SKETCHES OF NONCOMMUTATIVE TOPOLOGY
ALEXEY KUZMIN
Contents
1. C˚-algebras 2
2. C0pXq-structures 3
3. Actions of groups and crossed products 3
4. KK-theory 4
5. Rieffel deformation 5
6. Classification of Kirchberg algebras 6
7. Noncommutative tori 7
8. Wick algebras 7
9. Pseudodifferential operators 9
9.1. Pseudodifferential operators 9
9.2. Schwartz kernels 10
9.3. Adjoints and products 10
9.4. Elliptic operators 11
9.5. Pseudodifferential operators on manifolds 12
9.6. Symbol of an operator 12
10. Functional analytic properties of a pseudodifferential operator 12
10.1. Boundedness 12
10.2. Compactness 12
10.3. Selfadjointness, normalness and unitarity 12
10.4. Essential normality 13
10.5. Spectrum 13
10.6. Fredholmness 13
11. Pseudodifferential operators and BDF theory 13
11.1. BDF theory 13
11.2. Classification of classical pseudodifferential operators 14
11.3. Conjectural approximation of abstract essentially normal operators
by pseudodifferential operators 15
12. Dirac operators 15
12.1. Clifford algebra 15
12.2. Operators of Dirac type 16
13. Many incarnations of the Atiyah-Singer index theorem 17
13.1. Atiyah-Singer 17
13.2. Toeplitz index theorem 18
13.3. Gauss-Bonnet 18
13.4. Riemann-Roch 18
Date: November 4, 2022.
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13.5. Hirzebruch index theorem 19
13.6. Gromov-Lawson-Rosenberg conjecture 19
13.7. Stolz conjecture 19
14. Summary of Paper I 20
15. Summary of Paper II 20
16. Summary of Paper III 21
17. Summary of paper IV 23
18. Summary of paper V 23
References 24
1. C˚-algebras
Given a compact Hausdorff space X, the set of continuous complex-valued func-
tions CpXq on X forms an algebra under pointwise multiplication. The algebra is
unital as the constant function 1 is the multiplicative identity. Taking the point-
wise complex conjugate of a function defines a ˚-operation that makes CpXq into
a ˚-algebra. We also define the supremum norm ‖f‖ P r0,8q of f P CpXq by
‖f‖ “ max |fpxq|.
xPX
Since a continuous real function on a compact set attain its maximum value, it is
immediate that ‖f‖ exists. Any Cauchy sequence of continuous functions in CpXq
converges to a continuous function, so the normed space CpXq is complete. For
f, g P CpXq we have
‚ ‖fg‖ ď ‖f‖‖g‖, so CpXq is a Banach algebra.
‚ ‖f˚‖ “ ‖f‖, so CpXq is a Banach ˚-algebra.
‚ ‖f˚f‖ “ ‖f‖2, so CpXq is a C˚-algebra.
Definition 1.1. A C˚-algebra is a Banach ˚-algebra over the field of complex
numbers with the following properties:
‚ ‖xy‖ ď ‖x‖‖y‖;
‚ ‖x˚‖ “ ‖x‖;
‚ ‖xx˚‖ “ ‖x‖2.
CpXq is an example of a commutative unital C˚-algebra. The Gelfand theorem
(see [1]) states that every commutative unital C˚-algebra is isomorphic to CpXq
for some compact Hausdorff space X.
Given a set G, a relation on G is a set R consisting of pairs pp, ηq where p is a
˚-polynomial on G and η is non-negative real number. A representation of pG,Rq
on a Hilbert space H is function ρ from G to the algebra of bounded operators on
H such that ‖p ˝ ρpRq‖ ď η for all pp, ηq in R. The pair is called admissible if a
representation exists and the direct sum of representations is also a representation.
Then
‖a‖univ “ supt‖ρpaq‖ : ρ is a representation of pG,Rqu
is finite and defines a seminorm satisfying the C˚-norm condition on the free algebra
on G.
Definition 1.2. The completion of the quotient of the free algebra by the ideal
ta : ‖a‖univ “ 0u in norm ‖¨‖univ is called the universal C˚-algebra of pG,Rq.
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Example 1.3. Let H be a separable Hilbert space. The algebra KpHq of com-
pact operators on H is a norm closed subalgebra of BpHq. It is also closed under
involution; hence it is a C˚-algebra.
Example 1.4 ([9]). Let n ě 2 and H be a separable Hilbert space. Consider the
C˚-algebra generated by n isometries S1, . . . , Sn acting on H satisfying
n
ÿ
S S˚i i “ 1.
i“1
The concrete C*-algebra generated by S1, . . . , Sn is isomorphic to the universal
C*-algebra On generated G “ t1, s1, . . . , snu and R “ t1 “ 1˚ “ 12, 1s1 “ s11 “
s , . . . , 1s “ s 1 “ s , s˚
řn
1 n n n 1s1 “ 1, . . . , s˚s “ 1, s ˚n n i“1 isi “ 1u, where η “ 0 for
each polynomial described in R. On is called the Cuntz algebra.
Example 1.5. Given a skew-symmetric matrix nˆ n matrix Θ the noncommuta-
tive torus CpTnΘq is defined as the universal C˚-algebra generated by n unitaries
u1, . . . , un subject to the relations
uiu “ e´2πiΘijj ujui.
Properties of this C˚-algebra will be discussed in Section 7.
2. C0pXq-structures
Let X be a locally compact Hausdorff space and let C0pXq be the C˚-algebra of
continuous functions on X that vanish at infinity.
Definition 2.1 ([30]). A C0pXq-structure on a C˚-algebra A is a monomorphism
Φ : C0pXq Ñ ZMpAq
such that the ideal ΦpCpXqq ¨A is dense in A.
For x P X consider closed two-sided ideal
Ix “ tΦpfq ¨ a, a P A, f P C0pXq such that fpxq “ 0u.
The fiber Apxq of A over x is defined as
Apxq “ A{Ix,
and the canonical quotient map evx : AÑ Apxq is called the evaluation map at x.
Definition 2.2. An action α of a locally compact group G on a C0pXq-C˚-algebra
A is said to be fibrewise if
αgpΦpfqaq “ Φpfqpαgpaqq, g P G, a P A, f P C0pXq.
If an action α is fibrewise then it induces an action αx of G on Apxq for every
x P X making the fiber restriction equivariant.
3. Actions of groups and crossed products
LetG be a locally compact group with a choice of (left) invariant Haar measure λ,
A be a C˚-algebra and α : GÑ AutpAq be an action of G on A. A ˚-homomorphism
ϕ : A Ñ B between C˚-algebras with G-actions α and β is called equivariant if
ϕpαgpaqq “ βgpϕpaqq for every a P A, g P G.
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Definition 3.1. A covariant representation of pG,A, αq on a Hilbert space H is a
pair pv, πq consisting of a unitary representation v : GÑ UpHq and a representation
π : AÑ BpHq, satisfying the covariance condition
vpgqπpaqvpgq˚ “ πpαgpaqq
for all g P G and a P A.
Let CcpG,A, αq be the linear space of compactly supported continuous A-valued
functions on G. Given f, g P CcpG,A, αq we define multiplication as the following
twisted convolution product:
ż
pf ´11 ¨α f2qpgq “ f1phqαhpf2ph gqqdλphq.
