximately inner flip, Pacific J. Math. 77 (1978),
417–443.
[6] L. Fuchs. Infinite abelian groups, vol. 1, Academic Press, New York and London, 1970.
[7] I. Hirshberg, M. Rørdam and W. Winter. C0(X)-algebras, stability and strongly self-absorbing C
algebras, arXiv preprint math.OA/0610344 (2006). To appear in Math. Ann.
[8] E. Kirchberg. The classification of purely infinite C∗-algebras using Kasparov’s theory, preprint (1994).
http://arxiv.org/abs/math/0611405
http://arxiv.org/abs/math/0610344
12 MARIUS DADARLAT AND WILHELM WINTER
[9] M. V. Pimsner. A topology on the Kasparov groups, draft.
[10] N. C. Phillips. A classification theorem for nuclear purely infinite simple C∗-algebras, Documenta
Math. 5 (2000), 49–114.
[11] M. Rørdam. Classification of Nuclear C∗-Algebras, Encyclopaedia Math. Sci., vol. 126, Springer,
Berlin, 2002.
[12] C. Schochet. The fine structure of the Kasparov groups I. Continuity of the KK-pairing, J. Funct.
Anal. 186 (2001), 25–61.
[13] K. Thomsen. The homotopy type of the group of automorphisms of a UHF-algebra, J. Funct. Anal. 72
(1987), 182–207.
[14] A. Toms andW.Winter. Strongly self-absorbing C∗-algebras, arXiv preprint math.OA/0502211 (2005).
To appear in Trans. Amer. Math. Soc.
[15] A. Toms and W. Winter. Z-stable ASH algebras, arXiv preprint math.OA/0508218 (2005). To appear
in Can. J. Math.
[16] W. Winter. On the classification of simple Z-stable C∗-algebras with real rank zero and finite decom-
position rank, J. London Math. Soc. 74 (2006), 167–183.
[17] W. Winter. Simple C∗-algebras with locally finite decomposition rank, J. Funct. Anal. 243 (2007),
394–425.
[18] W. Winter. Localizing the Elliott conjecture, in preparation.
Department of Mathematics, Purdue University, West Lafayette,, IN 47907, USA
E-mail address: mdd@math.purdue.edu
Mathematisches Institut der Universität Münster, Einsteinstr. 62, D-48149 Münster,
Germany
E-mail address: wwinter@math.uni-muenster.de
http://arxiv.org/abs/math/0502211
http://arxiv.org/abs/math/0508218
	0. Introduction
	1. Strongly self-absorbing C*-algebras
	2. Asymptotic vs. approximate unitary equivalence
	3. The group KK(D,AD) and some applications
	References

  Let $\Dh$ and $A$ be unital and separable $C^{*}$-algebras; let $\Dh$ be
strongly self-absorbing. It is known that any two unital $^*$-homomorphisms
from $\Dh$ to $A \otimes \Dh$ are approximately unitarily equivalent. We show
that, if $\Dh$ is also $K_{1}$-injective, they are even asymptotically
unitarily equivalent. This in particular implies that any unital endomorphism
of $\Dh$ is asymptotically inner. Moreover, the space of automorphisms of $\Dh$
is compactly-contractible (in the point-norm topology) in the sense that for
any compact Hausdorff space $X$, the set of homotopy classes $[X,\Aut(\Dh)]$
reduces to a point. The respective statement holds for the space of unital
endomorphisms of $\Dh$. As an application, we give a description of the
Kasparov group $KK(\Dh, A\ot \Dh)$ in terms of $^*$-homomorphisms and
asymptotic unitary equivalence. Along the way, we show that the Kasparov group
$KK(\Dh, A\ot \Dh)$ is isomorphic to $K_0(A\ot \Dh)$.

