Fundamental groups of C∗ - algebras

June 15, 2013

Petr R. Ivankov*e-mail: * monster.ivankov@gmail.com

Abstract

Gelfand - Naimark theorem supplies contravariant functor from category of com-mutative C∗− algebras to category of locally compact Hausdorff spaces. ThereforeC∗− algebra is altrernative representation of a topological space. Similarly category of(noncommutative) C∗− algebras may be regarded as category of generalized (noncom-mutative) locally compact Hausdorff spaces. Generalizations of topological invariantsmay be defined by algebraic methods. For example Serre Swan theorem [24] statesthat complex topological K - theory coincides with K - theory of C∗ - algebras. How-ever algebraic topology have rich set of invariants. Some invariants do not have non-commutative generalizations. This article contains several steps towards definition ofnoncommutative generalization of fundamental group.

Contents

1 Motivation. Preliminaries 3

2 Fundamental groupoid and fundamental group 62.1 Fundamental groupoid/group in algebraic topology . . . . . . . . . . . . . . 62.2 Fundamental groupoid/group of schemes . . . . . . . . . . . . . . . . . . . . 62.3 Construction of the Fundamental group . . . . . . . . . . . . . . . . . . . . . . 9

1

3 Hopf - Galois extensions 133.1 Coaction of Hopf algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Action of finite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 Noncommutative finite covering projection . . . . . . . . . . . . . . . . . . . . 173.4 Fundamental groupoid of covering projection . . . . . . . . . . . . . . . . . . 183.5 Covering projection of continuous trace C∗-algebras . . . . . . . . . . . . . . . 193.6 Finite coverings of noncommutative torus . . . . . . . . . . . . . . . . . . . . . 21

4 Invariants of finite covering projections 22

5 Morita equivalences induced by bimodules over Hopf-Galois extensions 245.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6 Category of finite noncommutative covering projections 27

7 Covering projections of noncommutative torus 27

8 Relative invariants 30

9 Groupoid group 31

10 Generalization of fundamental group functor 31

11 Abelian covering projections 3211.1 Abelian fundamental group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3211.2 Canonical constructions of cyclic covering projection . . . . . . . . . . . . . . 3311.3 Relations with K groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3611.4 Further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.5 K theoretic cyclic covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3811.6 Analogy with Kummer extensions . . . . . . . . . . . . . . . . . . . . . . . . . 40

12 Generalization of infinite covering 4212.1 Noncommutative generalization of R→ S1 covering . . . . . . . . . . . . . . 42

12.1.1 R→ S1 covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4212.1.2 Generalization of R→ S1 covering . . . . . . . . . . . . . . . . . . . . 43

12.2 Generalization of arbitrary infinite covering . . . . . . . . . . . . . . . . . . . . 4412.2.1 Commutative infinite covering from algebraic viewpoint . . . . . . . . 4512.2.2 Noncommutative algebraic generalization of infinite covering . . . . . 46

13 Generalization of Hurewicz homomorphism 4713.1 Generalization of homology group H1 . . . . . . . . . . . . . . . . . . . . . . . 4813.2 Generalization of pointed space . . . . . . . . . . . . . . . . . . . . . . . . . . . 4813.3 Hurewicz homomorphism with respect to covering . . . . . . . . . . . . . . . 4913.4 Noncommutative Hurewicz homomorphism . . . . . . . . . . . . . . . . . . . 5013.5 Construction Hurewicz homomorphism generalization . . . . . . . . . . . . . 52

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1 Motivation. Preliminaries

Following Gelfand-Naimark theorem [27] states that category of locally compact Haus-dorff topological spaces is equivalent to category of commutative C∗− algebras.

Theorem 1.1. Let Haus be the category of locally compact Hausdorff spaces with continuousproper maps as morphisms. And, let C∗Comm be the category of commutative C -algebras withproper *-homomorphisms (send approximate units into approximate units) as morphisms. Thereis a contravariant functor C : Haus → C∗Comm which sends each locally compact Hausdorffspace X to the commutative C∗ -algebra C0(X) (C(X) if X is compact). Conversely, there is acontravariant functor Ω : C∗Comm → Haus which sends each commutative C∗ -algebra A tothe space of characters on A (with the Gelfand topology).The functors C and Ω are an equivalence of categories.

So any (noncommutative) C∗− algebra may be regarded as generalized (noncommutative)locally compact Hausdorff topological space. We may summarize several properties of theGelfand Naimark cofunctor with the following dictionary.

TOPOLOGY ALGEBRALocally compact space C∗ - algebra

Compact space Unital C∗ - algebraContinuous map *-homomorpfism

Minimal compactification UnitizationMaximal compactification Algebra if multpicators

Closed subset IdealMorphism of covering ?Pointed space (X, x0) ?Fundamental group ?Singular homology ?

Hurewicz homomorphism ?

Involutive homomorphism in C∗Comm is just continuous map. However ∗ - homomor-phism is not always good nontion of morphism in category of (noncommutative) C∗ - al-gebras (See [30], [29]). Morita equivalence and Kasparov intersection product [22] are alsomorphisms of C∗ - algebras. Isomorphisms may be regarded as particular case of Moritaequivalence and every *- homomorphism defines unique Kasparov intersection product.If two commutative C∗− algebras are Morita equivalent then they are isomorphic. SoMorita equivalence does not yield substantially new results for commutative case. Moritaequivalent rings have isomorphic centers. This implies that Morita equivalent Abelianrings are already isomorphic. Thus Morita equivalence is essentially a non-commutativephenomenon. This gives another reason why so many homology functors are Moritainvariant: Usually, they arise as extensions of functors defined on a category of commu-tative algebras to a category of noncommutative algebras. But Morita invariance imposesno restrictions whatsoever on functors defined on a category of commutative algebras, so

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that we can hope for a Morita invariant extension. Examples show that if a functor canbe extended “naturally”, then the extension tends to be indeed Morita invariant. Moritaequivalent rings also have equivalent categories of right modules and bimodules. It isalso easy to see that they have equivalent lattices of ideals, so that the properties of be-ing Noetherian, Artinian, or simple are Morita invariant (cf. [5]). They have isomorphiccategories of projective modules and thus equivalent K-theories. More generally, a decent(co)homology theory should be Morita invariant, and this is indeed true for cyclic homol-ogy, Hochschild homology. Moreover, Morita equivalent rings have isomorphic centers.This implies that Morita equivalent Abelian rings are already isomorphic. Thus Moritaequivalence is essentially a non-commutative phenomenon. This gives another reasonwhy so many homology functors are Morita invariant: Usually, they arise as extensions offunctors defined on a category of commutative algebras to a category of noncommutativealgebras. But Morita invariance imposes no restrictions whatsoever on functors definedon a category of commutative algebras, so that we can hope for a Morita invariant ex-tension. Examples show that if a functor can be extended “naturally”, then the extensiontends to be indeed Morita invariant. A generalization of *- homomorphism will be usedbelow for definition of fundamental group. This generalization uses some ideas related toMorita equivalence.This article assumes elementary knowlege of following subjects.

