Homological index formulas for elliptic operators over C*-algebras

0.29 MB·English

by  Charlotte Wahl

Homological index formulas for elliptic operators
9
0 over C -algebras
0 ∗
2

n Charlotte Wahl ∗
a
J

3

] Abstract
T
K Weproveindexformulasforellipticoperatorsactingbetweensections
. ofC∗-vectorbundlesonaclosedmanifold. TheformulasinvolveKaroubi’s
h Cherncharacter from K-theoryof aC∗-algebra todeRhamhomology of
t
a smoothsubalgebras. Weshowhowtheyapplytothehigherindextheorem
m for coverings and to flat foliated bundles, and prove an index theorem
[ for C∗-dynamical systems associated to actions of compact Lie groups.
In an Appendix we relate the pairing of odd K-theory and KK-theory
2
to the noncommutative spectral flow and prove the regularity of elliptic
v pseudodifferential operators over C∗-algebras.
4
9
6
3 1 Introduction
0
6
0 OneofthegeneralizationsoftheAtiyah-Singerindextheoremistoellipticpseu-
/ dodifferential operators associated to C∗-vector bundles. Mishenko–Fomenko
h
introduced these operators and their index, an element in the K-theory of the
t
a C∗-algebra [MF]. Furthermore they defined a Chern character for C∗-vector
m
bundlesandusedittoformulateandproveananalogueoftheAtiyah–Singerin-
v: dextheorem. However,ingeneralitisnotclearhowtocalculatetheMishenko–
i Fomenko Chern character of a C∗-vector bundle: Its definition is based on the
X
map K (C(M, )) C K (C(M)) K ( ) C K (C(M)) K ( ) C
0 0 0 1 1
r for a closed maAnifo⊗ld M→and a unital⊗C∗-algAeb⊗ra ⊕, which exists⊗by KAu¨nn⊗eth
a
A
formula.
In this paper we prove index theorems for the same situation using Karoubi’s
Chern character from the K-theory of a C∗-algebra to the de Rham homology
of smooth subalgebras [Ka]. Karoubi’s Chern character is a generalization of
the Chern characterin differential geometry and is closely relatedto the Chern
character in cyclic homology. Karoubi’s de Rham homology has been used
especially in noncommutative superconnections proof beginning with [Lo].
Wealsoprove(intheAppendix)thatthepairingK ( ) KK ( , ) K ( ),
1 1 1
A × A B → B
where , areunitalC∗-algebras,canbeexpressedintermsofthenoncommu-
A B
tative spectralflow,whichwasintroducedinthe contextoffamily index theory
by Dai–Zhang [DZ]. See [Wa] for further references and a systematic account.
The formula is well-known for =C and the ordinary spectral flow.
B
∗ThisresearchwasfundedbyagrantofAdvanceVT
1
Themainingredientoftheproofoftheindextheoremisaresultaboutthecom-
patibility of Karoubi’s Chern character with the tensor product in K-theory.
This allows the comparison of Karoubi’s Chern character with Mishenko-
Fomenko’s Chern character.
Our proof generalizes the derivation of Atiyah’s L2-index theorem from the
Mishenko–Fomenkoindex theoremin [Sc]. It is also closelyrelatedto the proof
of an index theorem for flat foliated bundles in [J], which is a special case of
Connes’ index theorem for foliated manifolds [C2, p. 273] and implies the C∗-
algebraic version of higher index theorem of Connes-Moscovici [CM]. As an
illustration we derive the the latter in detail from our formula. We also show
how to apply the formula to flat foliated bundles. In this context we introduce
a smooth subalgebra which is defined in more general situations than the one
in [J].
We also prove an index theorem for Toeplitz operators associated to a C∗-
dynamical system ( ,G,α) where G is a compact Lie group. The Chern char-
A
acterinvolvedherehasbeendefinedin[C1]. In[Le]asimilarindextheoremwas
proven for G = IR using Breuer-Fredholm operators. We relate both theorems
in the case where the IR-action is periodic.
