Almost Homomorphisms and KK-Theory Alain Connes and Nigel

32 Pages·2004·0.21 MB·English

Almost Homomorphisms and KK-Theory
Alain Connes and Nigel Higson
0. Introduction
The object of these notes is to introduce a notion of “almost homomorphism” for C∗-
algebras, and explore its consequences for K-theory and KK-theory.
Roughly speaking an almost homomorphism from A to B is a continuous family of
functions ϕ :A → B which asymptotically obey the axioms for ∗-homomorphisms between
t
C∗-algebras. Our definition has the following important features:
(i) An almost homomorphism {ϕ }:A → B induces a homomorphism Φ :K (A) →
t ∗ ∗
K (B) of K-theory groups.
∗
(ii) Almost homomorphisms can be composed, at the level of homotopy, and the compo-
sition is consistent with the action on K-theory.
(iii) The homotopy classes of almost homomorphisms from C (R) ⊗ A to C (R) ⊗ B ⊗
0 0
K form an abelian group, denoted E(A,B). This “E-theory” is a bifunctor, and
composition of almost homomorphisms gives rise to a product structure analagous
to the Kasparov product in KK-theory. There is in fact a natural transformation
KK(A,B) → E(A,B) preserving the product structure.
(iv) The natural transformation from KK-theory to E-theory is an isomorphism on the
class of K-nuclear C∗-algebras introduced by Skandalis. However, arbitrary short
exact sequences of C∗-algebras give rise to periodic six-term exact sequences in E-
theory, in both variables: no nuclearity hypothesis is required. The new theory is the
“universal” refinement of KK-theory with this excision property.
(v) An unbounded Kasparov (A,B)-bimodule induces in a natural way an almost homo-
morphism from C (R) ⊗ A to C (R) ⊗ B ⊗ K. Furthermore, the “assembly map”
0 0
K (C (TM)) → K (C∗(π M)), appearing in the Strong Novikov Conjecture and the
∗ 0 ∗ r 1
1
Baum-Connes Conjecture, is induced by an almost homomorphism from C (TM) to
0
C∗(π M) ⊗ K. Other examples appear to arise in the theory of almost flat vector
r 1
bundles.
The present notes are devoted to setting up the general theory of almost homomorphisms:
our goalwill be the definition of E-theoryand the characterization of the theory mentioned
in item (iv). The examples mentioned above will be discussed briefly in Section 8, but the
possible applications to the Novikov and Baum-Connes Conjectures will be considered
elsewhere.
1. Almost homomorphisms
There are a number of variations of the basic definition, but the following seems to
be the simplest and most natural.
Definition 1. Let A and B be C∗-algebras. An almost homomorphism from A to B is a
family of functions {ϕ } :A → B such that:
t t∈[1,∞)
(i) the functions t 7→ ϕ (a) are continuous for every a ∈ A; and
t
(ii) for every a,a′ ∈ A and α ∈ C,
′ ′
lim (ϕ (a)+αϕ (a )−ϕ (a+αa )) = 0
t t t
t→∞
′ ′
lim (ϕ (a)ϕ (a )−ϕ (aa )) = 0
t t t
t→∞
∗ ∗
lim (ϕ (a) −ϕ (a )) = 0.
t t
t→∞
Notice that the continuity condition imposed is very weak (in particular, it is not
requiredthat eachindividualϕ becontinuous). However, weshall show inthenext section
t
that it is possible to tighten the continuity requirements without essentially altering the
theory. In any case, there is the following “automatic continuity” property.
2
Lemma 1. Let {ϕ }:A → B be an almost homomorphism. Then for every a ∈ A,
t
limsupkϕ (a)k ≤ kak.
t
Proof. Adjoin units to A and B and extend ϕ to a function ϕ :A˜ → B˜ by setting
t t
ϕ (a+α1) = ϕ (a)+α1. If u ∈ A˜ is unitary then by Definition 1,
t t
∗
lim(ϕ (u)ϕ (u) −1) = 0,
t t
t→∞
from which we see that lim kϕ (u)k = 1. An arbitrary a ∈ A with kak < 1 can be
t→∞ t
˜
expressed as a convex combination of a finite number of unitaries in A, say a = α u ,
i i
P
and so
limsupkϕ (a)k ≤ α lim kϕ (u )k = 1.
t i t i
t→∞
X
The result follows from this ⊓⊔
Definition 2. Let B be a C∗-algebra. Denote by CB the C∗-algebra of bounded contin-
uous functions from [1,∞) to B; denote by C B the ideal in CB consisting of functions
0
which vanish at ∞; and denote by QB the quotient algebra CB/C B.