Introduction
A unital and separable C∗-algebra D 6= C is strongly self-absorbing if there is an
isomorphism D
→ D ⊗ D which is approximately unitarily equivalent to the inclusion
map D → D ⊗ D, d 7→ d ⊗ 1D ([14]). Strongly self-absorbing C
∗-algebras are known to
be simple and nuclear; moreover, they are either purely infinite or stably finite. The only
known examples of strongly self-absorbing C∗-algebras are the UHF algebras of infinite
type (i.e., every prime number that occurs in the respective supernatural number occurs
with infinite multiplicity), the Cuntz algebras O2 and O∞, the Jiang–Su algebra Z and
tensor products of O∞ with UHF algebras of infinite type, see [14]. All these examples
are K1-injective, i.e., the canonical map U(D)/U0(D) → K1(D) is injective.
It was observed in [14] that any two unital ∗-homomorphisms σ, γ : D → A ⊗ D are
approximately unitarily equivalent, were A is another unital and separable C∗-algebra.
If D is K1-injective, the unitaries implementing the equivalence may even be chosen to
Date: August 3, 2021.
2000 Mathematics Subject Classification. 46L05, 47L40.
Key words and phrases. Strongly self-absorbing C∗-algebras, KK-theory, asymptotic unitary
equivalence, continuous fields of C∗-algebras.
Supported by: The first named author was partially supported by NSF grant #DMS-0500693.
The second named author was supported by the DFG (SFB 478).
http://arxiv.org/abs/0704.0583v1
2 MARIUS DADARLAT AND WILHELM WINTER
be homotopic to the unit. When D is O2, O∞, it was known that σ and γ are even
asymptotically unitarily equivalent – i.e., they can be intertwined by a continuous path of
unitaries, parametrized by a half-open interval. Up to this point, it was not clear whether
the respective statement holds for the Jiang–Su algebra Z. Theorem 2.2 below provides
an affirmative answer to this problem. Even more, we show that the path intertwining σ
and γ may be chosen in the component of the unit.
We believe this result, albeit technical, is interesting in its own right, and that it will
be a useful ingredient for the systematic further use of strongly self-absorbing C∗-algebras
in Elliott’s program to classify nuclear C∗-algebras by K-theory data. In fact, this point
of view is our main motivation for the study of strongly self-absorbing C∗-algebras; see
[8], [10], [16], [17], [18] and [15] for already existing results in this direction.
For the time being, we use Theorem 2.2 to derive some consequences for the Kasparov
groups of the form KK(D, A ⊗ D). More precisely, we show that all the elements of the
Kasparov group KK(D, A ⊗ D) are of the form [ϕ] − n[ι] where ϕ : D → K ⊗ A ⊗ D
is a ∗-homomorphism and ι : D → A ⊗ D is the inclusion ι(d) = 1A ⊗ d and n ∈ N.
Moreover, two non-zero ∗-homomorphisms ϕ,ψ : D → K⊗A⊗D with ϕ(1D) = ψ(1D) = e
have the same KK-theory class if and only if there is a unitary-valued continuous map
u : [0, 1) → e(K ⊗ A ⊗ D)e, t 7→ ut such that u0 = e and limt→1 ‖ut ϕ(d)u
t − ψ(d)‖ = 0
for all d ∈ D. In addition, we show that KKi(D,D ⊗A) ∼= Ki(D ⊗A), i = 0, 1.
One may note the similarity to the descriptions of KK(O∞,O∞ ⊗ A) ([8],[10]) and
KK(C,C ⊗ A). However, we do not require that D satisfies the universal coefficient
theorem (UCT) in KK-theory. In the same spirit, we characterize O2 and the universal
UHF algebra Q using K-theoretic conditions, but without involving the UCT.
As another application of Theorem 2.2 (and the results of [7]), we prove in [4] an
automatic trivialization result for continuous fields with strongly self-absorbing fibres over
finite dimensional spaces.
The second named author would like to thank Eberhard Kirchberg for an inspiring
conversation on the problem of proving Theorem 2.2.
1. Strongly self-absorbing C∗-algebras
In this section we recall the notion of strongly self-absorbing C∗-algebras and some facts
from [14].