In the Appendix we explain how the pairing of K-theory with KK-theory is
relatedtoindextheoryandcollectsomeusefulfactsaboutpseudodifferentialop-
eratorsoverC∗-algebrasbeyondthoseprovenin[MF],inparticularthatelliptic
pseudodifferential operators are adjointable as bounded operators between ap-
propriateSobolevspacesandregularasunboundedoperatorsonafixedSobolev
space.
Ifnotspecified,tensorproductsbetweengradedspacesaregraded,andbetween
Fr´echet spaces they are completed projective.
Acknowledgements: I would like to thank Peter Haskell for helpful comments
on previous versions of this paper.
2 De Rham homology and the Chern character
2.1 Definition
In this section we recall and slightly extend the definition of Karoubi’s Chern
character and collect properties that are relevant for index theory. The main
reference is [Ka].
Let be a locally m-convex Fr´echet algebra.
∞
A
The left -module of differential forms of order k of is defined as
∞ ∞
A A
Ωˆ := ( /C)⊗k
k ∞ ∞ ∞
A A ⊗ A
and the ZZ-graded space of all differential forms is
∞
Ωˆ := Ωˆ .
∗ ∞ k ∞
A A
kY=0
2
There is a differential d on Ωˆ of degree one defined by
∗ ∞
A
d(a ... a )=1 a ... a
0 k 0 k
⊗ ⊗ ⊗ ⊗ ⊗
and a product determined by the properties that
a ... a =a da ...da
0 k 0 1 k
⊗ ⊗
and that Leibniz rule holds which says that for α Ωˆ , β Ωˆ
k ∞ ∗ ∞
∈ A ∈ A
d(αβ)=(dα)β+( 1)kαdβ .
−
With these structures Ωˆ is a graded differential locally m-convex Fr´echet
∗ ∞
A
algebra.
For a closed manifold M
Ωˆp,q(M, ):=Ωˆp(M,Ωˆ )=Ωˆp(M) Ωˆ ,
∞ q ∞ q ∞
A A ⊗ A
where Ωˆ∗(M) is the space of smooth differential forms on M.
We call an open subset U M regular if the compactly supported de Rham
⊂
cohomology H∗(U) is finite-dimensional and if there are open subsets U ,U
c 0 1
withU U andU U suchthatthereisasmoothhomotopyF :[0,1] U
0 1 1
⊂ ⊂ × →
U with F(0,x)=x for all x U and such that F−1( 1 U) 1 U and
1 0
∈ { }× ⊂{ }×
F−1( t U) t U for all t [0,1].
{ }× ⊂{ }× ∈
For a regular open subset U in M we define Ωˆp,q(U, ) to be the closure of
0 A∞
the subspace of Ωˆp,q(M, ) spanned by forms with support in U.
∞
A
The product on Ωˆ∗∗(U, ) is determined by the natural isomorphism
Ωˆ∗∗(U, ) = Ωˆ∗(U0) ΩˆA∞ . Here the right hand side is understood as
0 A∞ ∼ 0 ⊗ ∗A∞
a graded tensor product of graded algebras. Let d be the de Rham dif-
U
ferential on U. The differential of the total complex of the double complex
(Ωˆ∗∗(U, ),d ,d) is denoted by d and its homology by H∗(U, ). The
0 A∞ U tot 0 A∞
definition does not depend on the embedding of U into M as a regular subset.
For a closed manifold M we usually omit the suffix and write H∗(M, ).
∞
A
For F as above let f = F(t, ) : U U. Then f∗ : H∗(U) H∗(U) is
inverse to the map Ht∗(U) H· ∗(U),→since for a clos1ed for0m ω →Ωˆ∗(cU) the
c → 0 ∈ 0
form f∗ω is a closed form supported in U and f∗ω ω = d 1F∗ω. Hence
1 1 − U 0
H∗(U)=H∗(U). R
c ∼ 0
The isomorphism Ωˆ∗∗(U, )=Ωˆ∗(U) Ωˆ induces isomorphisms
0 A∞ ∼ 0 ⊗ ∗A∞
Ωˆ∗∗(U, )/[Ωˆ∗∗(U, ),Ωˆ∗∗(U, )] =Ωˆ∗(U) Ωˆ /[Ωˆ ,Ωˆ ] ,
0 A∞ 0 A∞ 0 A∞ s ∼ 0 ⊗ ∗A∞ ∗A∞ ∗A∞ s
Hn(U, )= Hp(U) Hq( )= Hp(U,H ( )) .