1. Set theory [32].

2. Category theory [28].

3. Algebraic topology [28].

4. C∗− algebras and operator theory [22], [23], [27], [41].

The tems "set", "family" and "collection" are synonyms, and the term "class" is reservedfor an aggregate which is not assumed to be a set [32]. A category [34] is said to be smallif whose class of objects is a set. Following table contains used in this paper notations.The set of primitive ideals is a topological space with the hull-kernel topology (or Jacob-son topology).

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Symbol MeaningN monoid of natural numbersZ ring of integers

Zm ring of integers modulo mR (resp. C) Field or real (resp. complex) numbers

H Hilbert spaceI = [0, 1] ⊂ R Closed unit interval

A′′ Bicommutant of C∗ algebra A [40]B(H) Algebra of bounded operators on Hilbert space H

K(H) or K Algebra of compact operators on Hilbert space HU(H) ⊂ B(H) Group of unitary operators on Hilbert space HU(n) = U(Cn) Group of unitary operators on Hilbert space Cn

U(A) ∈ A Group of unitary operators of algebra AM(A) Multiplier algebra of C∗− algebra A

Q(A) = M(A)/A Outer multiplier algebra of C∗− algebra AMs(A) = M(A)⊗K Stable multiplier algebra of C∗− algebra A

Qs(A) = (M(A)⊗K)/(A⊗K) Stable outer multiplier algebra of C∗− algebra AMn(A) The n× n matrix algebra over C∗− algebra A

A+ C∗− algebra A with adjointed identityA+ Positive cone of C∗− algebra A

Aut(A) Group * - automorphisms of C∗ algebra A(π, H) Representation of C∗ algebra A,

i.e. * - homomorphism A→ B(H)C∗ = U(1) = z ∈ C | |z| = 1 Group of unitary elements in C

C1 = C⊕C Complex Clifford algebra with standard odd grading [22]C(X) C∗ - algebra of continuous complex valued

functions on topological space XCb(X) C∗ - algebra of bounded continuous complex valued

functions on topological space XA Spectrum of C∗ - algebra A with the hull-kernel topology

(or Jacobson topology)

Definition 1.2. [28] Let p : X → X covering projection. It is clear that there is a groupof self-equivalences of p (a self-equivalence is a homeomorphism f : X → X such thatp f = p). We denote this group by G(X|X). This group is also called the group of coveringtransformations of p.

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2 Fundamental groupoid and fundamental group

2.1 Fundamental groupoid/group in algebraic topology

Algebraic topology concern with good spaces as well as Hausdorff, (locally) compact orcellular spaces etc. Paths and their homotopies yield satisfactory definition of fundamentalgroupoid/group for these spaces. Let us recall necessary algebraic topology definitions.A groupoid is a small category in which every morphism is equivalence Paths and theirhomotopies can be used in case of good spaces.

Theorem 2.1. [28] For each topological space X there is a category P(X) whose objects are pointsof X, whose morphisms from x0 to x1 are path classes with x0 as origin and x1 as end, and whosecomposite is product of path classes.

The category P(X) is called category of path classes of X or the fundamental groupoid of Xbecause of following theorem.

Theorem 2.2. [28] P(X) is a groupoid.

Definition 2.3. Let X be a topological space and let x0 ∈ X. The fundamental group of Xbased at x0, denoted by π(X, x0) is defined to be the group of path classes with x0 as originand end.

If topological space X is locally path connected and semilocally 1-path connected [28] thenexist universal covering projection p : X → X. In this case fundamental group is group ofcovering transformations G(X|X) of universal covering projection p.

2.2 Fundamental groupoid/group of schemes

Notion of fundamental group can be generalized in different ways. For example it isgeneralized for schemes [25]. The definition 2.3 does not generalize well to schemes -there are simply too few algebraically defined closed paths. But it it is generalizationcovering projection which yields generalization of fundamental group of scheme. Let Xbe projective variety over C and let Xan be associated analytical manifold. The Riemannexistence theorem states that functor, which associates with any finite étale map Y → Xthe covering projection of analytic manifolds Yan → Xan. So it is reasonably to regard anyfinite étale morphism of scheme as generalization of finitely listed covering projection.Following construction provides generalization of fundamental group of scheme. Let Xbe a connected scheme, and let x → X be a geometric point of X. Define a functorF : FEt/X → Sets, where FEt/X is the category of X - schemes finite and étale over X, bysetting F(Y) = HomX(x, Y). Thus to give an element of F(Y) is to give a point y ∈ Y lyingover x and k(x) homomorphism k(y)→ k(x). It may be shown that this functor is strictlyprorepresentable, that is, that there exists a directed set I, a projective system (Xi, φij)i,j∈Iin FEt/X in which the transition morphisms φij : Xj → Xi(i ≤ j) are epimorphisms andelements fi ∈ F(Xi) such that

1. fi = φij f j, and

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2. the natural map lim←− → F(Z) is an isomorphism for any Z ∈ FEt/X.