1.1 Definition: Let A, B be C∗-algebras and σ, γ : A → B be ∗-homomorphisms.
Suppose that B is unital.
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 3
(i) We say that σ and γ are approximately unitarily equivalent, σ ≈u γ, if there is a
sequence (un)n∈N of unitaries in B such that
‖unσ(a)u
n − γ(a)‖
for every a ∈ A. If all un can be chosen to be in U0(B), the connected component of
1B of the unitary group U(B), then we say that σ and γ are strongly approximately
unitarily equivalent, written σ ≈su γ.
(ii) We say that σ and γ are asymptotically unitarily equivalent, σ ≈uh γ, if there is
a norm-continuous path (ut)t∈[0,∞) of unitaries in B such that
‖utσ(a)u
t − γ(a)‖
for every a ∈ A. If one can arrange that u0 = 1B and hence (ut ∈ U0(B) for
all t), then we say that σ and γ are strongly asymptotically unitarily equivalent,
written σ ≈suh γ.
1.2 The concept of strongly self-absorbing C∗-algebras was formally introduced in [14,
Definition 1.3]:
Definition: A separable unital C∗-algebra D is strongly self-absorbing, if D 6= C and
there is an isomorphism ϕ : D → D ⊗D such that ϕ ≈u idD ⊗ 1D.
1.3 Recall [14, Corollary 1.12]:
Proposition: Let A and D be unital C∗-algebras, with D strongly self-absorbing. Then,
any two unital ∗-homomorphisms σ, γ : D → A⊗D are approximately unitarily equivalent.
In particular, any two unital endomorphisms of D are approximately unitarily equivalent.
We note that the assumption that A is separable which appears in the original statement
of [14, Corollary 1.12] is not necessary and was not used in the proof.
1.4 Lemma: Let D be a strongly self-absorbing C∗-algebra. Then there is a sequence
of unitaries (wn)n∈N in the commutator subgroup of U(D ⊗ D) such that for all d ∈ D
‖wn(d⊗ 1D)w
n − 1D ⊗ d‖ → 0 as n→ ∞.
Proof: Let F ⊂ D be a finite normalized set and let ε > 0. By [14, Prop. 1.5] there is a
unitary u ∈ U(D⊗D) such that ‖u(d⊗1D)u
∗−1D⊗d‖ < ε for all d ∈ F . Let θ : D⊗D → D
be a ∗-isomorphism. Then ‖(θ(u∗) ⊗ 1D)u(d ⊗ 1D)u
∗(θ(u) ⊗ 1D) − 1D ⊗ d‖ < ε for all
d ∈ F . By Proposition 1.3 θ ⊗ 1D ≈u idD⊗D and so there is a unitary v ∈ U(D ⊗ D)
such that ‖θ(u∗) ⊗ 1D − vu
∗v∗‖ < ε and hence ‖(θ(u∗) ⊗ 1D)u − vu
∗v∗u‖ < ε. Setting
w = vu∗v∗u we deduce that ‖w(d ⊗ 1D)w
∗ − 1D ⊗ d‖ < 3ε for all d ∈ F .
1.5 Remark: In the situation of Proposition 1.3, suppose that the commutator subgroup
of U(D) is contained in U0(D). This will happen for instance if D is assumed to be K1-
injective. Then one may choose the unitaries (un)n∈N which implement the approximate
4 MARIUS DADARLAT AND WILHELM WINTER
unitary equivalence between σ and γ to lie in U0(A⊗D). This follows from [14, (the proof
of) Corollary 1.12], since the unitaries (un)n∈N are essentially images of the unitaries
(wn)n∈N of Lemma 1.4 under suitable unital
∗-homomorphisms.
2. Asymptotic vs. approximate unitary equivalence
It is the aim of this section to establish a continuous version of Proposition 1.3.