0 A∞ ∼⊕p+q=n 0 ⊗ 0 A∞ ∼⊕p+q=n 0 q A∞
These isomorphisms have been proven in [Ka, 4.7, 4.8] in a slighly different
§§
situation. The proof carries over. It uses completed tensor products, therefore
we use Ωˆ∗(U) instead of compactly supported forms for the definition of coho-
0
mology. The proof uses furthermore the fact that H∗(U) is finite-dimensional.
0
WecallasmoothpossiblynoncompactmanifoldM regularifthereisacovering
(U ) by regular subsets with U U .
n n∈IN n n+1
⊂
3
Extending a form by zero induces a well-defined push forward map
H∗(U , ) H∗(U , ) so that we can define
0 n A∞ → 0 n+1 A∞
H∗(M, )= lim H∗(U , ) .
c A∞ 0 n A∞
n−→→∞
ItisclearthatH∗(M)agreeswiththe compactlysupporteddeRhamcohomol-
c
ogy of M.
If M B is a fiber bundle of regular oriented manifolds, then integration over
→
the fiber yields a homomorphism
:H∗(M, ) H∗−dimMb(B, ).
Z c A∞ → c A∞
Mb
The de Rham homology of is
∞
A
H ( ):=H∗( , ) ,
∗ ∞ ∞
A ∗ A
where is the point.
∗
If M is a closed manifold, we usually write H∗(M, ) for H∗(M, ). Note
that then the quotient map Ωˆ (C∞(M, )) A∞ Ωˆp,q(0M, A∞) induces
n ∞ p+q=n ∞
A → ⊕ A
a homomorphism
H (C∞(M, )) H∗(M, ) .
∗ ∞ ∞
A → A
We proceed with the definition and the properties of the Chern character.
Let be a unital C∗-algebra and let be a dense subalgebra that
∞
A A ⊂ A
is closed under involution and holomorphic functional calculus in . Assume
A
that is endowed with the topology of a locally m-convex algebra such that
∞
A
֒ is continuous. We call such a subalgebra a smooth subalgebra of .
∞
A →A A
Let M be a regular manifold. Recall that K (C (M, )) =
0 0
A
Ker(K (C (M, )+) K (C)), where C (M, )+ denotes the unitalization of
0 0 0 0
A → A
C (M, ). Since C∞(M, )+ is dense and closed under holomorphic func-
0 A c A∞
tionalcalculusin C (M, )+, we havethat K (C∞(M, ))=K (C (M, )).
0 A 0 c A∞ ∼ 0 0 A
The Chern characterform of a projectionP M (C∞(M, )+) is defined as
∈ n c A∞
∞ ( 1)k
chM (P):= − trP(d P)2k .
A∞ (2πi)kk! tot
Xk=0
Thenormalizationdiffersfromthenormalizationin[Ka]andischosensuchthat
the Chern character of the Bott element B K (C ((0,1)2)) integrated over
0 0
∈
(0,1)2equals1. (ThereisalsosomeambiguityaboutthesignoftheBottelement
B in the literature. Here we take B = 1 [H] Ker(K (C(S2)) K (C)),
0 0
− ∈ →
where H is the Hopf bundle.)
In the following proposition we denote by P M (C) the image of P
∞ n
∈ ∈
M (C∞(M, )+) under “evaluation at infinity”.
n c A∞
Proposition 2.1. 1. chM (P) is closed.
A∞
4
2. Let P :[0,1] M (C∞(M, )+) be a differentiable path of projections
→ n c A∞
and let U M be such that supp(P(t) P (t)) U for all t [0,1].