G
On a locally compact group G there exists unique scalar function ∆ such that
for every Borel subset S Ă G it holds that λpg´1Sq “ ∆pgqλpSq. We define the
˚-operation on CcpG,A, αq by
f˚pgq “ ∆pgq´1αgpfpg´1q˚q.
Definition 3.2. The integrated form of a covariant representation pv, πq is the
representation v ˆ π : CcpG,A, αq Ñ BpHq given by
ż
pv ˆ πqpfqξ “ πpfpgqqvpgqξdλpgq.
G
Definition 3.3. The crossed product A ¸α G is the completion of the ˚-algebra
CcpG,A, αq by the norm
‖f‖u :“ supt‖pv ˆ πqpfq‖ : pv, πq is a covariant representationu.
If A “ C then we call C¸G the group C˚-algebra C˚pGq.
4. KK-theory
KK-theory (see [1]) is a bivariant functor that jointly generalizes operator K-
theory and K-homology: for two C˚-algebras A,B, the KK-group KKpA,Bq is
a homotopy equivalence class of pA,Bq-Hilbert C˚-bimodules equipped with an
additional Fredholm module structure. The KK-group KKpA,Bq behave as K-
homology of A in the first argument and as operator K-theory of B in the second.
For further references see [1].
Definition 4.1. For a C˚-algebra B, a Hilbert C˚-module over B is a complex
vector space E equpped with an action of B from the right and a sesquilinear map
x¨, ¨y : E ˆ E Ñ B such that
(1) xx, yy˚ “ xy, xy;
(2) xx, xy ě 0;
(3) xx, xy “ 0 precisely if x “ 0;
(4) xx, y ¨ by “ xx, yy ¨ b;
1
(5) E is complete with respect to the norm ‖x‖E “ ‖xx, xy‖ 2B .
Definition 4.2. For a C˚-algebra B and E a Hilbert C˚-module over B, a C-linear
operator F : E Ñ E is called adjointable if there is an adjoint operator F˚ : E Ñ E
with respect to the A-valued inner product in the sense that
xF paq, by “ xa, F˚pbqy, for all a, b P E.
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Adjointable operators are automatically bounded and B-linear. We denote the
set of adjointable operators on a Hilbert B-module E by BBpEq.
Definition 4.3. For a Hilbert C˚-module E, an adjointable operator T : E Ñ E
is of finite rank if it is of the form
n
ÿ
T : v ÞÑ wixvi, vy
i“1
for vi P E and wi P E. T is called B-compact operator if it is in the norm-closure
of finite-rank operators.
We denote the set of B-compact operators on a Hilbert C˚-module E over B by
KBpEq.
Definition 4.4. For C˚-algebras A,B an pA,Bq-Hilbert C˚-bimodule is a Hilbert
C˚-module over B equipped with an action of A from the left such that all a P A
are adjointable for the B-valued inner product
xa˚ ¨ x, yy “ xx, a ¨ yy.
Definition 4.5. For C˚-algebras A,B a Kasparov pA,Bq-bimodule is a Z2-graded
pA,Bq-Hilbert bimodule E equipped with an adjointable oddly-graded bounded
operator F P BBpEq such that for all a P A
(1) pF 2 ´ 1qπpaq P KBpEq.
(2) rF, πpaqs P KBpEq.
(3) pF ´ F˚qπpaq P KBpEq.
Definition 4.6. A homotopy between two Kasparov pA,Bq-bimodules is a pA,Cpr0, 1s, Bqq-
bimodule that interpolates between the two.
A homotopy of Kasparov bimodules is an equivalence relation.
Definition 4.7. We write KKpA,Bq for the set of equivalence classes of Kasparov
pA,Bq-bimodules under homotopy.
KKpA,Bq is an abelian group under direct sum of bimodules and operators.
There is a composition operation KKpA,Bq ˆKKpB,Cq Ñ KKpA,Cq called the
Kasparov product. We will denote by ˝ : KKpA,Bq ˆ KKpB,Cq Ñ KKpA,Cq
the Kasparov product, and by b : KKpA,Bq ˆKKpC,Dq Ñ KKpAb C,B bDq
the exterior tensor product. We say that two C˚-algebras A and B are KK-
isomorphic if there exists an invertible element in KKpA,Bq with respect to the
Kasparov product. Given a homomorphism φ : AÑ B, put rφs P KKpA,Bq to be
the induced KK-morphism. For more details see [1, 15].
5. Rieffel deformation
Given a C˚-algebra A with an action α of Rn and a skew-symmetric matrix
Θ one can consider the Rieffel deformation AΘ of A, see [31]. Let C nbpR , Aq be
the C˚-algebra of bounded continuous A-valued functions on Rn equipped with
the supremum norm. There is an action of Rn on C pRnb , Aq by translations. De-
note BApRnq to be the set of smooth elements for the translation action of Rn
on C pRnb , Aq. For f, g P BApRnq we define product by the following oscillatory
integral
ż
pf ˆΘ gqptq :“ fpt`Θpzqqgpt` sqe2πiz¨sdzds,
RnˆRn
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where the integral is understood in the sense of Proposition 1.6 of [31]. Denote by
BApRnΘ q the algebra equipped with multiplication ˆΘ.
Let SApRnq denote the space of A-valued Schwartz functions. There is an A-
valued inner product on SApRnq given by
ż
xf |gyA :“ fpsq˚gpsqds.
Rn
1
We denote by X the completion of SApRnq in the norm ‖f‖A :“ ‖xf |fy‖ 2A. Then
X is a right Hilbert A-module. Let f P BAΘpRnq act on SApRnq as
πΘpfqg “ f ˆΘ g.
Then πΘpfq is an adjointable bounded operator on X.
Equip BAΘpRnq with the pre-C˚-norm
‖f‖Θ :“ ‖πΘpfq‖,
where ‖¨‖ is the operator norm on B A n ˚ApXq. Denote by BΘpR q the C -completion
of BAΘpRnq with respect to this norm.
Definition 5.1. Let α be an action of Rn on A and let A8 Ă A be the subalgebra
of smooth elements for the action α. For a P A denote fa to be the function
in BAΘpRnq defined by faptq “ α´tpaq. For a P A8 we have f P BAa ΘpRnq. The
Rieffel deformation of A with respect to pα,Θq is the C˚-algebra AΘ obtained by
completing A8 in the norm
‖a‖Θ :“ ‖πΘpfaq‖.
Thus, AΘ is a C˚-algebra with multiplication given by
ż ż
a ¨ b :“ α 2πixx,yyΘ Θpxqpaqαypbqe dxdy, a, b P A8. (1)
Rn Rn
Example 5.2. Let α : Rn Ñ AutpCpTnqq be the action defined by
αt1,...,tnpuiq “ e2πitiui.
Then for the C˚-algebra defined in the Example 3 it holds that
CpTnΘq » CpTnq
Θ
2 .
6. Classification of Kirchberg algebras
For more details on the definitions below see [2]
Definition 6.1. A C˚-algebra A is nuclear if there exists a sequence of finite-
dimensional C˚-algebras F1, F2, . . . and completely positive maps ϕn : A Ñ Fn,
ψn : Fn Ñ A such that
lim‖ψn ˝ ϕnpaq ´ a‖ “ 0.
n
Definition 6.2. A C˚-algebra A is simple if it contains no nontrivial closed two-
sided ideals.
Definition 6.3. A simple unital C˚-algebra A of dimension at least 2 is purely
infinite if for every non-zero element a P A there are elements x, y P A such that
xay “ 1.