2.1 Lemma: Let D be separable unital strongly self-absorbing C∗-algebra. For any finite
subset F ⊂ D and ε > 0, there are a finite subset G ⊂ D and δ > 0 such that the following
holds:
If A is another unital C∗-algebra and σ : D → A⊗D is a unital ∗-homomorphism, and
if w ∈ U0(A⊗D) is a unitary satisfying
‖[w, σ(d)]‖ < δ
for all d ∈ G, then there is a continuous path (wt)t∈[0,1] of unitaries in U0(A ⊗ D) such
that w0 = w, w1 = 1A⊗D and
‖[wt, σ(d)]‖ < ε
for all d ∈ F , t ∈ [0, 1].
Proof: We may clearly assume that the elements of F are normalized and that ε < 1.
Let u ∈ D ⊗D be a unitary satisfying
(1) ‖u(d ⊗ 1D)u
∗ − 1D ⊗ d‖ <
for all d ∈ F . There exist k ∈ N and elements s1, . . . , sk, t1, . . . , tk ∈ D of norm at most
one such that
(2) ‖u−
si ⊗ ti‖ <
(3) δ :=
k · 10
(4) G := {s1, . . . , sk} ⊂ D.
Now let w ∈ U0(A⊗D) be a unitary as in the assertion of the lemma, i.e., w satisfies
(5) ‖[w, σ(si)]‖ < δ
for all i = 1, . . . , k. We proceed to construct the path (wt)t∈[0,1].
By [14, Remark 2.7] there is a unital ∗-homomorphism
ϕ : A⊗D ⊗D → A⊗D
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 5
such that
(6) ‖ϕ(a⊗ 1D)− a‖ <
for all a ∈ σ(F) ∪ {w}.
Since w ∈ U0(A⊗D), there is a path (w̄t)t∈[ 1
,1] of unitaries in A⊗D such that
(7) w̄ 1
= w and w̄1 = 1A⊗D.
For t ∈ [1
, 1] define
(8) wt := ϕ((σ ⊗ idD)(u)
∗(w̄t ⊗ 1D)(σ ⊗ idD)(u)) ∈ U(A⊗D);
then (wt)t∈[ 1
,1] is a continuous path of unitaries in A ⊗ D. For t ∈ [
, 1] and d ∈ F we
‖[wt, σ(d)]‖
= ‖wtσ(d)w
t − σ(d)‖
< ‖wtϕ(σ(d) ⊗ 1D)w
t − ϕ(σ(d) ⊗ 1D)‖+ 2 ·
≤ ‖((σ ⊗ idD)(u))
∗(w̄t ⊗ 1D)((σ ⊗ idD)(u(d ⊗ 1D)u
∗))(w̄∗t ⊗ 1D)
·((σ ⊗ idD)(u)) − ((σ ⊗ idD)(d⊗ 1D))‖ +
< ‖((σ ⊗ idD)(u))
∗(w̄t ⊗ 1D)((σ ⊗ idD)(1D ⊗ d))(w̄
t ⊗ 1D)
·((σ ⊗ idD)(u)) − ((σ ⊗ idD)(d⊗ 1D))‖ +
= ‖(σ ⊗ idD)(u
∗(1D ⊗ d)u− d⊗ 1D)‖+
6 MARIUS DADARLAT AND WILHELM WINTER
where for the last equality we have used that the w̄t are unitaries and that σ is a unital
∗-homomorphism. Furthermore, we have
(7),(8)
= ‖ϕ(((σ ⊗ idD)(u))
∗(w ⊗ 1D)((σ ⊗ idD)(u))) − w‖
< ‖ϕ(((σ ⊗ idD)(u))
∗(w ⊗ 1D)(
σ(si)⊗ ti))− w‖+
≤ ‖ϕ(((σ ⊗ idD)(u))
σ(si)⊗ ti)(w ⊗ 1D))− w‖
‖[w, σ(si)]‖ · ‖ti‖+
(5),(4),(2)
< ‖ϕ(w ⊗ 1D)− w‖+ k · δ + 2 ·
(6),(3)
+ 2 ·
The above estimate allows us to extend the path (wt)t∈[ 1
,1] to the whole interval [0, 1]
in the desired way: We have ‖w 1
w∗ − 1D‖ <
< 2, whence −1 is not in the spectrum
of w 1
w∗. By functional calculus, there is a = a∗ ∈ A ⊗ D with ‖a‖ < 1 such that
w∗ = exp(πia). For t ∈ [0, 1
) we may therefore define a continuous path of unitaries
wt := (exp(2πita))w ∈ U(A⊗D).