∞
⊂ − ⊂ ∈
Then there is a form α Ωˆ∗∗(U, )/[Ωˆ∗∗(U, ),Ωˆ∗∗(U, )] such
that d α=chM (P(1))∈ch0M (PA(0∞)). 0 A∞ 0 A∞ s
tot A∞ − A∞
3. The Chern character form induces a well-defined homomorphism
K (C (M, )) H∗(M, ) .
0 0 A → c A∞
Proof. For M compact the proofs are standard. We include the proof of 2) in
order to show that it works in the noncompact case as well:
From Leipniz rule on deduces that the terms PP′P, (1 P)P′(1
− −
P), P(d P)P, (1 P)(d P)(1 P) all vanish.
tot tot
− −
Hence
tr(P(d P)2k)′ = trP′(d P)2k+trP((d P)2k)′
tot tot tot
= trP((d P)2k)′
tot
2k−1
= trP(d P)i(d P)′(d P)2k−i−1 .
tot tot tot
Xi=0
This vanishes for k =0.
For i even and k =0
6
trP(d P)i(d P)′(d P)2k−i−1
tot tot tot
= tr(d P)iP(d P′)(d P)2k−i−1
tot tot tot
= tr(d P)i(d (PP′))(d P)2k−i−1 tr(d P)i(dP)P′(d P)2k−i−1
tot tot tot tot tot
−
= tr(d P)i(d (PP′))(d P)2k−i−1
tot tot tot
= d trP(d P)i−1(d (PP′))(d P)2k−i−1 .
tot tot tot tot
Note that trP(d P)i−1(d (PP′))(d P)2k−i−1 vanishes on U for k =0.
tot tot tot
6
For i odd the argument is similar.
We define the odd Chern character via the following diagram:
K (C ((0,1) M, )) ∼= K (C (M, ))
0 0 1 0
× A −−−−→ A
chA(0∞,1)×M chMA∞ (2.1.1)
 
Hev((0,1)y M, ) R01 Hodd(My, ) .
c × A∞ −−−−→ c A∞
Note that 1 : H∗((0,1) M, ) H∗(M, ) is an isomorphism by
0 c × A∞ → c A∞
Hc∗((0,1)×RM,A∞)∼=Hc∗((0,1))⊗Hc∗(M,A∞)∼=Hc∗(M,A∞) .
InthefollowingwederiveaformulafortheoddCherncharacter. Itisanalogous
tothosewell-knownindeRhamcohomologyandcyclichomology(comparewith
[Gl]).
5
Proposition 2.2. For u U (C∞(M, )+) in H∗(M, )
∈ n c A∞ c A∞
∞ 1 k (k 1)!
chM ([u])= − − u∗(d u)((d u∗)(d u))k−1 .
A∞ (cid:18)2πi(cid:19) (2k 1)! tot tot tot
Xk=1 −
Proof. Here we use that the Chern character can be defined in terms of non-
commutative connections [Ka].
Let P M (C) be the projection onto the first n components. Let
n 2n
∈
W(t) C∞([0,1],U (C∞(M, )+)) withW(0)=1 andW(1)=diag(u,u∗).
∈ 2n c A∞
Then the isomorphism K (C (M, )) K (C((0,1) M, )) maps [u] to
1 0 0
A → × A
[WP W∗] [P ]. The Chern character is independent of the choice of the
n n
−
connection [Ka, Th. 1.22], thus we may use the connection WP (d +
n tot
dx∂ +xW(1)∗d (W(1)))W∗ onthe projectiveC∞((0,1) M, )+-module
x tot c × A∞
WP W∗(C∞((0,1) M, )+)n for its calculation. It follows that
n c × A∞
chM (u)
A∞
= ch(0,1)×M(WP W∗)
A∞ n
∞ 1 ( 1)k
= − tr(x2u∗(d u)u∗(d u)+x(d u∗)(d u)+dx u∗(d u))k
Z (2πi)kk! tot tot tot tot tot
Xk=0 0
∞ ( 1)k 1
= − tr((x x2)(d u∗)(d u)+dx u∗(d u))k
(2πi)kk!Z − tot tot tot
Xk=0 0
∞ ( 1)k 1
= − dx (x x2)k−1u∗(d u)((d u∗)(d u))k−1
(2πi)kk!Z − tot tot tot
Xk=1 0
∞ k
1 (k 1)!