Definition 6.4. A C˚-algebra is called Kirchberg algebra if it is separable, nuclear,
unital, purely infinite and simple.
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Theorem 6.5 (Kirchberg-Philips, [27]). Let A and B be Kirchberg C˚-algebras, and
suppose that there exists an invertible element η P KKpA,Bq such that rιAs ˝ η “
rιBs, where ιA : C Ñ A is λ ÞÑ λ1A, and similarly for ιB. Then A and B are
isomorphic as C˚-algebras.
7. Noncommutative tori
By now no one has given a satisfactory definition for a noncommutative manifold,
however a number of naturally arising examples are known. One of the most well-
studied examples of a noncommutative manifold is noncommutative tori. Given a
skew-symmetric n ˆ n matrix Θ the noncommutative torus CpTnΘq is defined as a
universal C˚-algebras generated by n unitaries u1, . . . , un subject to the relations
u u “ e´2πiΘiji j ujui.
Various different aspects of CpTnΘq has been studied and it appeared to be useful
not just as a toy-object used to study effects which appear in non-commutative
geometry, but also outside of the subject of noncommutative geometry, for example
in mathematical physics: quantum diffusion [14], quantum Hall effect [7], String
theory [22], Yang-Mills theory [8], in number theory: Generalized theta functions
as holomorphic elements of projective modules [32], parallel between the theory of
elliptic curves with complex multiplication and the theory of noncommutative tori
with real multiplication [24].
We call Θ irrational if whenever pΘZn Ă Z for some p P Zn it follows that p “ 0.
If Θ is irrational then CpTnΘq is simple.
One has an action α of Tn on CpTnΘq given by
αzpuiq “ ziui.
The action is ergodic, i.e. CpTnΘqα » C. The conditional expectation with respect
to this action is the trace τ defined from the following property
ż
1 ¨ τpaq “ αzpaqdz, a P CpTnΘq.
Tn
One can show that in case of that CpTnΘq is simple it is the unique trace up to scalar
multiple.
The problem of classifying CpTn q up to C˚Θ -isomorphism has been solved in the
case when Θ is irrational, see [28]. In particular, for n “ 2, CpT2θ q » CpT2θ q iff1 2
θ1 “ ˘θ2 mod Z. When Θ satisfies certain rationality condition a classification is
given in [5] for general n.
8. Wick algebras
Consider a set of numbers tT klij , i, j, k, l “ 1, du Ă C satisfying the condition
T klij “
lk
T ji . Let H “ Cd and e1, …, ed be the standard orthonormal basis of H.
Construct
d
ÿ
T : Hb2 Ñ Hb2, T e b e “ T ljk l ikei b ej .
i,j“1
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The Wick algebra W pT q, see [16], is the ˚-algebra generated by elements a , a˚j j ,
j “ 1, . . . , d subject to the relations
d
ÿ
a˚i aj “ δij1` T klij a ˚lak .
k,l“1
It was studied in [16] how the properties of W pT q depend on a self-adjoint operator
T . Notice that the subalgebra of W pT q generated by ta djuj“1 is free and can be
À
identified with the full tensor algebra F “ 8 Hbnn“0 via
ai . . . a ÑÞ e b ¨ ¨ ¨ b e bk1 ik i1 i P H .k
Definition 8.1. The Fock representation πF,T of W pT q is a representation on
Hilbert space H such that the following conditions hold: (i) πF,T is an irreducible
˚-representation; (ii) there exists a unit vector Ω P H (vacuum vector), such that
πF,T pa˚j qΩ “ 0, j “ 1, d.
The Fock representation, if it exists, is unique up to a unitary equivalence. In
general, the problem of existence of πF,T is non-trivial and is one of the central
problems in representation theory of Wick algebras. Some sufficient conditions are
collected in the following theorem, see [4, 17, 16].
Theorem 8.2. The Fock representation πF,T ofW pT q exists if one of the conditions
below is satisfied
‚ The ope?rator of coefficients T ě 0;
‚ ||T || ă 2´ 1;
‚ T is braided, i.e. p1bT qpTb1qp1bT q “ pTb1qp1bT qp1bT q on Hb3 and
||T || ď 1. Moreover, if ||T || ă 1 then πF,T is a faithful representation of
W pT q and ||πF,T pajq|| ă p1´ ||T ||q´
1
2 . If ||T || “ 1, one can not guarantee
boundedness of πF,T and in this case kerπF,T is a ˚-ideal I2 generated as a
˚-ideal by kerp1` T q. Hence πF,T is a faithful representation of W pT q{I2.
An important question in the theory of Wick algebras is the question of stability
of isomorphism classes of WpT q “ C˚pW pT qq for the case ||T || ă 1. The following
problem was posed in [18].
Conjecture 8.3. Let T : Hb2 Ñ Hb2 be a self-adjoint braided operator and
||T || ă 1. Then WpT q »Wp0q.
In particu?lar, the authors of [18] have shown that the conjecture holds for the
case ||T || ă 2´ 1, for more results on the subject see [10], [19].
Consider the case T “ 0 in a few more details. If d “ dimH “ 1, then W p0q
is generated by a single isometry s, s˚s “ 1. In this case the universal C˚-algebra
E of W p0q exists and is isomorphic to the C˚-algebra generated by the unilateral
shift S in l2pZ`q. Notice also that πF,0psq “ S, so the Fock representation of the
C˚-algebra E is faithful. The ideal I in E , generated by 1 ´ ss˚ is isomorphic to
the algebra of compact operators and E{I » CpS1q, see [6]. When d ě 2, W p0q is
generated by s , s˚j j , such that
s˚i sj “ δij1, i, j “ 1, d.
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The Fock representation πF,d acts on F :“ Fd as follows
πF,dpsjqΩ “ ej , πF,dpsjqei b ¨ ¨ ¨ b ei “ ej b ei b ¨ ¨ ¨ b ei , k ě 1,1 k 1 k
π ˚ ˚F,dpsj qΩ “ 0, πF,dpsj qei b ¨ ¨ ¨ b ei “ δji ei b ¨ ¨ ¨ b ei , k ě 1.1 k 1 2 k
The universal C˚-algebra generated by W p0q with d ě 2 exists and coincides with
the Cuntz-Toeplitz agebra Op0qd . It is isomorphic to C˚pπF,dpW p0qqq, so the Fock
representation of Op0qd is faithful, see [9]. Further, the ideal I generated by 1 ´
řd ˚
j“1 sjsj is the unique largest ideal in O
p0q
d . It is isomorphic to the algebra of
compact operators on Fd. The quotient Op0qd {I is the Cuntz algebra Od.
9. Pseudodifferential operators
9.1. Pseudodifferential operators. Theory of Pseudodifferential operators starts
from the following idea. Write the Fourier inversion formula:
ż
fpxq “ fppξqeix¨ξdξ,
where
ż
fppξq “ 1 ´ix¨ξp2πq fpxqe dx.n
After differentiation one obtains:
ż
Dαfpxq “ ξαfppξqeix¨ξdξ,
where Dα “ Dα11 ¨Dαnn , D “ 1 Bj i Bx . Hence, ifj
ÿ
ppx,Dq “ aαpxqDα
|α|ďk
is a differential operator, we have
ż
ppx,Dqfpxq “ ppx, ξqfpξqeix¨ξp dξ,
where
ÿ
ppx, ξq “ aαpxqξα.
|α|ďk
One uses the Fourier integral representation to define pseudodifferential operators,
taking the function ppx, ξq to belong to one of a number of different classes of
symbols.