It is clear that w0 = w and wt → w 1
as t→ (1
)−, whence (wt)t∈[0,1] is a continuous path
of unitaries in A satisfying w0 = w and w1 = 1A ⊗D. Moreover, it is easy to see that
‖wt − w‖ ≤ ‖w 1
− w‖ <
for all t ∈ [0, 1
), whence
‖[wt, σ(d)]‖ < ‖[w 1
, σ(d)]‖ +
for t ∈ [0, 1
), d ∈ F .
We have now constructed a path (wt)t∈[0,1] ⊂ U(A) with the desired properties.
2.2 Theorem: Let A and D be unital C∗-algebras, with D separable, strongly self-
absorbing and K1-injective. Then, any two unital
∗-homomorphisms σ, γ : D → A⊗D are
strongly asymptotically unitarily equivalent. In particular, any two unital endomorphisms
of D are strongly asymptotically unitarily equivalent.
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 7
Proof: Note that the second statement follows from the first one with A = D, since
D ∼= D ⊗D by assumption.
Let A be a unital C∗-algebra such that A ∼= A ⊗ D and let σ, γ : D → A be unital
∗-homomorphisms. We shall prove that σ and γ are strongly asymptotically unitarily
equivalent. Choose an increasing sequence
F0 ⊂ F1 ⊂ . . .
of finite subsets of D such that
Fn is a dense subset of D. Let 1 > ε0 > ε1 > . . . be a
decreasing sequence of strictly positive numbers converging to 0.
For each n ∈ N, employ Lemma 2.1 (with Fn and εn in place of F and ε) to obtain a
finite subset Gn ⊂ D and δn > 0. We may clearly assume that
(10) Fn ⊂ Gn ⊂ Gn+1 and that δn+1 < δn < εn
for all n ∈ N.
Since σ and γ are strongly approximately unitarily equivalent by Proposition 1.3 and
Remark 1.5, there is a sequence of unitaries (un)n∈N ⊂ U0(A) such that
(11) ‖unσ(d)u
n − γ(d)‖ <
for all d ∈ Gn and n ∈ N. Let us set
wn := u
n+1un, n ∈ N.
Then wn ∈ U0(A) and
‖[wn, σ(d)]‖
= ‖wnσ(d)w
n − σ(d)‖
≤ ‖u∗n+1unσ(d)u
nun+1 − u
n+1γ(d)un+1‖
+‖u∗n+1γ(d)un+1 − σ(d)‖
for d ∈ Gn, n ∈ N. Now by Lemma 2.1 (and the choice of the Gn and δn), for each n there
is a continuous path (wn,t)t∈[0,1] of unitaries in U0(A) such that wn,0 = wn, wn,1 = 1A and
(12) ‖[wn,t, σ(d)]‖ < εn
for all d ∈ Fn, t ∈ [0, 1].
Next, define a path (ūt)t∈[0,∞) of unitaries in U0(A) by
ūt := un+1wn,t−n if t ∈ [n, n+ 1).
8 MARIUS DADARLAT AND WILHELM WINTER
We have that
(13) ūn = un+1wn = un
and that
ūt → un+1
as t → n + 1 from below, which implies that the path (ūt)t∈[0,∞) is continuous in U0(A).
Furthermore, for t ∈ [n, n+ 1) and d ∈ Fn we obtain
‖ūtσ(d)ū
t − γ(d)‖
= ‖un+1wn,t−nσ(d)w
n,t−nu
n+1 − γ(d)‖
< ‖un+1σ(d)u
n+1 − γ(d)‖ + εn
(11),(10)
< 2εn.