= − − u∗(d u)((d u∗)(d u))k−1 .
tot tot tot
(cid:18)2πi(cid:19) (2k 1)!
Xk=1 −
2.2 Chern character and tensor products
From now on assume that M is a closed manifold.
Let K ( ) :=K ( ) C.
i C i
A A ⊗
InthefollowingweprovethecompatibilityoftheCherncharacterwiththeBott
periodicity map K (C ((0,1), )=K ( ) and with the Ku¨nneth formulas
1 0 A ∼ 0 A
K (C(M)) K ( ) K (C(M)) K ( ) =K (C(M, ))
0 C⊗ 0 A C⊕ 1 C⊗ 1 A C ∼ 0 A C
and
K (C(M)) K ( ) K (C(M)) K ( ) =K (C(M, )) .
0 C⊗ 1 A C⊕ 1 C⊗ 0 A C ∼ 0 A C
These isomorphisms are defined via the tensor product
K (C(M)) K ( ) K (C(M, )), i,j ZZ/2 .
i j i+j
⊗ A → A ∈
The tensor product is injective, hence we may consider K (C(M)) K ( ) as
i j
⊗ A
a subspace of K (C(M, )).
i+j
A
6
First recall the definition of the tensor product. For i,j =0 the tensor product
is induced by the tensor product of projections. The remaining three cases
are derived from the tensor product of projections using Bott periodicity, for
example
K (C(M)) K ( ) = K (C ((0,1) M)) K (C ((0,1)) )
1 ⊗ 1 A ∼ 0 0 × ⊗ 0 0 ⊗A
K (C ((0,1)2 M, ))
0 0
→ × A
= K (C(M, ))
∼ 0 A
and
K (C(M)) K ( ) = K (C(M)) K (C ((0,1), ))
0 ⊗ 1 A ∼ 0 ⊗ 0 0 A
K (C ((0,1) M, ))
0 0
→ × A
= K (C(M, )) .
∼ 1 A
A standard calculation (see [Ka, Th. 1.26]) shows that the tensor product for
i = j = 0 is compatible with the Chern character, namely for a K (C(M))
0
∈
and b K ( )
0
∈ A chM (a b)=chM(a)ch (b) .
A∞ ⊗ A∞
In the following proposition β : K (C(M, )) K (C ((0,1)2 M, ), a
0 0 0
A → × A 7→
a B is the Bott periodicity map.
⊗
Proposition 2.3. 1. For a K (C ((0,1)2 M, ))
0 0
∈ × A
chM β−1(a)= ch(0,1)2×M(a) .
A∞ Z A∞
(0,1)2
2. For a K (C ((0,1) M)) and b K (C ((0,1), ))
0 0 0 0
∈ × ∈ A
chM β−1(a b)= ch(0,1)×M(a)ch(0,1)(b) .
A∞ ⊗ Z A∞
(0,1)2
Proof. WeconsiderK (C ((0,1)2 M, ))asasubgroupofK (C(T2 M, )).
0 0 0
× A × A
1) Let b K (C(M, )) with a=B b. Then
0
∈ A ⊗
ch(0,1)2×M(B b) = chT2×M(B b)
Z A∞ ⊗ Z A∞ ⊗
(0,1)2 T2
= chM (b) chT2(B)
A∞ Z
T2
= chM (b) .
A∞
2) The assertion follows from the commutative diagram
K (C ((0,1) M)) K (C ((0,1), )) K (C(S1 M)) K (C(S1, ))
0 0 0 0 0 0
× ⊗ A −−−−→ × ⊗ A
⊗
 
K0(C0((0,1y)2 M, )) ֒→ K0(C(T2y M, )) .
× A −−−−→ × A
chMA∞◦β−1 RT2chAT2∞×M
 
H∗(My, ∞) = H∗(My, ∞)
A −−−−→ A
7
Sincethehorizontalarrowsareinclusions,thefirstverticalmaponthelefthand
side is determined by the first vertical map on the right hand side. The second
square commutes by 1).