Definition 9.1. Assume ρ, δ P r0, 1s, m P R, define Smρ,δ to consist of C8-functions
ppx, ξq satisfying
|Dβ αxDξ ppx, ξq| ď C p1` |ξ|2q
1
2 pm´ρ|α|`δ|β|qαβ ,
for all α, β.
Usually the case of interest is ρ “ 1, δ “ 0.
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Definition 9.2. Suppose there are smooth pm´jpx, ξq, homogeneous in ξ of degree
m´ j for |ξ| ě 1, that is, pm´jpx, rξq “ rm´jpm´jpx, ξq for r, |ξ| ě 1 such that for
all N
N
ÿ
ppx, ξq ´ p px, ξq P Sm´N´1m´j 1,0 .
j“0
Then we say that ppx, ξq P Smcl is classical symbol of order m.
9.2. Schwartz kernels. To an operator ppx,Dq P Smρ,δ corresponds a Schwartz
kernel K P D1pRn ˆ Rnq, satisfying
ż ż ż ż ż
xu¨v,Ky “ 1upxqppx, ξqv ix¨ξ ipx´yq¨ξppξqe dξdx “ p q upxqppx, ξqe vpyqdydξdx.2π n
Proposition 9.3. Given symbol ppx, ξq the integral kernel which corresponds to it
is
ż
1
K “ p q ppx, ξqe
ipx´yq¨ξdξ.
2π n
Important theorem for the theory of Pseudodifferential operators:
Theorem 9.4. If ρ ą 0, then K is C8 off the diagonal ∆ Ă Rn ˆ Rn.
Theorem 9.5. If ρ ą 0 and δ ă 1 then ppx,Dq has the pseudolocal property:
sing suppppx,Dqpuq Ă sing suppu, u P E 1pRnq.
Definition 9.6. Define H#µ pRnq to be the space of distributions on Rn, homoge-
neous of degree µ, which are smooth on Rnz0.
Theorem 9.7. Assume L P S 1pRn ˆ Rnq is a smooth function of x with values
in S1 pRnq X L10 pRnq. Let j “ 1, 2, 3, . . .. Then Kpx, yq “ Lpx, x ´ yq defines an
operator in S´jcl if and only if
ÿ
Lpx, zq „ pqlpx, zq ` plpx, zq log |z|q,
lě0
where each Dβxqlpx, ¨q is a bounded continuous function of x with values in H
#
j`l´n,
and plpx, zq is a polynomial homogeneous of degree j ` l ´ n in z, with coefficients
that are bounded, together with all their x-derivatives.
9.3. Adjoints and products. Given ppx, ξq P Smρ,δ, we obtain that the adjoint is
given by
ż
ppx,Dq˚v “ 1 ppy, ξq˚eipx´yq¨ξp q vpyqdydξ.2π n
However presence of ppy, ξq˚ makes the expression be in a nonstandard form. In-
stead there is an asymptotic expansion.
Proposition 9.8. If ppx,Dq P Smρ,δ then
|α|
ÿ
pp ix, ξq˚ „ Dα α ˚ξDxppx, ξq .α!
αě0
There is also an asymptotic expansion formula for the symbol of product of two
pseudodifferential operators.
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Proposition 9.9. Given p1px,Dq P Sm1ρ and p2px,Dq P S
m2 , suppose
1,δ1 ρ2,δ2
0 ď δ2 ă ρ ď 1, ρ “ minpρ1, ρ2q.
Then
p1px,Dqp2px,Dq “ qpx,Dq P Sm1`m2ρ,δ ,
with δ “ maxpδ1, δ2q and
|α|
ÿ
p q „ iq x, ξ Dαξ p1px, ξqDαxp2px, ξq.
ě α!α 0
9.4. Elliptic operators. We say that ppx,Dq P Smρ,δ is elliptic if for some r ă 8
|ppx, ξq´1| ď Cp1` |ξ|2q´m2 , |ξ| ě r.
Thus if ψpξq P C8pRnq is equal to 0 for |ξ| ď r, 1 for |ξ| ě 2r, it follows easily from
the chain rule that
ψpξqppx, ξq´1 “ q0px, ξq P S´mρ,δ .
Applying asyptotic expansion formula for products we obtain
q0px,Dqppx,Dq “ I ` r0px,Dq,
ppx,Dqq0px,Dq “ I ` rr0px,Dq,
with
r0px, ξq, r px, ξq P S´ρ`δr0 ρ,δ .
Using the formal expansion
I ´ r0px,Dq ` r0px,Dq2 ´ . . . „ I ` spx,Dq P S0ρ,δ
and setting qpx,Dq “ pI ` spx,Dqqq ´m0px,Dq P Sρ,δ , we have
qpx,Dqppx,Dq “ I ` rpx,Dq, rpx, ξq P S´8.
Similarly, we obtain qrpx,Dq P S´mρ,δ satisfying
ppx,Dqqrpx,Dq “ I ` rpx,Dq, px, ξq P S´8r .
But evaluating
pqpx,Dqppx,Dqqqrpx,Dq “ qpx,Dqpppx,Dqqrpx,Dqq
yields qpx,Dq “ qrpx,Dq mod S´8, so in fact
qpx,Dqppx,Dq “ I mod S´8,
ppx,Dqqpx,Dq “ I mod S´8.
We say that qpx,Dq is a two-sided parametrix for ppx,Dq.
The parametrix can establish the local regularity of a solution to
ppx,Dqu “ f.
Suppose u, f are tempered distributions and ppx,Dq P Smρ,δ is elliptic. Constructing
qpx,Dq P S´mρ,δ we have
u “ qpx,Dqf ´ rpx,Dqu.
Thus u “ qpx,Dqf mod C8.
Proposition 9.10. If ppx,Dq P Smρ,δ is elliptic then for any u being tempered
distribution,
sing supp ppx,Dqu “ sing supp u.
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9.5. Pseudodifferential operators on manifolds.
Definition 9.11. If X is a smooth manifold and C8 8c pXq Ă C pXq is the space
of C8 functions of compact support, then, for any m P R, ΨmpXq is the space of
linear operators
A : C8c pXq Ñ C8pXq
with the following properties. First, if φ, ψ P C8pXq have disjoint supports then
exists K P C8pX ˆX,ΩRq such that for all u P C8c pXq,
ż
φAψu “ Kpx, yqupyq,
X
and secondly if F : W Ñ Rn is a coordinate system in X and ψ P C8c pXq has
support in W then there exists B in Ψm8pRnq with support in F pW q ˆ F pW q such
that ψAψu restricted to W is F˚pBppF´1q˚pψuqqq for all u P C8c pXq.
9.6. Symbol of an operator. Let D be a differential operator on manifold M of
order k. For px, ξq P T˚xM take g P C8pM,Rq with gpxq “ 0 and dgpxq “ ξ. Then
1
σDpx, ξq “ Dp gkqpxq.
k!
10. Functional analytic properties of a pseudodifferential operator
Many functional analytic properties of a pseudodifferential operators can be
extracted from its symbol.
10.1. Boundedness.
Theorem 10.1. Let a : Rn ˆ Rn Ñ C be a continuous function whose derivatives
Bαx B
β
ξ a in the distribution sense satisfy the following condition: there is a constant
C ą 0 such that
‖BαBβx ξ a‖L8pRnˆRnq ď C,
where α “ pα1, . . . , αnq, β “ pβ1, . . . , βnq with αj “ 0 or 1, βj “ 0 or 1.