The projective system X = (Xi, φij) will play role of the universal covering space of atopological space, and fundamental group can be defined as automorphism group.For any X - scheme Y we write AutX(Y) for the group of X - automorphisms of Y actingon the right. For any Y ∈ EFt/X, AutX(Y) acts on F(Y) (on the right), and if Y is con-nected, then this action is faithful, that is, for any g ∈ F(Y) σ 7→ σg : Aut(Y) → F(Y) isbijective, then Y is said to be Galois over X. For any Y ∈ EFt/X there is Y′ ∈ EFt/X that isGalois and an X - morphism Y′ → Y. It follows that the objects Xi ∈ X may be assumed tobe Galois over X. Now given j ≥ i, we can define a map ψij : AutX(Xj)→ AutX(Xi) by re-quiring that ψij(σ) fi = φij σ f j. We define π1(X, x) to be profinite group lim←−AutX(Xi).The notion of Galois sheme can be generalized by following way. We say that Y → X isGalois if following map

AutX(Y)×Y → Y×X Y;

(g, y) 7→ (y, gy); (g ∈ G, y ∈ Y)

is bijection.Let us remind theorem about that functor F(Y) = HomX(x, Y) is pro-representable. In-deed we prove more general theorem which concern Galois category. First of all weremind notion of Galois category [26].

Definition 2.4. A sequence of sets

E1h−→ E2

h1⇒h2

E3

is exact if h is an injection and h(E1) = x ∈ E2 : h1(x) = h2(x).

Definition 2.5. Let

E1h1⇒h2

E2h−→ E3

be a sequence in category C. The sequence is exact if ∀X ∈ C following sequence is exact

Hom(E3, X)h∗−→ Hom(E2, X)

h∗1⇒h∗2

Hom(E1, X).

Definition 2.6. A morphism f : S′ → S is said to be an effective epimorphism if the sequence

S′′ = S′ ×S S′p1⇒p2

S′f−→ S

is exact i.e., if the sequence

Hom(S, Y)→ Hom(S′, Y)p∗1⇒p∗2

Hom(S′′, Y)

is exact, as a sequence of sets, ∀Y.

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Galois category contains category C and functor F which satisfies following axioms:

1. (C0) C has an initial object ∅ and a final object S .

2. (C1) Finite fibre-products exist in C, i.e., if X → Z and Y → Z are morphisms in C ,then X×Z Y exists in C.

3. (C2) If X, Y ∈ C , then the disjoint union X ä Y ∈ C.

4. (C3) Any morphism u : X → Y in C admits a factorization of the form

Xu1

u // Y

Y1

/

j??

where u1 is an effective epimorphism, j is a monomorphism and Y = Y1 ä Y2, Y2 ∈C. Further, this factorization of u into an epimorphism and a monomorphism isessentially unique in the sense that if

Xu′1

u // Y

Y′1/

j′??

is another such factorization, then there exists an isomorphism ω : Y1 → Y′1 suchthat u′1 = ω u1 and j = j′ ω. (Because a factorization into a product of an effectiveepimorphism and a monomorphism is unique).

5. (C4) If X → C and G is a finite group of automorphisms of X acting, say, to theright on X, then the quotient X/G of X by G exists in C and the natural morphism

Xη−→ X/G is an effective epimorphism.

1. (F0) F(X) = ∅⇔ X = ∅.

2. (F1) F(S) = a set with one element;

F(X×Z Y) = F(X)×F(Z) F(Y), ∀X, Y, Z ∈ C.

3. (F2) F(X1 ä X2);

4. (F3) If X u−→ Y is an effective epimorphism in C, the map F(u) : F(Z)→ F(Y) is onto.

5. (F4) Let X ∈ C and G a finite group of S -automorphisms of X (acting to the righton X). The natural map η : X → X/G (see (C3)) defines a surjection F(η) : F(X) →F(X/G) (see (F3)).

Definition 2.7. A category which has the properties (C0), . . . , (C4) of and from whichthere is given a functorF into finite sets with the above properties (F0), . . . , (F5) is calleda Galois category; the functor F itself is known as a fundamental functor.

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Definition 2.8. A covariant functor G from a category C to Ens is representable if ∃ anobject Y ∈ C such that:

HomC(Y, X)∼→G(X), ∀X ∈ C.

Suppose now that C is any category and G : C → Ens is a covariant functor. If X ∈ C and

ξ ∈ G(X), we write, as a matter of notation, Gξ−→ X If G

ξ−→ X, and Gη−→ Y and X u−→ Y is a

C -morphism, we say that the diagram

G X

Y

[[]ηw

ξ

u

is commutative if G(u)(ξ) = η. If Gξ−→ X, then for any Z ∈ C , we have a natural map

HomC(X, Z)→ G(Z) defined by u 7→ G(u)(ξ).

Definition 2.9. We say that G is pro-representable if ∃ a projective system (Si, φij)i,j∈I ofobjects of C and elements τi ∈ G(Si) (called the canonical elements of G(Si)) such that

1. the diagrams

G Si

Sj

[[]τjw

τi

ϕij(j≥i)

are commutative.

2. for any Z ∈ C , the (natural) map

lim←−i∈I

HomC(Si, Z)→ F(Z)

is bijective.

2.3 Construction of the Fundamental group

Theorem 2.10. 1. Let C be a Galois category with a fundamental functor F. Then there exists apro-finite group π (i.e., a group π which is a projective limit of finite discrete groups providedwith the limit topology) such that F is an equivalence between C and the category C(π) offinite sets on which π acts continuously.

2. If C F′−→ C(π′) is another such equivalence, then π′ is continuously isomorphic to π and thisisomorphism between π and π′ is canonically determined up to an inner automorphism ofπ.

Definition 2.11. The profinite group π, whose existence is envisaged in assertion (1) ofthe theorem 2.10 will be called the fundamental group of the Galois category C

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The theorem 2.10 is a consequence of the following series of lemmas.

Definition 2.12. A category C is Artinian if any "decreasing" sequenceT1 T2?

_

j1oo T3?

_

j2oo ...? _

j3oo

of monomorphisms in C is stationary, i.e., the jr are isomorphisms for large r. A (covariant)functor F : C → Ens is left-exact if it commutes with finite products i.e., if F(X × Y) =F(X)× F(Y) and if, for every exact sequence

X u−→ Yu1⇒u2

Z

in C, the sequence

F(X)F(u)−−→ F(Y)

F(u1)⇒

F(u2)F(Z)

is exact as a sequence of sets. If

Yu1⇒u2

Z

are morphisms in C , a kernel for u1, u2 in C is a pair (X, u) with X ∈ C and u : X → Y inC such that

X u−→ Yu1⇒u2

Z

is exact in C. Clearly a kernel is determined uniquely up to an isomorphism in C.

Lemma 2.13. Let C be a category in which finite products exists. Then finite fibre-products existin C⇔ kernels exist in C .