Corollary 2.4. The diagram
K (C ((0,1) M, ) ∼= K (C(M, ))
1 0 0
× A −−−−→ A
chA(0∞,1)×M chMA∞
 
Hodd((0,1)y M, ) R01 Hev(My, )
c × A∞ −−−−→ A∞
commutes.
Proof. Consider the diagram
K (C ((0,1) M, )) ∼= K (C ((0,1)2 M, )) β−1 K (C(M, ))
1 0 0 0 0
× A ←−−−− × A −−−−→ A
chA(0∞,1)×M chA(0∞,1)2×M chMA∞ .
  
Hodd((0,1)y M, ) R01 Hev((0,1)2y M, ) R(0,1)2 Hev(My, )
c × A∞ ←−−−− c × A∞ −−−−→ A∞
The first square commutes by diagram 2.1.1 applied to (0,1) M. The second
×
square commutes by the first part of the previous proposition.
We denote by
R :K (C(M, )) K (C(M)) K ( ) K (C(M, ))
jk i C j C k C i C
A → ⊗ A ⊂ A
the projections induced by the Ku¨nneth formulas.
We have a tensor product
chM ch :K (C(M)) K ( ) H∗(M) H ( )=H∗(M, ) .
⊗ A∞ i C⊗ i A C → ⊗ ∗ A∞ ∼ A∞
Proposition 2.5. 1. On K (C(M, ))
0 C
A
chM =(chM ch ) R +(chM ch ) R .
A∞ ⊗ A∞ ◦ 00 ⊗ A∞ ◦ 11
2. On K (C(M, ))
1 C
A
chM =(chM ch ) R +(chM ch ) R .
A∞ ⊗ A∞ ◦ 01 ⊗ A∞ ◦ 10
Proof. 1) follows fromthe previous proposition: Let a b K (C(M, )) with
0
⊗ ∈ A
a K (C(M))) and b K ( ). Let a correspond to a˜ K (C ((0,1) M))
1 1 0 0
an∈d b to˜b K (C ((0,∈1), ))A. ∈ ×
0 0
∈ A
ThenbydefinitionchM(a)= 1ch(0,1)×M(a˜)andch (b)= 1ch(0,1)(˜b). Now
0 A∞ 0 A∞
by the previous lemma R R
chM (a b) = ch(0,1)×M(a˜)ch(0,1)(˜b)
A∞ ⊗ Z A∞
(0,1)2
1 1
= ch(0,1)×M(a˜) ch(0,1)(˜b)
Z Z A∞
0 0
= chM(a)ch (b) .
A∞
8
2) follows applying by 1) to K (C ((0,1) M, )) since the Chern character
0 0
× A
interchangesthe suspensionisomorphismsinK-theoryandde Rhamhomology.
Define ChM as the map
A
K (C(M, )) K (C(M, ))
0 0 C
A → A
= K (C(M)) K ( ) K (C(M)) K ( )
∼ 0 C⊗ 0 A C⊕ 1 C⊗ 1 A C
chM Hev(M) K ( ) Hodd(M) K ( ) .
−→ dR ⊗ 0 A ⊕ dR ⊗ 1 A
and analogously for K (C(M, )). This is the Chern character introduced by
1
A
Mishenko-Fomenko [MF].
The previous Proposition is equivalent to the equation
ch ChM = chM . (2.2.1)
A∞◦ A A∞
2.3 Pairing with cyclic cocycles
In the noncommutative geometry the Chern character with values in the cyclic
homologyis more common than the one with values in the de Rham homology.
DeRhamhomologycanbepairedwithnormalizedcycliccocycles;inthispairing
both Chern characters agree up to normalization:
Let Cλ( ) be the quotient of the algebraic tensor product ( /C)⊗n+1 by
the acntioAn∞of ZZ/(n+1)ZZ. Let A∞
λ λ
b:C ( ) C ( ) ,
n A∞ → n−1 A∞
b(a ...a ) = ( 1)na a ... a
0 n n 0 n−1
⊗ − ⊗ ⊗
∞
+ ( 1)ia ... a a ...a .