Then apx,Dq is continuous from L2pRnq to L2pRnq with its norm bounded by
Cn‖a‖ where Cn is a constant depending only on n and ‖a‖ is the smallest C such
that the inequality above holds.
10.2. Compactness.
Theorem 10.2. Let apx,Dq P S0ρ,δ such that the kernel of apx,Dq has compact
support and supx |apx, ξq| Ñ 0 as ξ Ñ 8. Then apx,Dq extends to a compact
operator on L2.
Theorem 10.3. Classical pseudodifferential operator on a compact manifold is
compact if and only if it has negative order.
10.3. Selfadjointness, normalness and unitarity.
Theorem 10.4. Let apx,Dq P S01,0. Then it is self-adjoint operator on L2pRnq iff
for all ξ, η P Rn
ż
e2πixξ´η,yypapy, ξq ´ apy, ηqqdy “ 0.
Rn
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Theorem 10.5. Let apx,Dq P S01,0. Then it is normal operator on L2pRnq iff for
all ξ, η P Rn
ż
e2πixξ´η,yypapy, ξqapy, ηq ´ a˚py, ξqa˚py, ηqqdy “ 0.
Rn
Theorem 10.6. Let apx,Dq P S01,0. Then it is unitary operator on L2pRnq iff for
all ξ, η P Rn
ż
e2πixξ´η,yypapy, ξqapy, ηq ´ a˚py, ξqa˚py, ηqqdy “ δξ,η.
Rn
Theorem 10.7. Let apx,Dq P S01,0. Then it is unitary operator on L2pRnq iff for
all ξ, η P Rn
tap¨, ξq, ξ P Rnu and ta˚p¨, ξq, ξ P Rnu
are orthonormal bases for L2pRnq.
10.4. Essential normality.
Theorem 10.8. Let apx,Dq P S01,0pMq for a compact manifoldM . Then apx,Dq is
automatically essentially normal! To understand this you can read article of Shahla
Molahajloo called ”A Characterization of Compact Pseudo-Differential Operators on
S1”, where in Proposition 2.2 he gives an argument for M “ S1, but the argument
works for arbitrary compact manifold.
10.5. Spectrum.
Theorem 10.9. Suppose A is a self-adjoint elliptic pseudodifferential operator on
a smooth manifold M . Then there exists a complete orthonormal system of C8pMq
functions which are eigenfunctions of A.
Theorem 10.10. Suppose A P Smρ,δ is elliptic with m ą 0, 1 ´ ρ ď δ ă ρ. Then
for the spectrum σpAq there are two possibilities:
(1) σpAq “ C (which, in particular, is the case when ind pAq ‰ 0).
(2) σpAq is discrete.
10.6. Fredholmness.
Theorem 10.11. Let M be a closed manifold and A P Smρ,δpMq is elliptic, 1´ ρ ď
δ ă ρ. For any s P R construct the operator As P BpHspMq,Hs´mpMqq - the
extension of A be continuity to Sobolev spaces. Then
(1) As is Fredholm.
(2) kerA 8s Ă C pMq, therefore kerAs does not depend on s.
(3) indexpAsq does not depend on s.
(4) if D P Sm1ρ,δ, where m1 ă m then indexpA`Dq “ indexpAq.
11. Pseudodifferential operators and BDF theory
11.1. BDF theory. The story of Brown–Douglas–Fillmore theory begins with the
Weyl–von Neumann theorem, which, in one of its formulations, says that a bounded
self-adjoint operator T “ T˚ on an infinite-dimensional separable Hilbert space H
is determined up to compact perturbations, modulo unitary equivalence, by its
essential spectrum. (The essential spectrum is the spectrum σpπpT qq) of the image
πpT q of T in the Calkin algebra QpHq “ BpHq{KpHq; it is also the spectrum of
the restriction of T to the orthogonal complement of the eigenspaces of T for the
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eigenvalues of finite multiplicity. In other words, unitary equivalence modulo the
compacts KpHq washes out all information about the spectral measure of T , and
only the essential spectrum remains. This result was extended to normal operators
by I.D. Berg and W. Sikonia, working independently. However, the theorem is not
true for operators that are only essentially normal, in other words, for operators T
such that T˚T ´ TT˚ P KpHq. Indeed, the ”unilateral shift” S satisfies S˚S “ 1
and SS˚ “ 1 ´ P , where P is a rank-one projection, yet S cannot be a compact
perturbation of a normal operator since its Fredholm index is non-zero. L.G. Brown,
R.G. Douglas and P.A. Fillmore (known to operator theorists as ”BDF” ) showed
that this is the only obstruction: an operator T in BpHq is a compact perturbation
of a normal operator if and only if T is essentially normal and ind pT ´ λq “ 0 for
every λ R σpπpT qq.
11.2. Classification of classical pseudodifferential operators. Let M be a
closed Riemannian manifold.
Theorem 11.1. The following diagram commutes, has exact rows and the vertical
maps are injections:
0 S´1 0
σ0 8
cl pMq SclpMq C pS˚Mq 0
0 KpL2pMqq BpL2pMqq BpL2pMqq{KpL2pMqq 0
Every 0-order classical pseudodifferential operator is essentially normal. Thus
by BDF theory, up to unitary equivalence modulo compact they are classified by
(1) The essential spectrum.
(2) The set pind pT ´ λqqλRσesspT q.
The following proposition follow from the commutative diagram above.
Proposition 11.2. Let A P S0clpMq. Then
σesspAq “ imσ0pAq.
Proposition 11.3. Let M be closed connected manifold. Let A,B P S0clpMq.
Then there exists a unitary operator U on L2pMq and K P KpL2pMqq such that
UAU˚ “ B `K iff
(1) imσ0A “ imσ0B.
(2) For some λ R imσ0A holds ind pA´ λq “ ind pB ´ λq.
Proposition 11.4. Let M be a closed connected manifold. Then up to scaling
there is unique elliptic classical pseudodifferential operator of order 0 with discrete
spectrum up to unitary equivalence modulo compacts.
Proof. So:
(1) Pseudodifferential operators of order 0 are essentially normal.
(2) Essential spectrum is a subset of spectrum.
(3) If M is connected, then the essential spectrum is connected. Thus essential
spectrum is a point in C.
(4) By BDF theory, up to unitary equivalence modulo compacts the operator
is determined by index (which is 0) and the essential spectrum (which is
one point), so up to scaling the operator is unique.
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‹
Corollary 11.5. Let M be a closed connected manifold. Then scalar multiples of
identity are the only elliptic pseudodifferential operators of order 0 with discrete
spectrum modulo compact operators.
Corollary 11.6. LetM be a closed connected manifold. If elliptic pseudodifferential
operator of order 0 has index 0 then it is scalar multiple of identity modulo compact
operator.
Corollary 11.7. If M is a closed connected odd-dimensional manifold then every
elliptic A P Ψ0pMq has form
A “ λI `K, K P Ψ´1pMq.
Corollary 11.8. If M is a compact connected manifold and P P Ψ0pMq is a
projection then either P or 1´ P is compact.
Corollary 11.9. On a compact connected manifold there is unique pseudodifferen-
tial projection up to unitary equivalence modulo compacts.
Corollary 11.10. On a compact connected manifold there are 2ℵ0 projections
modulo unitary equivalence:
P0, 1´ P0, P1, 1´ P1, . . . ,
in every pair one projection is compact and the other is elliptic.