Proof. ⇒ Let

Yu1⇒u2

Z

be morphisms in C. We have a commutative diagram:

Z

Y Y

Y×Z Y

(Y×Z Y)×(Y×Y) Y Y×Y

Y

44444446u1

u2

p1

4444446p2

(p1,p2)

44446

4444446

diagonal

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and it easily follows that (Y×Z Y)×Y×Y Y is a solution for the kernel of u1 and u2.

⇐ Suppose Xf−→ Z and Y

g−→ Z are morphisms in C. If p and q are the canonical projections

X×Yp−→ X, X×Y

q−→ Y, we have an exact sequence:

ker( f p, gq)f p⇒gq

Z.

It follows that ker( f p, gq) is a solution for the fibre-product X×Z Y.

In fact, we have shown that finite fibre products and kernels can be expressed in terms ofeach other. Hence, F commutes with finite fibre products⇔ it is left-exact.

Corollary 2.14. A fundamental functor is left-exact (see (F1))

Lemma 2.15. A Galois category is Artinian.

Proof. LetT1 T2?

_

j1oo T3?

_

j2oo ...? _

j3oo

be a decreasing sequence of monomorphisms in C . We have then:

Tr+1jr−→ Tr is a monomorphism

⇔ Tr+1∼→∆

Tr+1 ×Tr Tr+1

⇒ F(Tr+1)∼→∆

F(Tr+1)×F(Tr) F(Tr+1) by ((F1), (F5))

⇔ F(Tr+1)F(jr)−−→ F(Tr) is a monomorphism.

Since the F(Tr) are finite, this implies that the F(jr) are isomorphisms for large r; we arethrough by (F5).

Lemma 2.16. Let C be a Galois category with a fundamental functor F. Then F is strictly pro-representable

Proof. With the notations of 2.8 consider the set E of pairs (X, ξ) with Fξ−→ X. We order E

as follows:(X, ξ) ≥ (X′, ξ ′)⇔ ∃ a commutative diagram :

F X

X′

[[]ξ ′

w

ξ

We claim that E is filtered for this ordering; in fact, if (X, ξ), (X′, ξ ′) ∈ E in view of (F1)we get a commutative diagram:

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X

F X× X′

X

ξ

[[[]

ξ ′

w

(ξ,ξ ′)

[[[ p

p′

where p and p are the natural projections. We say that a pair (X, ξ) ∈ E is minimal in E iffor any commutative diagram

Fη

ξ // X

Y/

j??

with a monomorphism j, one necessarily has that j is an isomorphism.

1. (*) Every pair in E is dominated, in this ordering, by a minimal pair in E. Observethat C is Artinian (Lemma 2.15).

2. (**) If (X, ξ) ∈ E is minimal and (Y, η) ∈ E then a u ∈ HomC(X, Y) in a commutativediagram

F X

Y

[[]η

w

ξ

u

is uniquely determined. In fact, if u1, u2 ∈ HomC(X, Y) such that the diagrams

F X

Y

[[]η

w

ξ

u1

F X

Y

[[]η

w

ξ

u2

are commutative then by (C1) and Lemma 2.13 ker(u1, u2) exists; since F is left exactwe get a commutative diagram

F

ξ

ξ

((

η // Y

u2

u1

ker(u1, u2)

j // X

12

with a monomorphism j. As (X, ξ) is minimal j must be an isomorphism, i.e.,u1 = u2. From (*), (**) it follows that the system I of minimal pairs of E is directed. If(X, ξ) ∈ I, (Y, η) ∈ E and u ∈ HomC(Y, X) appears in a commutative diagram

F X

Y

[[]η

w

ξ

u

then u must be an effective epimorphism.

In fact, be (C3) we get a factorization

Fξ //

ξ

η

X1 ä X2 = X

X1+

j

99

Yu1

>>

By minimality of (X, ξ), it follows that j is an isomorphism; thus u is an effectiveepimorphism. In particular:

3. (***) The structure morphisms occurring in the projective family I are effective epi-morphisms.

Consider now the natural map

lim−→i∈I

HomC(Si, X)F(X), X ∈ C.

By (*) this is onto; by (**) it is injective. From (***) it thus follows that F is strictly pro-representable.

3 Hopf - Galois extensions

3.1 Coaction of Hopf algebras

Let H be a Hopf algebra over a commutative ring C, with bijective antipode S. We use theSweedler notation [42] for the comultiplication on H : ∆(h) = h(1) ⊗ h(2). MH (respec-tively HM) is the category of right (respectively left) H-comodules. For a right H-coactionρ (respectively a left H-coaction λ) on a C -module M, we denote

ρ(m) = m[0] ⊗m[1]; λ(m) = m[1] ⊗m[0].

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The submodule of coinvariants McoH of a right (respectively left) H-comodule M consistsof the elements m ∈ M satisfying

ρ(m) = m⊗ 1 (1)

respectivelyλ(m) = 1⊗m. (2)

Definition 3.1. Let A be associative algebra and A ∈ MH . Algebra A is said to be H -comodule algebra if H - coaction ρ : A→ A⊗ H satisfies following conditions:

ρ(ab) = a[0]b[0] ⊗ a[1]b[1]; ∀a, b ∈ A; (3)

a⊗ ∆(h) = ρ(a)⊗ h. (4)

Let A be a right H-comodule algebra.AMH and MHA are the categories of left and right

relative Hopf modules. We have two pairs of adjoint functors (F1 = A ⊗AcoH −, G1 =(−)coH) and (F2 = ⊗AcoH A, G2 = (−)coH) between the categories AcoH M and AMH , andbetween MAcoH and MH

A . The unit and counit of the adjunction (F1, G1)are given by theformulas

η1,N : N → (A⊗AcoH N)coH , η1,N(n) = 1⊗ n;

ε1,M : A⊗AcoH McoH → M, ε1,M(a⊗m) = am.

The formulas for the unit and counit of (F2, G2) are similar. Consider the canonical maps

can : A⊗AcoH A→ A⊗ H, can(a⊗ b) = ab[0] ⊗ b[1]; (5)

can′ : A⊗AcoH A→ A⊗ H, can′(a⊗ b) = a[0]b⊗ a[1]. (6)

Theorem 3.2. [16] Let A be a right H-comodule algebra. Consider the following statements:

1. (F2, G2) is a pair of inverse equivalences;

2. (F2, G2) is a pair of inverse equivalences and A ∈AcoH M is flat;

3. can is an isomorphism and A ∈AcoH M is faithfully flat;

4. (F1, G1) is a pair of inverse equivalences;

5. (F1, G1) is a pair of inverse equivalences and A ∈ MAcoH is flat;

6. can′ is an isomorphism and A ∈ MAcoH is faithfully flat.