0 i i+1 n
− ⊗ ⊗ ⊗
Xi=0
λ
The homology of the complex (C ( ),b) is the reduced cyclic homology
∗ A∞
HC ( ). Using the completed projective tensor product instead of the alge-
∗ ∞
A top
braic one we obtain the topologicalreduced cyclic homology HC ( ). Fur-
∗ A∞
sep λ
thermore we denote by HC ( ) the topological homology of (C ( ),b),
∗ A∞ n A∞
i.e. weusethecompletedprojectivetensorproductandquotientouttheclosure
of the range of b.
∗
The reducedcyclic cohomologyHC ( ) is the homologyofthe dualcomplex
∞
(Cn( ),bt) (in the algebraic sense).AElements of Cn( ) are called normal-
λ A∞ λ A∞ ∗
ized cochains. The continuous reduced cyclic cohomology HC ( ) is the
top A∞
homology of the topological dual complex.
∗ top ∗
ThepairingHC ( ) HC ( ) CdescendstoapairingHC ( )
top A∞ ⊗ ∗ A∞ → top A∞ ⊗
HCsep( ) C. Furthermore the quotient map Ωˆ ( ) Cn( ) induces
∗ A∞ → sep n A∞ → λ A∞
an homomorphism H ( ) HC ( ), which is an embedding for n 1
n A∞ → n A∞ ≥
9
(see [Ka, 4.1 and 2.13]). In degree zero there is a pairing of H ( ) =
0 ∞
§§ A
/[ , ] with traces on .
∞ ∞ ∞ ∞
A A A A
The Chern character chλ :K ( ) HC ( ) is defined by
0 ∞ ∗ ∞
A → A
∞
chλ(p)= ( 1)mtrp⊗2m+1
−
mX=0
for a projection p M ( ). Hence the composition
n ∞
∈ A
chλ sep
K ( ) HC ( ) HC ( )
0 A∞ −→ ∗ A∞ → ∗ A∞
agrees up to normalization with the map
K ( )chA∞ H ( )֒ HCsep( ) .
0 A∞ −→ ∗ A∞ → ∗ A∞
m
In particular if φ HC ( ), then
∈ top A∞
φ chλ =(2πi)mm! φ ch .
◦ ◦ A∞
3 Index theorems
InthefollowingwegiveaformulationoftheMishenko–Fomenkoindextheorem,
whichisdifferentfromtheoriginaloneandadaptedtotheapplications. Further-
morewetranslateitsproofinthelanguageofKK-theory: Weshowthecompat-
ibilityoftheCherncharacterwiththepairingK (C(M, )) KK (C(M),C)
i j
K ( ) for i,j ZZ/2, where on KK ((C(M),C) we uAse⊗the Chern charact→er
i+j j
A ∈
from K-homology to de Rham homology of M. We refer to Appendix 5.1 for
some facts about the connection of KK-theory to index theory.
Lemma 3.1. 1. For x K (C(M)) K ( ) K (C(M, )) and y
i i 0
∈ ⊗ A ⊂ A ∈
KK (C(M),C) with i=j
j
6
x y =0 K ( ) .
C(M) j
⊗ ∈ A
2. Forx K (C(M)) K ( ) K (C(M, )) andy KK (C(M),C)with
i j 1 j
∈ ⊗ A ⊂ A ∈
i=j
6
x y =0 K ( ) .
C(M) i
⊗ ∈ A
It follows that for x K (C(M, )) and y KK (C(M),C)
i j
∈ A ∈
x y =R (x) y K ( ) .
C(M) j,i+j C(M) i+j C
⊗ ⊗ ∈ A
Proof. 1) Let B KK (C ((0,1)2),C) be the Bott element. By the standard
1 0 0
∈
isomorphism K ( ) = KK (C, ) and the fact that the tensor product in K-
i A ∼ i A
theory is a special case of the Kasparovproduct all we have to show is that for
a KK (C,C ((0,1), )) and b KK (C,C ((0,1) M))
j 0 j 0
∈ A ∈ ×
((a b) B ) y =0 .
⊗ ⊗C0((0,1)2) 1 ⊗C(M)
10