And so on! There are numerous ways to play with BDF theory and pseudodif-
ferential operators.
11.3. Conjectural approximation of abstract essentially normal operators
by pseudodifferential operators. Since mapping rDs ÞÑ indD is surjective,
one can approximate every essentially normal operator N with connected essential
spectrum with a sequence of pseudodifferential operators A1, A2, . . ., for which we
have topological index formula. Thus we get topological index formula for abstract
operators in functional analysis:
ż
ind pNq “ lim τ ˝ chrσ s ^ TdpT˚Mq.
nÑ8 AnM
There are several problems of course:
(1) How to realize arbitrary operator as operator on L2pMq?
(2) How to find sequence of pseudo-differential approximations?
(3) What should we do with non-connected essential spectrum?
12. Dirac operators
12.1. Clifford algebra. Let V be a finite-dimensional, real vector space, g a qua-
dratic form on V . We allow g to be definite or indefinite if nondegenerate; we even
allow g to be degenerate.
Definition 12.1. The Clifford algebra ClpV, gq is the quotient algebra of the tensor
algebra
â
V “ R‘ V ‘ pV b V q ‘ pV b V b V q ‘ . . .
Â
by the ideal I Ă V generated by
tv b w ` w b v ´ 2xv, wy ¨ 1 : v, w P V u,
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where x¨, ¨y is the symmetric bilinear form on V arising from g.
Thus, in ClpV, gq, V occurs naturally as a linear subspace, and there is the
anti-commutation relations
vw ` wv “ 2xv, wy ¨ 1, v, w, P V.
Definition 12.2. Vector space E is a Clifford module if there exists ν : V Ñ
EndpEq a linear map from V into the space of endomorphisms of a vector space E
such that
νpvq2 “ xv, vy ¨ I, v P V.
In this case ν extends uniquely to an algebra homomorphism
ν : ClpV, gq Ñ EndpEq, νp1q “ I.
Definition 12.3. Clifford module E with a Hermitean metric is a Hermitean Clif-
ford module if νpvq “ νpvq˚.
Definition 12.4. Hermitean Clifford module E with a Z{2Z-grading E “ E0‘E1
is a graded Hermitean Clifford module if νpvqpEiq Ă Ei`1.
12.2. Operators of Dirac type. Let M be a Riemannian manifold, Ej Ñ M
vector bundles with Hermitean metrics.
Definition 12.5. A first-order, elliptic differential operator
D : C8pM,E0q Ñ C8pM,E1q
is said to be of Dirac type provided D˚D has scalar principal symbol, i.e.
σD˚Dpx, ξq “ gpx, ξqI : E0,x Ñ E0,x,
where gpx, ξq is a positive quadratic form on T˚xM .
If E0 “ E1 and D “ D˚, we say D is a symmetric Dirac-type operator. Given a
general operator D of Dirac type, if we set E “ E0‘E1 and define Dr on C8pM,Eq
as
ˆ ˙
“ 0 D
˚
Dr ,
D 0
then D is a symmetric Dirac-type operator.
Let νpx, ξq denote the principal symbol of a symmetric Dirac-type operator.
With x P M fixed, set νpξq “ νpx, ξq. Thus ν is a linear map from T˚xM into
EndpExq satisfying
νpξq “ νpξq˚, νpξq2 “ xξ, ξyI.
Example 12.6. If M is a Riemannian manifold, the exterior derivative operator
d : ΛjM Ñ Λj`1M
has a formal adjoint
δ “ d˚ : Λj`1M Ñ ΛjM,
so d` δ acts on Λ˚M . One can show that
pd` δq˚pd` δq “ ´∆,
where ∆ is the Hodge Laplacian, so d` δ is a symmetric Dirac-type operator.
This operator is used to deduce Gauss-Bonnet formula from the Atiyah-Singer
index formula.
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Example 12.7. Suppose dimM “ 2k is even. In terms of the Hodge star operator,
δ is defined on ΛjM as follows:
δ “ d˚ “ p´1qjpn´jq`j ˚ d˚ “ ˚d ˚ .
On complexification Λ˚CM define
α : ΛjCM Ñ Λ
n´j
C M, α “ i
jpj´1q`k ˚ .
Then α2 “ 1 and αpd` δq “ ´pd` δqα, so we can write eigenspaces of α:
Λ˚C “ Λ`M ‘ Λ´M
and
D˘H “ d` δ : C
8pM,Λ˘q Ñ C8pM,Λ¯q
are operators of Dirac type.
This operators are used to deduce Hirzebruch signature formula from the Atiyah-
Singer index formula.
Example 12.8. LetM be Riemannian manifold, T˚xM has an induced inner prod-
uct, giving rise to bundle ClpMq ÑM of Clifford algebas. We suppose that E ÑM
is a Hermitian vector bundle such that each fiber is a Hermitian ClxpMq-module.
Let E ÑM have a connection ∇, so
∇ : C8pM,Eq Ñ C8pM,T˚ b Eq.
If Ex is a ClxpMq-module, the inclusion T˚x Ñ Clx gives rise to a linear map
m : C8pM,T˚ b Eq Ñ C8pM,Eq,
called Clifford multiplication. Let
D “ i ¨m ˝∇ : C8pM,Eq Ñ C8pM,Eq.
For v P Ex,
σDpx, ξqv “ mpξ b vq “ ξ ˝ v,
so σDpx, ξq is |ξ|x times an isometry on Ex. Hence D is of Dirac type.
13. Many incarnations of the Atiyah-Singer index theorem
13.1. Atiyah-Singer.
Theorem 13.1. Let D be an elliptic differential operator on a closed manifold M .
Then
ż
indpDq “ τ ˝ chprσ ˚Dsq ^ TdpT Mq,
M
where
‚ TdpT˚Mq P H˚pMq is the Todd class.
‚ rσ s P K pT˚D ˚ Mq is a K-theory class of the elliptic complex given by mul-
tiplication with σ on T˚D M .
‚ chprσ sq P H˚D pT˚Mq is the image of the Chern character.
‚ τ : H˚pT˚Mq Ñ H˚pMq is the Thom isomorphism.
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13.2. Toeplitz index theorem. Let M be an odd dimensional closed Spinc man-
ifold with Dirac operator D acting on sections of the spinor bundle S. If E is a
smooth C vector bundle on M , DE denotes D twisted by E. The closure ED of
DE is an unbounded self-adjoint operator on the Hilbert space L2pM,S b Eq of
L2-sections of Sb EE. D has discrete spectrum with finite dimensional eigenspaces.
Denote by L2`pM,SbEq the Hilbert space direct sum of the eigenspaces of
E
D for
eigenvalues λ ě 0. PE` denotes the orthogonal projection
PE` : L
2pM,S b Eq Ñ L2`pM,S b Eq.
Suppose that α is an automorphism of E, and ISbα the resulting automorphism of
S bE. Mα is the bounded invertible operator on L2pS bEq obtained from IS bα.
The Toeplitz operator Tα is the composition ofMα : L2` Ñ L2 with PE` : L2 Ñ L2`,
Tα “ PE` ˝Mα : L2`pM,S b Eq Ñ L2`pM,S b Eq.
The Toeplitz operator Tα is a Fredholm operator.
Theorem 13.2. Let M be an odd dimensional compact Spinc manifold without
boundary. If E is a smooth C vector bundle on M , and α is an automorphsim of
E, then
ind pTαq “ pchpE,αq X TdpMqqrM sq.