These the six conditions are equivalent.

Definition 3.3. If conditions of theorem 3.1 are hold, then A is said to be left faithfully flatH-Galois extension

It is well-known that can is can isomorphism if and only if can′ is an isomorphism.

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3.2 Action of finite group

Let G be a finite group. A set H = Map(G, C) has a natural structure of Hopf algebra(See [20]). Addition (resp. multiplication) on H is pointwise addition (resp. pointwisemultiplication).. Let δg ∈ H, (g ∈ G) be such that

δg(g′)

1 g′ = g0 g′ 6= g

(7)

Comultiplication ∆ : H → H ⊗ H is induced by group multiplication

∆ f (g) = ∑g1g2=g

f (g1)⊗ f (g2); ∀ f ∈ Map(G, C), ∀g ∈ G.

i.e.∆δg = ∑

g1g2=gδg1 ⊗ δg2 ; ∀g ∈ G.

Action G× A→ A, (g, a) 7→ ga naturally induces coaction A→ A⊗ H (H = Map(G, C)).

a 7→ ∑g∈G

ga⊗ δg (8)

Equations (3), (4) are equivalent to following conditions of group action

g(a1a2) = (ga1)(ga2), ∀g ∈ G, a1, a2 ∈ A,

(g1g2)a = g1(g2a), ∀g1, g2 ∈ G, a ∈ A.

Any element x ∈ A⊗ H can be represened as following sum

x =

(∑

g∈Gag ⊗ δg

).

Let a ∈ A be such that ga = a, ∀g ∈ G then

a 7→ ∑g∈G

a⊗ δg = a⊗ 1. (9)

From (9) it follows that AcoH = AG, where AG = a ∈ A : ga = a; ∀g ∈ G is an algebraof invariants. There is a bijective natural map

A⊗ H ≈−→ Map(G, A) (10)

∑g∈G

ag ⊗ δg 7→(

g 7→ ag)

.

From 9 it follows that 5) can be represented in terms of group action by following way

15

can

(∑

i=1,...,nai ⊗ bi

)= ∑

i=1,...,ng∈G

ai(gbi)⊗ δg. (11)

There is the unique map canG : A⊗AG A→ Map(G, A)

∑i=1,...,n

ai ⊗ bi 7→ (g 7→ ∑i=1,...,n

ai(gbi)), (ai, bi ∈ B, ∀g ∈ G) (12)

From bijection of (10) it follows that bijectivity of can is equivalent to bijectivity of canG,i.e

A⊗AG A ≈ Map(G, A). (13)

Lemma 3.4. Let A be an unital algebra. Suppose that finite group G acts on A. Then followingstatements:

1. canG : A⊗AG A→ Map(G, A) defined by (12) is bijection;

2. There are elements bi, ai ∈ A (i = 1, ..., n) such that

∑i=1,...,n

aibi = 1A, (14)

∑i=1,...,n

ai(gbi) = 0 ∀g ∈ G (g is nontrivial); (15)

are equivalent.

Proof. 1. => Denote by e ∈ G unity of G. Let f ∈ Map(G, A) be such that

f (e) = 1A;

f (g) = 0; (g 6= e).

From bijcection of A⊗AG A→ Map(G, A) it follows that there are elements a1, ..., an, b1, ..., bn ∈A such that ∑i=1,...,n ai ⊗ bi corresponds to f i.e.

f (g) = ∑i=1,...,n

ai(gbi).

It is clear that elements a1, ..., an, b1, ..., bn satisfy conditions (14), (15)

2. <= Let us enumerate elements of G, i.e G = g1, ..., g|G|. a1, ..., an, b1, ..., bn satisfyconditions (14), (15), and let be f ∈ Map(G, A) be any map from G to A; andx ∈ A⊗AG A is defined as

x = ∑i=1,...,|G|

f (g)ai ⊗ g−1bi.

From (14), (15) it follows that f = canG(x) So canG is map onto.

Definition 3.5. Let G be a finite group. Suppose that H = Map(G, C) and H has a naturalstructure of Hopf algebra. Any H - Galois extension A → B is said to be G - Galoisextension.

16

3.3 Noncommutative finite covering projection

Definition 3.6. [13] Let A be C∗ - algebra. A *- automorphism α is said to be generalizedinner if is obtained by conjugating with unitaries from multiplier algebra M(A).

Definition 3.7. [13] Let A be C∗ - algebra. A *- automorphism α is said to be partly inner ifits restricion tio some non-zero α- invariant two-sided ideal is generalized inner. We callautormphism purely outer if it is not partly inner.

Definition 3.8. Suppose that finite gooup G acts on Mn(A). A projector p ∈ Mn(A) issaid to be an G - equivariant projector if gp = p for all g ∈ G. A left A−G module P is saidto be an equivariant G - projective if P is an image if G - equivariant projector.

Definition 3.9. An algebra A is said to be connected if A is not a direct sum of two non-trivial algebras.

Remark 3.10. Any connected commutative C∗ - algebra corresponds to a connected topo-logical space.

3.11. If ρ : A → B(H) a irreducible representation then denote by ρα : A → B(H) arepesentation defined by following way

ρα(a) = ρ(α(a)).

Definition 3.12. Let A be C∗ - algebra, α is *- automorhism of A. A repesentation π : A→B(H) is said to be α -invariant representation if ρα is unitary equivalent to ρ.

Definition 3.13. Let A be C∗ - algebra, α ∈ Aut(A), π : A → B(H) is α - invariantrepresentation. We say that α is inner with respect to π if there exists unitary elementu ∈ U(M(A)) such that π(α(a)) = π(u)π(a)π(u∗) (∀a ∈ A). If α is not inner for all α -invariant representations then α is said to be strictly outer automorphism.