0
It follows from Lemma 1 that an almost homomorphism {ϕ }:A → B determines a
t
function A → CB, and it follows from the definition of almost homomorphism that by
passing to the quotient CB/C B, we obtain a ∗-homomorphism Φ:A → QB. Conversely,
0
starting off with a ∗-homomorphism Φ:A → QB, by composing with any (set-theoretic)
section QB → CB, and then with the evaluation maps e :CB → B (t ∈ [1,∞)), we obtain
t
an almost homomorphism {ϕ }:A → B. This is a one-to-one correspondence, modulo the
t
following equivalence relation on almost homomorphisms.
Definition 3. Two almost homomorphisms {ϕ } and {ϕ′} are asymptotically equivalent
t t
if lim kϕ (a)−ϕ′(a)k = 0 for every a ∈ A.
t→∞ t t
Often our almost homomorphisms will be defined only up to asymptotic equivalence.
For example, suppose that A is a dense ∗-subalgebra of A and suppose that we are given a
3
family of maps {ϕ } :A → B which satisfies the conditions of Definition 1, as well as
t t∈[1,∞)
the conclusion of Lemma 1. Then the family defines a continuous ∗-homomorphism from
A to QB. This extends to A and so we obtain an almost homomorphism, up to asymptotic
equivalence. In connection with this, the following observation is useful.
Lemma 2. Let {ϕ } and {ϕ′} be two almost homomorphism such that lim kϕ (a)−
t t t→∞ t
ϕ′(a)k = 0 for every element a in a dense subset of A. Then {ϕ } and {ϕ′} are asymptot-
t t t
ically equivalent.
Proof. Observe that {ϕ } and {ϕ′} determine the same homomorphism from A to QB
t t
(by continuity of ∗-homomorphisms), and hence are asymptotically equivalent. ⊓⊔
2. The Homotopy Category of Almost Homomorphisms
Definition 1. Two almost homomorphisms, {ϕi}:A → B (i = 0,1), are said to be
t
homotopic if there exists an almost homomorphism {ϕ }:A → B[0,1] from which {ϕ0}
t t
and {ϕ1} are obtained by composition with evaluation at 0 and 1.
t
Remark. In this definition, B[0,1] denotes the continuous functions from [0,1] to B. We
note that Q(B[0,1]) is not isomorphic to (QB)[0,1], and so a homotopy of almost homo-
morphisms is not the same thing as a homotopy of the corresponding ∗-homomorphisms
from A to QB. When we write expressions such as “QB[0,1]” we shall mean “Q(B[0,1]),”
and so on.
Here are some examples of homotopies.
(i) Any two asymptotically equivalent almost homomorphisms are homotopic via the
straight line path between them.
(ii) In particular, if lim kϕ (a)k = 0 for every a ∈ A then {ϕ } is homotopic to the
t→∞ t t
zero almost homomorphism.
(iii) If r:[1,∞)→ [1,∞) is a continuous function such that lim r(t) = ∞ then {ϕ } is
t→∞ t
homotopic to {ϕ }.
r(t)
4
Homotopy is an equivalence relation on the set of all almost homomorphisms A to B;
we shall denote the set of equivalence classes by [[A,B]]. Our goal in this section is to define
composition of almost homomorphisms at the level of homotopy, and so organize the sets
[[A,B]] into a category. The idea is simple enough: given {ϕ }:A → B and {ψ }:B → C we
t t
wish to reparameterize {ψ } to obtain a composition of the form {ψ ◦ϕ } which satisfies
t r(t) t
the axioms for an almost homomorphism. However, there are one or two subtleties here
(connected with proving the associativity of composition), and so we shall proceed with
care.
For the restof this article, all C∗-algebras A, B, C, etc, will be assumed to be separable.
Definition 2. A generating system for a C∗-algebra A is an increasing family of compact
setsA ⊂ A ⊂ ···suchthatA A ⊂ A ,A +A ⊂ A ,A∗ ⊂ A ,andαA ⊂ A
1 2 k k k+1 k k k+1 k k+1 k k+1
for |α| ≤ 1, and such that A is dense in A.
k
S
Note that A is a ∗-subalgebra of A. It will be useful to have a notion of almost
k
S
homomorphism defined on generating systems.