Since the Fn are nested and the εn converge to 0, we have
(14) ‖ūtσ(d)ū
t − γ(d)‖
for all d ∈
n=0Fn; by continuity and since
n=0Fn is dense in D, we have (14) for all
d ∈ D. Since ū0 ∈ U0(A) we may arrange that ū0 = 1A.
3. The group KK(D, A⊗D) and some applications
3.1 For a separable C∗-algebra D we endow the group of automorphisms Aut (D) with the
point-norm topology.
Corollary: Let D be a separable, unital, strongly self-absorbing and K1-injective C
algebra. Then [X,Aut(D)] reduces to a point for any compact Hausdorff space X.
Proof: Let ϕ,ψ : X → Aut (D) be continuous maps. We identify ϕ and ψ with unital
∗-homomorphisms ϕ,ψ : D → C(X) ⊗ D. By Theorem 2.2, ϕ is strongly asymptotically
unitarily equivalent to ψ. This gives a homotopy between the two maps ϕ,ψ : X →
Aut (D).
3.2 Remark: The conclusion of Corollary 3.1 was known before for D a UHF algebra of
infinite type and X a CW complex by [13], for D = O2 by [8] and [10], and for D = O∞
by [2]. It is new for the Jiang–Su algebra.
3.3 For unital C∗-algebras D and B we denote by [D, B] the set of homotopy classes of
unital ∗-homomorphisms from D to B. By a similar argument as above we also have the
following corollary.
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 9
Corollary: Let D and A be unital C∗-algebras. If D is separable, strongly self-absorbing
and K1-injective, then [D, A⊗D] reduces to a singleton.
3.4 For separable unital C∗-algebras D and B, let χi : KKi(D, B) → KKi(C, B) ∼= Ki(B),
i = 0, 1 be the morphism of groups induced by the unital inclusion ν : C → D.
Theorem: Let D be a unital, separable and strongly self-absorbing C∗-algebra. Then for
any separable C∗-algebra A, the map χi : KKi(D, A ⊗ D) → Ki(A ⊗ D) is bijective, for
i = 0, 1. In particular both groups KKi(D, A⊗D) are countable and discrete with respect
to their natural topology.
Proof: Since D is KK-equivalent to D ⊗ O∞, we may assume that D is purely infinite
and in particular K1-injective by [11, Prop. 4.1.4]. Let CνD denote the mapping cone C
algebra of ν. By [3, Cor. 3.10], there is a bijection [D, A⊗ D] → KK(CνD, SA⊗ D) and
hence KK(CνD, SA⊗D) = 0 for all separable and unital C
∗-algebras A as a consequence
of Corollary 3.3. Since KK(CνD, A ⊗ D) is isomorphic to KK(CνD, S
2A ⊗ D) by Bott
periodicity and the latter group injects in KK(CνD, SC(T) ⊗ A ⊗ D) = 0, we have that
KKi(CνD,D ⊗ A) = 0 for all unital and separable C
∗-algebras A and i = 0, 1. Since
KKi(CνD,D ⊗A) is a subgroup of KKi(CνD,D ⊗ Ã) = 0 (where Ã is the unitization of
A) we see that KKi(CνD,D ⊗ A) = 0 for all separable C
∗-algebras A. Using the Puppe
exact sequence, where χi = ν
KKi+1(CνD, A⊗D) // KKi(D, A⊗D)
// KKi(C, A⊗D) // KKi(CνD, A⊗D)
we conclude that χi is an isomorphism, i = 0, 1. The map χi = ν
∗ is continuous since it
is given by the Kasparov product with a fixed element (we refer the reader to [12], [9] or
[1] for a background on the topology of the Kasparov groups). Since the topology of Ki is
discrete and χi is injective, it follows that the topology of KKi(D, A⊗D) is also discrete.