Definition 3.14. Let A be a C∗ - algebra and G ⊂ Aut(A) is a finite subgroup of * -automorphisms. An injective * - homomorphism f : AG → A is said to be noncommutativefinite covering projection (or noncommutative G - covering projection) if f satisfies followingconditions:

1. Both AG and A are connected algebras.

2. A is a finitely generated equivariant G - projective left AG Hilbert C∗-module.

3. If α ∈ G then α is strictly outer

4. f is a G - Galois extension.

G is said to be covering transformation group of f . Denote by G(B|A) covering transforma-tion group of covering projection A→ B.

17

3.15. Let f : B→ A be a noncommutative G - covering projection. Let ρ : A→ B(H) be airreducible representation i. e. ρ ∈ A. Let g ∈ G and ρg : A→ B(H) is such that

ρg(a) = ρ(ga).

So it is an action of G on A such that

g 7→ (ρ 7→ ρg); ∀g ∈ G, ∀ρ ∈ A.

Let us enumerate elements of G by integers, i. e. g1, ..., gn ∈ G, n = |G| and define actionof σ : G× i, ..., n → i, ..., n such that σ(g, i) = j ⇔ gj = ggi Let ρ⊕ = ⊕g∈Gρg : A →B(Hn) be such that

ρ⊕(a)(h1, ..., hn) = (ρ(g1a)h1, ..., (ρ(gna)hn). (16)

Let us define such linear action of G on Hn that

g(h1, ..., hn) = (hσ(g−1,1), ..., hσ(g−1,n)). (17)

From (16), (17) it follows that

g(ah) = (ga)(gh); ∀a ∈ A, ∀g ∈ G, ∀h ∈ Hn,

i.e. Hn ∈A MG. From AMG ≈B M it follows that ρ⊕ defines representation η : B →B(K). K = (Hn)G. If (η, H) is not irreducidle then there is a nontrivial B - submoduleN K . From AMG ≈B M it follows that A⊗B N Hn is a nontrivial A - submodule.If we identify H with first summand of Hn then (A ⊗B K) ∩ H H is a nontrivial A -submodule. This fact contradicts with that (ρ, H) is irreducible. So (η, K) is an irreduciblerepresentation. In result there is a natural map f : A→ B (ρ 7→ η).

3.4 Fundamental groupoid of covering projection

3.16. Let f : B → A be a noncommutative G- covering projection, (y0, y1) ∈ B2, Xy0,y1 =

(x0, x1) ∈ A2 : f (x0) = y0 & f (x1) = y1. Suppose that (x′0, x′1) ∼ (x′′0 , x′′1 ) if and only if∃g ∈ G such that x′′0 = xy′0, x′′1 = gx′1. An element of factorset ω : Xy0,y1 / ∼ is said to bea path from y0 to y1 with respect to f . Denote by Paths(y0, y1) the set of paths from y0 toY1. Let ω′ = (x0, x1) ∈ Paths(y0, y1), ω′′ = (x′1, x2) ∈ Paths(y1, y2) be to paths. There isan element g ∈ G such that x1 = gx′1. A path ω = (x1, gx2) ∈ Paths(y0, y2) is said to becomposition of ω′ and ω′′. This composition is also denoted as ω = ω′ ω′′.

Definition 3.17. Let f : B→ A be a noncommutative G - covering projection. Fundamen-tal groupoid of B with respect to f (Grpd(B, f )) is a small category. Objects of Grpd(B, f )are elements of B. Morphisms are paths with respect to f .

18

3.5 Covering projection of continuous trace C∗-algebras

3.18. Let A be a C∗ - algebra. For each x ∈ A+ the (canonical) trace of π(x) dependsonly on the equivalence class of an irreducible representation (π, H) of A, so that wemay define a function x : A → [0, ∞] by x(t) = Tr(π(x)) whenever (π, H) ∈ t. FromProposition 4.4.9 [41] it follows that x is lower semicontinuous function on a in Jacobsontopology.

Definition 3.19. [41] We say that element x ∈ A has continuous trace if x ∈ Cb(A). We saythat C∗ - algebra has continuous trace if set of elements with continuous trace is dense inA.

Definition 3.20. [41] A positive element in C∗ - algebra A is abelian if subalgebra xAx ⊂ Ais commutative.

We say that a C∗ - algebra A is of type I if each non-zero quotinent of A contains non-zeroabelian element. If A is even generated (as C∗ - algebra) by its abelian elements we saythat it is of type I0.

Proposition 3.21. [41] A positive element x in C∗ - algebra A is abelian if dimπ(x) ≤ 1 forevery irreducible representation (π, H) of A.

Theorem 3.22. (Theorem 5.6 [41]) For each C∗ - algebra A there is a dense hereditary ideal K(A),which is minimal among dense ideals.

Proposition 3.23. [41] Let A be a C∗ - algebra with continuous trace Then

1. A is of type I0;

2. A is a locally compact Hausdorff space;

3. For each t ∈ A there is an abelian element x ∈ A such that x ∈ K(A) and x(t) = 1.

The last condition is sufficient for A to have continuous trace.

Remark 3.24. From [8], Proposition 10, II.9 it follows that a continuous trace C∗-algebra isalways a CCR-algebra, a C∗-algebra where for every irreducible representation (π, H) ofA and for every element x ∈ A, π(x) is a compact operator.

Definition 3.25. Let A be C∗ - algebra, and Prim(A) is a set of primitive ideals. For anysubset F ∈ A there exist subset F− such that

F− = t ∈ Prim(A) : F ∈ t.

Prim(A) is topological space such that for any closed subset X ∈ Prim(A), ∃F ⊂ A, X =F−.

Proposition 3.26. [6] If a topological group G act prperly on a topological space then orbit spaceX/G is Hausdorff. If also G is Hausdorff, then X is Hausdorff.

19

Lemma 3.27. Let G be a finite group, f : B → A is a noncommutative G- covering projection.Suppose that B is separable continuous trace algebra. Then A is also separable continuous tracealgebra. Then it is natural (topological) covering projection f : A → B of topological spaces, Gacts freely on A and it is natural homeomorphism B ≈ A/G.

Proof. Suppose that G does not act freely on A. Then there are xinA and g ∈ G such thatgx = x. By defintion 3.14 g should be strictly outer. Let (ρ, H) be representative of x. Then(ρg, H) is also representative of x. So ρ is unitary equivalent to ρg, i. e. there is unitaryU ∈ U(H) such that ρ(a) = Uρg(a)U∗ (∀a ∈ A). However since A is continuous tracealgebre ρ(A) = K, ρ(M(A)) = B(H), ρ(U(M(A))) = U(H). So it is u ∈ M(A) such thatρ(u) = U and we have ρg(a) = ρ(u)ρ(a)ρ(u∗). It means that g is inner with respect to(ρ, H), so action of g is not strictly outer. This contadiction proves the lemma.