Definition 3. Let {A } be a generating system for A and let B be a C∗-algebra. A
k
uniformalmosthomomorphismfrom{A }toB (abbreviatedu.a.h.) isafamilyoffunctions
k
{ϕ¯ } : A → B such that:
t t∈[1,∞) k
S
(i) for every a ∈ A the map t 7→ ϕ¯ (a) is continuous;
k t
S
(ii) for every k ∈ N and every ǫ > 0 there exists T ∈ [1,∞) such that
′ ′
kϕ¯ (a)ϕ¯ (a )−ϕ¯ (aa )k < ǫ
t t t
′ ′
kϕ¯ (a)+αϕ¯ (a )−ϕ¯ (a+αa )k < ǫ
t t t
∗ ∗
kϕ¯ (a) −ϕ¯ (a )k < ǫ
t t
kϕ¯ (a)k < kak+ǫ
t
for all t > T, a,a′ ∈ A , and |α| ≤ 1; and
k
5
(iii) for every k ∈ N, ϕ¯ (a) is jointly continuous in t ∈ [1,∞) and a ∈ A .
t k
A uniform almost homomorphism {ϕ¯ }:{A } → B gives rise to a ∗-homomorphism
t k
from A to QB which is continuous, and hence extends to a ∗-homomorphism Φ:A →
k
S
QB. Therefore we obtain from {ϕ¯ } an almost homomorphism {ϕ }:A → B. We shall
t t
refer to this as the underlying almost homomorphism of C∗-algebras (strictly speaking,
we should refer to the underlying asymptotic equivalence class determined by {ϕ¯ }). Con-
t
versely, we shall say that {ϕ¯ } represents {ϕ }.
t t
Lemma 1. Let {ϕ }:A → B be an almost homomorphism and let {A } be a generating
t k
system for A. There exists a u.a.h. {ϕ¯ }:{A } → B which represents {ϕ }. In fact, there
t k t
exists an almost homomorphism {ϕ¯ }:A → B in the asymptotic equivalence class of {ϕ }
t t
which consists of an equicontinuous family of functions.
Proof. Pass from {ϕ } to the corresponding ∗-homomorphism Φ:A → QB, and compose
t
this with a continuous section QB → CB (such a section exists by the Bartle-Graves
Theorem). We obtain a map from A to CB, and composing with the evaluation maps
e :CB → B,(t ∈ [1,∞)),weobtainanalmosthomomorphism{ϕ¯ }whichisasymptotically
t t
equivalent to {ϕ }, and which consists of an equicontinuous family of maps. This will
t
restrict to a u.a.h. on any generating system. ⊓⊔
Definition 4. Let {A } and {B } be generating systems for A and B. Two uniform
k n
¯
almost homomorphisms {ϕ¯ }:{A } → B and {ψ }:{B } → C are composable if for every
t k t n
T ∈ [1,∞) and every k ∈ N the subset {ϕ¯ (a) : a ∈ A t ≤ T} of B is contained in some
t k
B .
n
Lemma 2. Suppose that {ϕ¯ } and {ψ¯ } are composable. There exists a continuous,
t t
¯
increasing function r:[1,∞) → [1,∞) such that {ψ ◦ ϕ¯ }:{A } → C is a u.a.h. for
s(t) t k
every continuous, increasing function s ≥ r, and such that the estimates in Definition 3
¯
hold uniformly for all families of the form {ψ ◦ϕ¯ }, where s ≥ r.
s(t) t
6
Proof. Choose an increasing sequence 1 < t < t < ···, converging to infinity, such that
1 2
if t ≥ t then
k
′ ′
kϕ¯ (a)ϕ¯ (a )−ϕ¯ (aa )k < 1/k
t t t
′ ′
kϕ¯ (a)+αϕ¯ (a )−ϕ¯ (a+αa )k < 1/k
t t t
∗ ∗
kϕ¯ (a) −ϕ¯ (a )k < 1/k
t t
kϕ¯ (a)k < kak+1/k
t
for all a,a′ ∈ A and |α| ≤ 1. Choose n < n < ... such that say {ϕ (a) : a ∈
k 1 2 t
A , t ≤ t } ⊂ B (putting in the term “100” gives us a large margin of safety).