13.3. Gauss-Bonnet. If you plug d ` d˚ into the Atiyah-Singer index formula,
you get Gauss-Bonnet.
Theorem 13.3. Let M be a Riemannian even-dimensional compact orientable
manifold, Ω - Riemannian curvature. Then the Euler characteristics χpMq can be
computed as
ż
χp 1Mq “ p PfpΩq.2πqn M
Corollary 13.4. In dimension 2n “ 4, we get
ż
χp q “ 1M |Riem|2 ´ 4|Ric|2 `R2,
8π2 M
where Riem is Riemannian curvature, Ric is the Ricci curvature and R is the scalar
curvature.
13.4. Riemann-Roch. Take X to be a complex manifold with a holomorphic vec-
tor bundle V . We let the vector bundles E and F be the sums of the bundles of
differential forms with coefficients in V of type p0, iq with i even or odd, and we let
the differential operator D be the sum
B ` B˚.
restricted to E. Then the analytical index of D is the holomorphic Euler charac-
teristic of V :
ÿ
indexpDq “ p´1qpdimHppX,V q.
p
The topological index of D is given by
indexpDq “ chpV qTdpXqrXs,
the product of the Chern character of V and the Todd class of X evaluated on the
fundamental class of X. By equating the topological and analytical indices we get
the Hirzebruch–Riemann–Roch theorem.
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13.5. Hirzebruch index theorem. The Hirzebruch signature theorem states that
the signature of a compact oriented manifold X of dimension 4k is given by the L
genus of the manifold. This follows from the Atiyah–Singer index theorem applied
to the following signature operator.
The bundles E and F are given by the `1 and 1 eigenspaces of the operator on
the bundle of differential forms of X, that acts on k-forms asikpk´1q
times the Hodge ˚ operator. The operator S is the Hodge Laplacian
D ” ∆ :“ pd` d˚q2
restricted to E, where d is the Cartan exterior derivative and d˚ is its adjoint.
The analytic index of D is the signature of the manifold X, and its topological
index is the L genus of X, so these are equal.
13.6. Gromov-Lawson-Rosenberg conjecture. A result of Lichnerowicz states
that there are spin manifolds which do not admit positive scalar curvature met-
rics. Indeed, by the Lichnerowicz formula, the existence of such a metric implies
that the index of the Dirac operator vanishes. This, combined with the Atiyah-
Singer index theorem implies that Ap genus, which is a linear combination of the
Pontrjagin classes of the manifold, vanishes. The Ap obstruction was generalized by
Hitchin to an obstruction αpMq P KOn, where α denotes the Atiyah-Bott-Shapiro
homomorphism. This agrees with Ap in dimensions 0 mod 4, but is in fact a strict
generalization, and indeed Hitchin constructed exotic spheres admitting no met-
ric of positive scalar curvature in dimensions n ” 1, 2 mod 8. Letting π denote
any fundamental group, the homomorphism α gives rise to a transformation of
cohomology theories
α : Ωspinn pBπq Ñ KOpBπq
and Gromov and Lawson conjectured that αpMq “ 0 was also a sufficient condition
for M to admit a metric of positive scalar curvature. Rosenberg later generalized
this further, showing that if a spin manifoldM with fundamental group π admitted
a metric of positive scalar curvature, then indprM,usq “ 0, where u is the classifying
map of the universal cover ofM , and ind maps to the real K−theory of the reduced
C˚ algebra of π:
ind “ A ˝ α : Ωspinn pBπq Ñ KOpC˚pπqredq.
This can be thought of as an equivariant generalized index, and the map A is
the assembly map of Baum-Connes. Modifying the Gromov-Lawson conjecture,
Rosenberg conjectured that the converse was true also; namely that a compact
spin manifold M with π1pMq “ π and n ě 5 admits a positive scalar curvature
metric if and only if indru :MBπs “ 0 P KOnpC˚pπqredq. The conjecture has been
proven in the simply connected case, if π has periodic cohomology, and if π is a free
group, free abelian group, or the fundamental group of an orientable surface. It is
also known to be false in general, for example if π “ Z4 ˆ Z3, and for a large class
of torsion free groups.
13.7. Stolz conjecture. The Stolz conjecture asserts that ifX is a closed manifold
with String structure which furthermore admits a Riemannian metric with positive
Ricci curvature, then its Witten genus vanishes.
One part of the reasoning that motivates the conjecture is the idea that string
geometry should be a “delooping” of spin geometry, and that the Witten genus is
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roughly like the index of a Dirac operator on loop space. Now for spin geometry
the Lichnerowicz formula implies that for positive scalar curvature there are no
harmonic spinors on a Riemannian manifold X, and hence that the index of the
Dirac operator vanishes. One might then expect that there is a sensible concept of
scalar curvature of smooth loop space obtained by integrating the Ricci curvature
on X along loops (transgression). Therefore, in this reasoning, a positive Ricci
curvature of X would imply a positive scalar curvature of the smooth loop space,
thus a vanishing of the index of the “Dirac operator on smooth loop space”, hence
a vanishing of the Witten genus.
14. Summary of Paper I
In Paper I, ”Faithfulness of the Fock representation of C*-algebra generated by
qij-commuting isometries” we consider universal C˚-algebra Isomq generated byij
n isometries a1, . . . , an such that
a˚i aj “ qijaja˚i .
This algebra has a distinguished representation called the Fock representation. It
is a unique up to unitary equivalence representation πF on the Fock space CtΩu ‘
Cn ‘ Cn2 ‘ . . . such that a˚i pΩq “ 0. This representation exists for every Wick
algebra and it is a question if πF is faithful. It is known however that it is faithful
˚-algebraically.
In order to prove faithfulness of πF for Isomq we examine the fixed point subal-ij
gebra under the action of Tn. It appears to be an AF-algebra with Bratelli diagram
which looks like multidimensional Pascal tetrahedron. Using integration trick we
reduce problem of faithfulness of πF on Isomqij to the problem of faithfulness of
πF restricted to
n
IsomTq . Since AF-algebras are limits of finite-dimensional alge-ij
bras and faithfulness is proven on algebraic level, we show faithfulness of πF on
C˚-algebraic level.
It is expected that Isomq is in fact isomorphic to the Cuntz-Toeplitz algebraij
KOn, which has unique maximal ideal K. For KOn generator is very simple, but
for Isomq it is not so obvious. We construct generated for ideal K explicitly.ij
15. Summary of Paper II
In Paper II, ”On q-tensor product of Cuntz algebras” we consider the C˚-algebra
Eqn,m generated by the Wick algebra for T described as follows. Let H “ Cn ‘Cm,
|q| ď 1 and
Tu1 b u2 “ 0, T v1 b v2 “ 0, u1, u2 P Cn, v1, v2 P Cm,
Tub v “ qv b u, Tv b u “ qub v, u P Cn, v P Cm.
We consider cases |q| ă 1 and |q| “ 1 separately. In the case |q| ă 1 we write
explicit formula for the isomorphism
Theorem 15.1. For any q P C, |q| ă 1, one has an isomorphism Eq 0n,m » En,m.
In the case |q| “ 1 the C˚-algebra Eqn,m decomposes into the following short
exact sequence
0ÑM Ñ Eq Θqq n,m Ñ pOn bOmq Ñ 0,
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ˆ ˙
φ
where for q “ e2πiφ we define Θ “ 0 2q ´φ and Mq is the maximal ideal2 0
described below.