Lemma 3.28. Let G be a finite group, f : B → A is a noncommutative G- covering projection.Suppose that B is separable continuous trace algebra. Then A is also separable continuous tracealgebra. Then it is natural (topological) covering projection f : A → B of topological spaces, Gacts freely on A and it is natural homeomorphism B ≈ A/G.

Proof. Since B is separable and A is finitely generated B - module, we see that A is sep-arable. From 3.15 follows that there is the natural map f : A → B (ρ 7→ η). From 3.24 itfollows that B is CCR algebra, so η(B) ≈ K(H). For any b ∈ B operator η(b) is compact,i.e. η(b) is norm limit of finite range operators sequence. Since A is finitely generatedthere are elements a1, ..., ap such that

∀a ∈ A, ∃b1, ..., bp; a = ∑1≤i≤p

biai. (18)

Let xij → ρ(bi) be convergent sequences of finite rank operators xij ∈ F (H). and ρ(bi) isnorm limit of finite rank operators. It clear that We have following convergent sequenceof finite rank operators

∑i

xijρ(ai)→ ρ(a) (19)

and ρ(A) ∈ K(H)Let ρ(a) ∈ ρ(A). We can define G - equivariant operator ρ⊕(a, ..., a) = ρ⊕(g(a, ..., a)). So(a, ..., a) is G invariant and ρ⊕((a, ..., a)) = ρ(b) It means it is canonical isomorphism

ρ(A) ≈ η(B). (20)

Thus ρ(B) = K(H) and A is a CCR algebra. Let t ∈ A and s = f (t) From 3.23 it followsthat there is an abelian element x ∈ B such that x ∈ K(A) and x(s) = 1. Element xmay be regarded as element of B and it is clear that x ∈ K(B). Denote by (ρ, H), (η, K)representations t and s respectively. From 3.21 it follows that dim η(x) ≤ 1. From 20 itfollows that dim ρ(x) ≤ 1. So x is element of B such that condition of 3.23 is satisfied.Thus B is continuous trace albebra. From 3.23 it follows that B is locally compact andHausdorff space. From 3.27 it follows that G acts freely on A and from B ≈ A/G itfollows that f is covering projection.

20

Remark 3.29. From lemma 3.28 it follows that finite covering projections of commutativealgebras are just covering projections of their character spaces. Really if B is commutativeand ρ : B → B(H) is irreducible then dim(H) = 1. From (20) it follows that any irre-ducible representation of A is one dimensional. So A is commutative and f : A → B iscommutative.

3.6 Finite coverings of noncommutative torus

Let us consider Galois extensions of noncommutative torus. Noncommutative torus Aθ

is C∗ - norm completion of algebra generated by two unitary elements u, v and followingconditions are hold:

uu∗ = u∗u = vv∗ = v∗v = 1;

uv = e2πiθvu,

where θ ∈ R. If θ = 0 then Aθ = A0 is commutative algebra of continuous functions oncommutative torus C(S1×S1) There is such trace τ0 on Aθ that τ0(∑−∞<i<∞,−∞<j<∞ aijuivj) =

a00. C∗ - norm of Aθ is defined by following way ‖a‖ =√

τ0(a∗a). Let us consider * -homomorphism f : Aθ → Aθ′ , where Aθ′ is generated by unitary elements u′ and v′.Homomorphism f is defined by following way:

u 7→ u′m;

v 7→ v′n;

It is clear thatθ′ =

θ + kmn

; (k = 0, ..., mn− 1). (21)

Let us show that *-homomorphism f is Galois extension. First of all note that commutativeC∗- subalgebras C(u′) ⊂ Aθ′ and C(v′) ⊂ Aθ′ generated by u′ and v′ respectively areisomorphic to algebra C(S1), where S1 is one dimensional circle. There are induced by f *-homomorphisms C(S1) = C(u)→ C(u′) = C(S1) , C(S1) = C(v)→ C(v′) = C(S1). These*-homomorphisms induces m and n listed covering projections respectively. Coveringgroups of these covering projections are G1 ≈ Zm and G2 ≈ Zn respectively. Generatorsof these groups are presented below:

u′ 7→ e2πim u′;

v′ 7→ e2πin v′.

Homomorphisms of commutative algebras C(u) → C(u′), C(v) → C(v′) correspond tocovering projection, it follows that there are elements xi ∈ C(u′) (i = 1, ..., r), yj ∈ C(v′)(j = 1, ..., s) such that

∑1≤i≤r

x2i = 1C(u′);

∑1≤i≤r

(g1xi)xi = 0; g1 ∈ G1;

21

∑1≤j≤s

y2i = 1C(v′);

∑1≤j≤s

(g2yi)yi = 0; g2 ∈ G2,

where g1 and g2 are nontrivial elements of G1 and G2.Actions of G1 and G2 induce action of G = G1 × G2 on Aθ′ . Let us set

ak = yjxi,

bk = xiyj,

where k = im + j.It is easy to check following equalities.

∑1≤k≤mn

akbk = 1Aθ′;

∑1≤k≤mn

(gak)bk = 0,

where g ∈ G is nontrivial element.

4 Invariants of finite covering projections

Let C be a category and F!, F! : C→ Groups covariant and contravariant functors from C tocategory to category of groups such that it is natural isomorphism F!(A) ≈ F!(A) for anyobject in C. If f is morphism in C then F! F!( f ) is endomorphism of F!(A)

Definition 4.1. Endomorphism F! F!( f ) is said to be invariant of f with respect to pair(F!, F!).

Definition 4.2. Let (F1, G1) , (F2, G2) be pairs of functors that above conditions are satis-fied. Class of homomorphisms φ(A) : F!1(A) → F!2(A) is said to be natural transformationfrom (F!1, F!

1) to (F!2, F!2) if diagram

F!1(A) F2(A)

F!1(A) F!2(A)u

F!1F!1( f )

w

φ(A)

uF!2F!

2( f )

w

φ(A)

is commutative for any C - morphism f .

