In this section we give a second generalisation of the well known Atiyah-Jänich bundle. Assume that A : X → CF(H) is a gap continuous family of Fredholm operators parametrised by a paracompact and locally compact spaceX. We define thesingular setof Ato be
Σ(A) ={x∈X : 0∈σ(Ax)},
which is closed by lemma 2.3.6. Since the topology in the domain has no influence on the properties required in the definition of a Fredholm operator, the bundle morphismA:D(A)→ X×H according to lemma 6.1.3 is fibrewise a Fredholm operator. Therefore we can regardA as an element inF(D(A), X×H)whose support is given by the singular set ofAaccording to remark 6.1.4. Now, if Σ(A) is compact, we obtain by the results of section 5.1 for any closed subspace A⊂X such thatA∩Σ(A) =∅ an element inK-theory
ind(A)∈K(X, A), which we call theindex bundle ofA.
Note at first that, ifA:X → BF(H)is a continuous family of bounded Fredholm operators, we obtain the classical Atiyah-Jänich bundle because of corollary 6.1.14 and the discussion at the end of section 5.2.
We now discuss the main properties of the index bundle in the case of families of generally un-bounded Fredholm operators. Of course most of them follow immediately from the corresponding properties in section 5.2 and the properties of the domain bundle established in section 6.1.
6.2.1 Lemma. LetA:X →GC(H)⊂ CF0(H). Then
ind(A) = 0∈K(X, A).
Proof. Since the induced Banach bundle morphism has empty support, this is an immediate consequence of lemma 5.2.1.
6.2.2 Lemma. Let Y be a paracompact and locally compact space, B ⊂ Y closed and f : (Y, B)→(X, A)proper. Then
ind(f∗A) =f∗ind(A)∈K(Y, B).
Proof. By lemma 6.1.8 we haveD(f∗A) =f∗D(A)and hencef∗Ais a bundle morphism between the bundlesf∗D(A)andf∗(X×H) =Y ×H. Now the assertion follows from lemma 5.2.2.
6.2.3 Lemma. Let H : I×X → CF(H) be gap continuous with a compact singular set such thatA∩Σ(Hλ) =∅for all λ∈I. Then
ind(H0) = ind(H1)∈K(X, A).
Proof. We consider the induced Banach bundle morphism
H :D(H)→(I×X)×H and obtain from lemma 5.2.3
ind(ι∗0H) = ind(ι∗1H)∈K(X, A), where
ι∗0:ι∗0D(H)→ι∗0((I×X)×H) =X×H, ι∗1:ι∗1D(H)→ι∗1((I×X)×H) =X×H are to be interpreted as bundle morphisms. But since
ι∗0D(H) =D(ι∗0H) =D(H0), ι∗1D(H) =D(ι∗1H) =D(H1) by lemma 6.1.8, we infer
ind(ι∗0H) = ind(H0), ind(ι∗1H) = ind(H1),
where the maps on the left hand sides are to be interpreted as bundle morphisms while the maps on the right hand sides are families of unbounded operators.
6.2.4 Corollary. If A → CFsa(H)is a gap continuous family of selfadjoint operators having a compact singular set such thatA∩Σ(A) =∅, then
ind(A) = 0∈K(X, A).
Proof. Consider the homotopy
H :I×X → CF(H), H(λ, x) =Ax+λiI
which is gap continuous by theorem 2.2.1. Since each Ax is selfadjoint we inferΣ(Hλ) =∅ for allλ >0and soind(Hλ) = 0∈K(X, A)by lemma 6.2.1. Now the assertion follows from the foregoing lemma 6.2.3.
6.2.5 Lemma. LetK:X→ K(H)be a continuous family of compact linear operators such that A+λK has a compact singular set inX×I andA∩Σ(A+λ·K) =∅ for allλ∈I. Then
ind(A+K) = ind(A)∈K(X, A).
Proof. SinceD(A+K) =D(A), the perturbed family defines a Banach bundle morphismA+K: D(A)→X×H which is the sum of a Fredholm operator and a compact operator in each fibre.
Now the assertion follows from corollary 5.2.5.
6.2.6 Lemma. Let A1 :X → CF(H) andA2 : X → CF(H) be gap continuous with compact singular sets such thatA∩(Σ(A1)∪Σ(A2)) =∅. Then
ind(A1⊕ A2) = ind(A1) + ind(A2)∈K(X, A).
Proof. Since obviously
D(A1⊕ A2) =D(A1)⊕D(A2), the result follows immediately from lemma 5.2.6.
Remembering the results of section 5.2, one may ask for the logarithmic property. Note that by theorem A.2.12 the product of two densely defined Fredholm operators is again Fredholm.
Moreover, in [Ne68, Cor.2.5] it is proved that the product of two gap continuous families of such operators is continuous as well. However, to establish a logarithmic property seems to be a quite delicate question. Because in order to use the corresponding result 5.2.7 for Banach bundle morphisms, we need to consider the productA2A1 of densely defined Fredholm operators as a composition of bundle morphisms
D(A2A1)−−→A1 D(A2)−−→A1 X×H.