We prove that pO bO qΘqn m is a Kirchberg algebra KK-isomorphic to OnbOm.
Then we use Kirchberg-Philips classification theorem in order to conclude the next
theorem
Theorem 15.2. The C˚-algebras pOn bOmqΘq and On bOm are isomorphic for
any |q| “ 1.
We show that the ideal M qq of En,m further decomposes as
0Ñ KÑMq Ñ Om bK‘On bKÑ 0.
The extension happens to be essential which allows us to use the Voiculescu theorem
to prove
Theorem 15.3. For any q P C, |q| “ 1, one has Mq »M1.
16. Summary of Paper III
In Paper III, ”Classification of irrational Θ-deformed CAR C˚-algebras” we con-
sider the C˚-algebra CARΘ defined as the universal enveloping C˚-algebra of ˚-
algebra generated by a1, . . . , an subject to the relations
a˚i ai ` aia˚i “ 1,
a˚a “ e2πiΘiji j a ˚jai ,
a a “ e´2πiΘiji j ajai.
CARΘ has an action α of Tn given by
αzpaiq “ ziai.
We express it as a Rieffel deformation of the tensor product of n copies of 1-
dimensional CAR with respect to the action α and the skew-symmetric matrix
Θ.
Proposition 16.1.
CARΘ » pCARbnq
Θ
2
1
Denote Cln to be the complex Clifford C˚-algebra on n generators. The C˚-
algebra CAR1 possess Cr0, 12 s-structure and its fibers are the following:
pCAR1qp0q » Cl2,
p 1CAR1qpxq » CpTq ¸σ Z2, 0 ă x ă , σpfqpzq “ fp´zq,
2
p 1CAR1qp q » CpTq.
2
Action α of T on CAR1 is fibrewise with respect to the Cr0, 12 s-structure, which
allows us to conclude the following:
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Proposition 16.2. CAR possess a Cr0, 1Θ 2 s
n-structure with fibers given by CARΘpxq »
pCARbnpxqqΘ . Given x “ px , . . . , x q P r0, 121 1 n 2 s
n, let
Lx “ ti P t1, . . . , nu : xi “ 0u,
“ t P t u ă ă 1Mx i 1, . . . , n : 0 xi u,
2
1
Rx “ ti P t1, . . . , nu : xi “ u.
2
Then
CAR pxq » Cl b CpT|Mx|`|Rx| |Mx|Θ 2|Lx| Θ q ¸ Z .Mx\Rx 2
Any irreducible representation of a C˚-algebraA equipped with a C0pXq-structure
factors through an irreducible representation of a fiber Apxq. This fact combines
with Proposition 2 give us the next theorem
Theorem 16.3. Any irreducible representation of CARΘ is unitary equivalent to a
 Â
representation τx, x P r0, 1 sMx2 , given on pbkPL C
2q pb 2
x kPM C q H byx
ź
τ pa q “ pe e˚ ` eπiΘi,k ˚x i k k ekekqei b 1, i P Lx,
kPLx
˜˜ ¸
ź ź
τxpaiq “ pe e˚ ` eπiΘi,ke˚ ˚ 2πiΘi,k ˚k k kekq b pekek ` e ekekq b 1H
kPLx kPMx,kăi
˜ ¸¸
? ź ?
ˆ xi pe˚kek ` e4πiΘi,ke e˚ ˚k kqei b vi ` 1´ xiei b 1H , i PMx,
kPMx,kěi
ź ź
τxpa ˚ πiΘi,k ˚iq “ pekek ` e ekekq b pe˚e ` e2πiΘ
1
i,ke e˚k k k kq b ? vi , i P Rx.
P 2k Lx kPMx
(2)
where pv q defines an irreducible representation of CpTMx\Rxi iPM \R Σ q on H, wherex x
$
4Θi,j i, j PM& x
Σi,j “ 2Θi,j pi, jq or pj, iq PMx ˆRx
%
Θi,j i, j P Rx
Moreover, two such irreducible representations τx and τy are unitary equivalent if
and only if x “ y and the corresponding representations of CpTMx\RxΣ q are unitarily
equivalent.
Finally, we give a partial classification result for CARΘ with irrational Θ
Theorem 16.4. Let Θ1 and Θ2 be irrational.
(1) If P is a signed permutation matrix then Θ1 “ PΘ2P t implies CARΘ1 »
CARΘ2 .
(2) If CARΘ » CARΘ then pΘ2qi,j “ ˘pΘ1qσpi,jq mod Z for a bijection σ of1 2
the set tpi, jq : i ă j, i, j “ 1, . . . , nu.
In particular, for n “ 2 we get
Corollary 16.5. If θ1, θ2 are irrational numbers then CARθ1 » CARθ2 iff θ1 “ ˘θ2
mod Z.
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17. Summary of paper IV
In paper IV, ”CCR and CAR algebras are connected via a path of Cuntz-Toeplitz
algebras” we prove conjecture of Jorgensen, Schmidt and Werner about indepen-
dence of C˚-isomorphism class of universal C˚-algebra ˆq generated by a1, . . . , an
subject to relations
a˚i aj “ δ ` qa a˚ij j i
from n?umber q. It was conjectured that ˆq » ˆ0, but the prove was done only for
|q| ă 2´ 1. We prove it for |q| ă 1 using recent developments in classification of
C˚-algebras, Kirchberg-Philips theory and works of Rosenberg from 70s.
18. Summary of paper V
Everybody knows Atiyah-Singer index formula, here it is:
ż
ind pDq “ τ ˝ chrσDs ^ TdpT˚Mq.
M
It works only for elliptic pseudodifferential operators on compact manifolds. But
there might be not so many of them! Eric van Erp gave (not for the first time in
history probably!) an example of non-elliptic differential operator which is hypoel-
liptic, Fredholm and everything else you might want from a differential operator,
you can observe it below:
D “ X2 ` Y 2 ` γZ,
where rX,Y s “ Z and γ is a function. It appears that D is Fredholm if and only if
image of γ does not contain odd integers. This differential operator lives naturally
in a nonclassical pseudodifferential calculus called Heisenberg calculus. Heisenberg
calculus can be introduced for contact manifolds, which locally look like Heisenberg
Lie algebra. Contact manifolds have a distinguished vector subbundle H Ă TM of
codimension 1, quotient TM{H let’s denote by Z. Eric van Erp and Paul Baum
modified Atiyah-Singer index formula to work for Heisenberg elliptic operators on
contact manifolds:
ż
ind pDq “ τ ˝ chprσHD s bHq ^ TdpZq.
Z
In paper V, ”Index theory of hypoelliptic operators on Carnot manifolds” we modify
this formula even further. Carnot manifolds are manifolds with filtration of TM
by subbundles:
t0u “ T 0M Ă T 1M Ă . . . Ă T kM “ TM,
such that rC8pM,T iMq, C8pM,T jMqs Ă C8pM,T i`jMq. Carnot manifolds lo-
cally resemble certain nilpotent Lie algebras. For such manifolds there is a cor-
responding pseudodifferential calculus and notion of ellipticity, called Rockland
condition. For pseudodifferential operators with Rockland condition our formula
now is
ż
ind pDq “ τ ˝ chprσFDs bHq ^ TdpΓq,
Γ
where Γ is a fiber bundle over the manifold with fiber being a dense stratum of
spectrum of the nilpotent Lie algebra associated to the Carnot manifold.
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