k+100 k+1 nk
Finally, choose r < r < ··· such that if s ≥ r then
1 2 k
¯ ¯ ′ ¯ ′
kψ (b)ψ (b )−ψ (bb )k < 1/k
s s s
¯ ¯ ′ ¯ ′
kψ (b)+βψ (b )−ψ (b+βb )k < 1/k
s s s
¯ ∗ ¯ ∗
kψ (b) −ψ (b )k < 1/k
s s
¯
kψ (b)k < kbk+1/k
s
for all b,b′ ∈ B , |β| ≤ 1. Any continuous, increasing function r such that r(t ) ≥ r
nk+100 k k
will suffice. ⊓⊔
¯
We shall refer to any uniform almost homomorphism {ψ ◦ϕ¯ } as in the Lemma as
r(t) t
a composition of {ϕ¯ } and {ψ¯ }.
t t
Lemma 3. The underlying C∗-algebra almost homomorphisms of any two compositions
are homotopic.
Proof. Let r and r′ satisfy the conclusions of Lemma 2. It suffices to show that the
underlyinghomomorphismofthecompositionsconstructedfromr andr+r′ arehomotopic.
The pathofparametrizationsr+λr′ (λ ∈ [0,1])givesrise toa familyofcompositionswhich
7
can be regarded as a u.a.h. from {A } to C[0,1]. The underlying almost homomorphism
k
from A to C[0,1] is a homotopy as desired. ⊓⊔
Lemma 4. Let {ϕ }:A → B and{ψ }:B → C bealmosthomomorphismsandlet{A }be
t t k
a generating system for A. There exist composable almost homomorphisms {ϕ¯ }:{A } →
t k
¯
B and {ψ }:{B } → C representing {ϕ } and {ψ }.
t n t t
Proof. By Lemma 1, there exists a u.a.h. {ϕ¯ }:{A } → B representing {ϕ }. Choose a
t k t
generating system {B } for B such that say {ϕ¯ (a) : a ∈ A t ≤ k} ⊂ B for every k
n t k k
(observe that this is possible since the sets {ϕ¯ (a) : a ∈ A t ≤ k} are compact). By
t k
¯
Lemma 1 again, there exists an almost homomorphism {ψ }:{B } → C which represents
t n
¯
{ψ }. Then {ϕ¯ } and {ψ } are composable. ⊓⊔
t t t
Lemma 5. Let {ϕ¯ }, {ψ¯ } and {ϕ¯′}, {ψ¯′} be two composable pairs of uniform almost
t t t t
homomorphisms representing the same C∗-algebra almost homomorphisms, {ϕ }:A → B
t
and {ψ }:B → C. There exists a continuous, increasing function r such that for every
t
s ≥ r both of the compositions {ψ¯ ◦ ϕ¯ } and {ψ¯′ ◦ ϕ¯′} are defined (in the sense of
s(t) t s(t) t
Lemma 2) and both represent the same almost homomorphism of C∗-algebras.
Proof. We may fix the pair {ϕ¯′}, {ψ¯′} to be a particular choice of representing almost
t t
homomorphisms: we shall choose them, as in the proof of Lemma 1, to be defined and
equicontinuous on all of A and B, respectively. Choose 1 < t < t ···, converging to
1 2
infinity, so that the first set of relations in the proof of Lemma 2 are satisfied for both
{ϕ¯ } and {ϕ¯′} (here we are considering the latter as being defined on {A }, the generating
t t k
system on which {ϕ¯ } is defined). We have that for every b ∈ B ,
t n
S
¯ ¯′ ¯ ¯′
lim (ψ (b)−ψ (b)) = lim (ψ (b)−ψ (b))+ lim (ψ (b)−ψ (b)) = 0
s s s s s s
s→∞ s→∞ s→∞
(1) (1) (1)
Infact, using a compactness argument, it iseasilyseen that there existr < r < r ···
1 2 3
(1)
such that if s ≥ r then
k
¯ ¯′
kψ (ϕ¯ (a))−ψ (ϕ¯ (a))k < 1/k
s t s t
8
¯
for all a ∈ A , t ≤ t . Let {B } be the generating system on which {ψ } is defined, and
k k n t
choose n < n < n ··· so that ϕ¯ (a) ∈ B for all a ∈ A and all t ≤ t . Choose