But it is neither clear that
A1:D(A2A1)→D(A2)
is a bundle morphism nor if its index bundle coincides with the index bundle of the original familyA1. Nevertheless we can at least prove the following result.
6.2.7 Lemma. Let U1, U2:X → U(H)be continuous families of unitary operators. Then
U1AU2:X → CF(H) is gap continuous and
ind(U1AU2) = ind(A)∈K(X, A).
Proof. The graph ofU1AU2is given by
graph(U1AU2) ={(u, U1AU2u) :u∈U2∗D(A)}
={(U2∗v, U1Av) :v∈ D(A)}
= U2∗ 0 0 U1
!
graph(A) =:Ugraph(A).
Since the orthogonal projection ontoUgraph(A)is given byU Pgraph(A)U∗, we obtain for any x, x0∈X
dG(U1,xAxU2,x, U1,x0Ax0U2,x0) =kPgraph(U1,xAxU2,x)−Pgraph(U1,x0Ax0U2,x0)k
=kUxPgraph(Ax)Ux∗−Ux0Pgraph(Ax
0)Ux∗
0k ≤ kUxPgraph(Ax)Ux∗−UxPgraph(Ax)Ux∗
0k +kUxPgraph(Ax)Ux∗
0−Ux0Pgraph(Ax)Ux∗
0k+kUx0Pgraph(Ax)Ux∗
0−Ux0Pgraph(Ax
0)Ux∗
0k
≤ kPgraph(Ax)−Pgraph(Ax
0)k+ 2kUx−Ux0k
≤dG(Ax,Ax0) + 2kUx−Ux0k,
where we use corollary 2.4.3 and lemma 2.1.3 in the last inequality. Hence the first assertion is proved.
In order to prove the second assertion, we use Kuiper’s theorem as already quoted in the proof of theorem 1.1.8 in order to choose two homotopies
H1, H2:I×X → U(H) such that
H1(0,·) =U1, H2(0,·) =U2, H1(1,·) =H2(1,·) =I.
Since the family
H :I×X → CF(H), H(t, x) =H1(t, x)AxH2(t, x)
is continuous with respect to the gap topology by our first assertion, we obtain by the homo-topy invariance property 6.2.3
ind(U1AU2) = ind(H(0)) = ind(H(1)) = ind(A)∈K(X, A).
6.2.8 Remark. As we already mentioned above, one can prove that the product of two gap continuous families of densely defined Fredholm operators is again continuous in the gap topology.
Since U(H) is a deformation retract of GL(H), we obtain that lemma 6.2.7 actually holds true if U1 andU2 are families of bounded invertible operators.
We finally want to mention three constructions that are related to our definition ofind(A) for gap continuous families of closed operators.
In [Ne68] Gerhard Neubauer studied certain metric spaces consisting of graphs of closed operators between Banach spaces with the metricˆδwe introduced in the first section of the second chapter.
He constructed a set valued index for families of such graphs parametrized by a compact space and showed that in the case that all graphs correspond to Fredholm operators, the obtained set is actually a group which turned out to be isomorphic toK(X). Finally, he even proved that his construction yields an isomorphism
[X,CF(H)]→K(X), (6.8)
showing thatCF(H)with the gap topology represents theK-functor (compare theorem 4.4.3).
We conjecture that Neubauer’s index coincides with our construction. However, compared to our definition his index is difficult to compute because every map into the space of graphs has to be deformed into a canonical form before the element inK(X)can be built. We do not know how complicate it can be to prove that both constructions yield the same element inK(X)and leave this question open for future research.
A further related construction is due to Patrick M. Fitzpatrick and Maria Testa [FT94] who generalised the so called parity to unbounded Fredholm operators that had been defined before in the bounded case in [FP88]. They considered gap continuous families A: X → CF0(H)of
Fredholm operators of index0parametrised by a compact metric space and constructed for any choice of a subspaceV ⊂H such that
im(Ax) +V =H, x∈X,
a continuous family of projectionsP :X → L(H)such thatimPx=A−1x (V).
We now can use proposition 1.2.6 in order to obtain a finite dimensional bundle over X from this family which has the same total space than ourE(A, V)as defined in section 5.1. Moreover, it seems that assuming the base space to have a metric and restricting to Fredholm operators of index0 is unnecessary in the construction in [FT94]. However, we loose the feature to view Aas a bundle morphism between Banach bundles and to obtainE(A, V)from it as the kernel of an explicit bundle epimorphism. Moreover, it seems to be hard to prove the properties of the index bundle from their definition. Conversely, since our construction transforms a gap continuous familyA:X → CF(H)to a family of bounded Fredholm operators, one should try to transfer the definition of parity from [FP88] to unbounded Fredholm operators by using the domain bundle instead of the family of projectionsP from above.
Finally we want to mention the work by Michael Joachim [Jo03] which we already quoted in theorem 4.4.3. He constructed a different bijection as in (6.8) which is however far from being elementary and relies on a result due to himself, Bunke and Stolz [BJS03]. Also here we do not know yet if this map coincides with the corresponding one induced by our index bundleindand leave this question open for future research.