The countability of KKi(D, A⊗D) follows from that of Ki(A⊗D), as A⊗D is separable.
3.5 Remark: In contrast to Theorem 3.4, if D is the universal UHF algebra, then
KK(D,C) ∼= Ext(Q,Z) ∼= QN has the power of the continuum [6, p. 221].
3.6 Let D and A be as in Theorem 3.4 and assume in addition that D is K1-injective and
A is unital. Let ι : D → A⊗D be defined by ι(d) = 1A ⊗ d.
Corollary: If e ∈ K ⊗ A⊗ D is a projection, and ϕ,ψ : D → e(K ⊗ A ⊗ D)e are two
unital ∗-homomorphisms, then ϕ ≈suh ψ and hence [ϕ] = [ψ] ∈ KK(D, A⊗D). Moreover:
KK(D, A⊗D) = {[ϕ]− n[ι] |ϕ : D → K⊗A⊗D is a ∗-homomorphism, n ∈ N}.
10 MARIUS DADARLAT AND WILHELM WINTER
Proof: Let ϕ, ψ and e be as in the first part of the statement. By [14, Cor. 3.1], the
unital C∗-algebra e(K⊗A⊗D)e is D-stable, being a hereditary subalgebra of a D-stable
C∗-algebra. Therefore ϕ ≈suh ψ by Theorem 2.2.
Now for the second part of the statement, let x ∈ KK(D, A ⊗ D) be an arbitrary
element. Then χ0(x) = [e]−n[1A⊗D] for some projection e ∈ K⊗A⊗D and n ∈ N. Since
e(K ⊗ A ⊗ D)e is D-stable, there is a unital ∗-homomorphism ϕ : D → e(K ⊗ A ⊗ D)e.
χ0([ϕ] − n[ι]) = [ϕ(1D)]− n[ι(1D)] = [e]− n[1A⊗D] = χ0(x),
and hence [ϕ]− n[ι] = x since χ0 is injective by Theorem 3.4.
In the remainder of the paper we give characterizations for the Cuntz algebra O2 and for
the universal UHF-algebra which do not require the UCT. The latter result is a variation
of a theorem of Effros and Rosenberg [5].
3.7 Proposition: Let D be a separable unital strongly self-absorbing C∗-algebra. If [1D] =
0 in K0(D), then D ∼= O2.
Proof: Since D must be nuclear (see [14]), D embeds unitally in O2 by Kirchberg’s
theorem. D is not stably finite since [1D] = 0. By the dichotomy of [14, Thm. 1.7] D must
be purely infinite. Since [1D] = 0 in K0(D), there is a unital embedding O2 → D, see [11,
Prop. 4.2.3]. We conclude that D is isomorphic to O2 by [14, Prop. 5.12].
3.8 Proposition: Let D, A be separable, unital, strongly self-absorbing C∗-algebras.
Suppose that for any finite subset F of D and any ε > 0 there is a u.c.p. map ϕ : D → A
such that ‖ϕ(cd) − ϕ(c)ϕ(d)‖ < ε for all c, d ∈ F . Then A ∼= A⊗D.
Proof: By [14, Thm. 2.2] it suffices to show that for any given finite subsets F of D, G of
A and any ε > 0 there is u.c.p. map Φ : D → A such that (i) ‖Φ(cd)− Φ(c)Φ(d)‖ < ε for
all c, d ∈ F and (ii) ‖[Φ(d), a]‖ < ε for all d ∈ F and a ∈ G. We may assume that ‖d‖ ≤ 1
for all d ∈ F . Since A is strongly self-absorbing, by [14, Prop. 1.10] there is a unital ∗-
homomorphism γ : A⊗A→ A such that ‖γ(a⊗1A)−a‖ < ε/2 for all a ∈ G. On the other
hand, by assumption there is a u.c.p. map ϕ : D → A such that ‖ϕ(cd) − ϕ(c)ϕ(d)‖ < ε
for all c, d ∈ F . Let us define a u.c.p. map Φ : D → A by Φ(d) = γ(1A ⊗ ϕ(d)). It is
clear that Φ satisfies (i) since γ is a ∗-homomorphism. To conclude the proof we check
now that Φ also satisfies (ii). Let d ∈ F and a ∈ G. Then
‖[Φ(d), a]‖
≤ ‖[Φ(d), a − γ(a⊗ 1A)]‖+ ‖[Φ(d), γ(a ⊗ 1A)]‖
≤ 2‖Φ(d)‖‖a − γ(a⊗ 1A)‖+ ‖[γ(1A ⊗ ϕ(d)), γ(a ⊗ 1A)]‖
< 2ε/2 + 0 = ε.