1 2 2 t nk k+100 k+1
(2) (2) (2) (2)
r < r < r < ··· so that if s ≥ r then
1 2 3 k
¯ ¯ ′ ¯ ′
kψ (b)ψ (b )−ψ (bb )k < 1/k
s s s
¯ ¯ ′ ¯ ′
kψ (b)+βψ (b )−ψ (b+βb )k < 1/k
s s s
¯ ∗ ¯ ∗
kψ (b) −ψ (b )k < 1/k
s s
¯
kψ (b)k < kbk+1/k
s
forallb,b′ ∈ B ,|β| ≤ 1. Let{B′}beageneratingsystemforB suchthatB′ contains
nk+100 k k
the set of all elements of the form ϕ¯ (a), ϕ¯′(a), or ϕ¯ (a)− ϕ¯′(a), where a ∈ A and
t t t t k+100
(3) (3) (3) (3)
t ≤ t . Choose r < r < r < ··· so that if s ≥ r then
k+1 1 2 3 k
¯′ ¯′ ′ ¯′ ′
kψ (b)ψ (b )−ψ (bb )k < 1/k
s s s
¯′ ¯′ ′ ¯′ ′
kψ (b)+βψ (b )−ψ (b+βb )k < 1/k
s s s
¯′ ∗ ¯′ ∗
kψ (b) −ψ (b )k < 1/k
s s
¯′
kψ (b)k < kbk+1/k
s
for all b,b′ ∈ B′, |β| ≤ 1. Writing ψ¯ (ϕ¯ (a))−ψ¯′(ϕ¯′(a)) as
k s t s t
¯ ¯′ ′ ¯ ¯′
ψ (ϕ¯ (a))−ψ (ϕ¯ (a)) = ψ (ϕ¯ (a))−ψ (ϕ¯ (a))
s t s t s t s t
(cid:8) (cid:9)
¯′ ¯′ ′ ¯′ ′
+ ψ (ϕ¯ (a))−ψ (ϕ¯ (a))+ψ (ϕ¯ (a)−ϕ¯ (a))
s t s t s t t
(cid:8) (cid:9)
¯′ ′
−ψ (ϕ¯ (a)−ϕ¯ (a))
s t t
we see that it suffices to choose any continuous, increasing function r such that r(t ) ≥
k
(1) (2) (3)
max{r ,r ,r }. ⊓⊔
k k k
Theorem 1. Define a composition of two almost homomorphisms {ϕ }:A → B and
t
{ψ }:B → C to be the underlying almost homomorphism of any composition of u.a.h.’s
t
9
representing {ϕ } and {ψ }. Then composition of almost homomorphisms passes to a well
t t
defined map [[A,B]]×[[B,C]] → [[A,C]].
Proof. It follows from Lemma 5 that the homotopy class of a product does not depend on
the choice of representing almost homomorphisms. It is then clear that it only depends on
the asymptotic equivalence class of the almost homomorophisms involved (since equivalent
almost homomorphisms have the same representatives). To see that it only depends on the
homotopy class of the almost homomorphisms involved, form compositions of homotopies.
(There is a small technical point that arises here: one must form an almost homomorphism
B[0,1] → C[0,1], starting from an almost homomorphism B → C. The obvious method of
doing so requires that, for example, the almost homomorphism B → C be comprised of an
equicontinuous family of maps. However, using the fact that composition is well defined
on asymptotic equivalence classes, by Lemma 1 we may assume this extra condition.) ⊓⊔
Theorem 2. The composition law [[A,B]]×[[B,C]] → [[A,C]] is associative.
Proof. Given almost homomorphisms {ϕ }:A → B, {ψ }:B → C, and {θ }:C → D,
t t t
by reparametrizing first {ϕ }, then {ψ }, and then {θ }, and choosing generating systems
t t t
appropriately, we can find representing uniform almost homomorphisms {ϕ¯ }:{A } → B,
t k
¯ ¯
{ψ }:{B } → C, and {θ }:{C } → D, such that
t k t k
′ ′
kϕ¯ (a)ϕ¯ (a )−ϕ¯ (aa )k < 1/k
t t t
′ ′
kϕ¯ (a)+αϕ¯ (a )−ϕ¯ (a+αa )k < 1/k
t t t
∗ ∗
kϕ¯ (a) −ϕ¯ (a )k < 1/k
t t
kϕ¯ (a)k < kak+1/k
t
for a,a′ ∈ A and t ≤ k; such that similar estimates hold for {ψ¯ } on {B }, and {θ¯}
k+100 t k t
on {C }; and such that
k
{ϕ¯ (a) : a ∈ A , t ≤ k} ⊂ B
t k+100 k
10