ON THE KK-THEORY OF STRONGLY SELF-ABSORBING C∗-ALGEBRAS 11
3.9 Proposition: Let D be a separable, unital, strongly self-absorbing C∗-algebra. Sup-
pose that D is quasidiagonal, it has cancellation of projections and that [1D] ∈ nK0(D)
for all n ≥ 1. Then D is isomorphic to the universal UHF algebra Q with K0(Q) ∼= Q.
Proof: Since D is separable unital and quasidiagonal, there is a unital ∗-representation
π : D → B(H) on a separable Hilbert space H and a sequence of nonzero projections
pn ∈ B(H) of finite rank k(n) such that limn→∞ ‖[pn, π(d)]‖ = 0 for all d ∈ D. Then
the sequence of u.c.p. maps ϕn : D → pnB(H)pn ∼= Mk(n)(C) ⊂ Q is asymptotically
multiplicative, i.e limn→∞ ‖ϕn(cd) − ϕn(c)ϕn(d))‖ = 0 for all c, d ∈ D. Therefore Q ∼=
Q⊗D by Proposition 3.8.
In the second part of the proof we show that D ∼= D ⊗Q. Let En : Q → Mn!(C) ⊂ Q
be a conditional expectation onto Mn!(C). Then limn→∞ ‖En(a)− a‖ = 0 for all a ∈ Q.
By assumption, for each n there is a projection e in D ⊗Mm(C) (for some m) such
that n![e] = [1D] in K0(D). Let ϕ : Mn!(C) → Mn!(C) ⊗ e(D ⊗ Mm(C))e be defined
by ϕ(b) = b ⊗ e. Since D has cancellation of projections and since n![e] = [1D], there
is a partial isometry v ∈ Mn!(C) ⊗ D ⊗Mm(C) such that v
∗v = 1Mn!(C) ⊗ e and vv
e11⊗1D⊗e11. Therefore b 7→ v ϕ(b) v
∗ gives a unital embedding ofMn!(C) into D. Finally,
ψn(a) = v (ϕ ◦ En(a)) v
∗ defines a sequence of asymptotically multiplicative u.c.p. maps
Q → D. Therefore D ∼= D ⊗Q by Proposition 3.8.
3.10 Remark: Let D be a separable, unital, strongly self-absorbing and quasidiagonal C∗-
algebra. Then D ⊗Q ∼= Q by the first part of the proof of Proposition 3.9. In particular
K1(D) ⊗ Q = 0 and K0(D) ⊗ Q ∼= Q by the Künneth formula (or by writing Q as an
inductive limit of matrices).
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http://arxiv.org/abs/math/0611405
http://arxiv.org/abs/math/0610344
12 MARIUS DADARLAT AND WILHELM WINTER
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Department of Mathematics, Purdue University, West Lafayette,, IN 47907, USA
E-mail address: mdd@math.purdue.edu
Mathematisches Institut der Universität Münster, Einsteinstr. 62, D-48149 Münster,
Germany
E-mail address: wwinter@math.uni-muenster.de
http://arxiv.org/abs/math/0502211
http://arxiv.org/abs/math/0508218
	0. Introduction
	1. Strongly self-absorbing C*-algebras
	2. Asymptotic vs. approximate unitary equivalence
	3. The group KK(D,AD) and some applications
	References