Example 4.3. Let C be category such that objects of C are pointed topological spacesand morphisms are covering projections. Let π1 be fundametal group functor. There iscontravariant functor π!

1 such that for any covering (Y, y0) → (X0) and any closed pathω : S1 → X the functor sets lift of ω in Y. It is clear that π1(X, x0) ≈ π!

1(X, x0) for allpointed spaces (X, x0).

22

Example 4.4. Let C be category from 4.3 and H1 is first singular homology functor. Liftof singular chains provides wrong way functoriality functor H!

1. So there are invariantswith respect to pair (H!

1, H1). It is known that H1(S1) ≈ Z. Let c ∈ H1(S1) be generatorof H1(S1). There is Hurewicz homomorphism φ : π1(X)→ H1(X) such that

[ω] 7→ H1(ω)(c),

where ω : S1 → X is closed path and [ω] ∈ π1(X, x0) is its homotopy class. Homomor-phism φ is said to be Hurewicz homomorphism [28], it induces natural transformationfrom φ from (πab, π!

ab) to (H1, H!1), i.e. diagram

πab(X) H1(X)

πab(X) H1(X)u

πabπ!ab( f )

w

φab(X)

uH1H!

1( f )

w

φab(X)

is commutative.

Example 4.5. Consider K1 - homology. Since K1(S1) ≈ Z there is natural transformationfrom (πab, π!

ab) to (K1, K!1), in category of commutative C∗ - algebras.

Example 4.6. There is a standard odd grading on A⊗ A for any algebra A: (A⊗ A)(0) =(a, a) : a ∈ A and (A⊗ A)(1) = (a,−a) : a ∈ A So any C∗ algebra may by regardedas Z2 - graded algebra. If A and B are Z2 - graded a C∗ algebras. Let E(A, B) is set oftriples of triples (E, φ, F) such that E is countably generated graded Hilbert module overB, φ is a graded *- homomorphism from A to B(E), and F is an operator in B(E) of degree1, such that [F, φ(a)], (F2 − 1)φ(a) and (F− F∗)φ(a) are all in K(E) for all a ∈ A. Triple(E, φ, F) is called a KK(A, B) cycle. There is equivalence relation ≈ on E(A, B) definedin [22] KK(A, B) = E(A, B)/ ≈. Elements of KK(A, B) are equivalence classes of triples(E, φ, F) such that E is countably generated graded Hilbert module over B, φ is a graded*- homomorphism from A to B(E), and F is an operator in B(E) of degree 1, such that[F, φ(a)], (F2 − 1)φ(a) and (F − F∗)φ(a) are all in K(E) for all a ∈ A. Triple (E, φ, F)is called a KK(A, B) cycle. If f : A1 → A2 is graded homomorphism then for any Bthere is group homomorphism f ∗ : KK(A2, B) → KK(A1, B) induced by map (E, φ, F) →(E, φ f , F). If f : A1 → A2 is injective graded homomorphism such that A2 is finitelygenerated A1 module, (E, φ, F) is KK(A1, B) cycle then (A2⊗ f E, 1⊗ φ, 1⊗ F) is KK(A2, B)cycle. So there is wrong way functoriality homomorphism f! : KK(A1, B) → KK(A2, B).It is clear that f ∗ f! ∈ EndKK(A1,A1)

(KK(A1, B)). If g : B1 → B2 is graded homomorphismthen for any A there is group homomorphism g∗ : KK(A, B1) → KK(A, B2) inducedby map (E, φ, F) 7→ (E⊗gB2, φ⊗1, F⊗1), where ⊗ means Hilbert tensor product. If B2is finitely generated B1 module then any KK(B2, A) cycle (E, φ, F) can be regarded asKK(B1, A) cycle (E, φ g, F). So there is wrong way functorality homomorphism g! :KK(A, B2)→ KK(A, B1). It is clear that g!g∗ ∈ EndKK(B1,B1)

(KK(A, B1)).

23

Definition 4.7. Let f : B→ A be a noncommutative finite covering and C is separable C∗ -algebra. Endomorphism f ∗ f! ∈ EndKK(B,B)(KK(B, C)) (resp f ! f∗) ∈ EndKK(B,B)(KK(C, B))is said to be contravariant (resp. covariant) invariant of f . By [ f , C] and [C, f ] denote theseinvariants.

Example 4.8. Let A be algebras of continuous trace, Qs(A) = M(A⊗K)/A⊗K is stablemultiplier algebra [22] of A. There is natural injective homomorphism C0(A) → Qs(A).So any KK(Qs(A), B) cycle may be regarded as KK(C0(A), B) cycle. If C1 ≈ C2 algebrawith odd grading then there is canonical homomorphism KK ∗ (C0(A), C1). Otherwisesince KK∗(S1, C1 ≈ Z we have canonical homomorphism π1(C0(A)) → KK∗(C0(A), C1).Thus it is canonical homomorphism φ : π1(C0(A)) → KK(Qs(A), C1). So φ is naturaltransformation from (π1, p!

1) to (F!, F!).

Example 4.9. Let θ ∈ R be irrational number, m, n ∈ N, mn > 1, θ′ = θ/mn, θ′′ =(θ + k)/mn (k 6= 0 mod mn). Let u, v ∈ Aθ , u′, v′ ∈ Aθ′ , u′′, v′′ ∈ Aθ′′ be generators ofnoncommutative toruses. Let f ′ : Aθ → Aθ′ (resp. f ′′ : Aθ → Aθ′′ ) be * - homomorphismu 7→ u′m, v 7→ v′n (resp. u 7→ u′′m, v 7→ v′′n). It is known that K1(Aθ) ≈ KK(C0(R), Aθ) ≈Z2, and [u], [v] ∈ K1(Aθ are generators of K1(Aθ). Covariant invariant [C0(R), f ′] satisfiesfollowing equation

[u] 7→ m[u], [v] 7→ n[v].

It is clear that [C0(R), f ′] = [C0(R), f ′′]. Covering transformation groups of both f ′ are f ′′

are isomorphic to Zm ×Zn. This example shows noncommutative torus has much morecoverings than commutative ones.

5 Morita equivalences induced by bimodules over Hopf-Galois extensions

According to example 4.9 there are a lot of noncommutative covering provections besidescommutative ones. This fact can substantially extend calculated by coverings fundamentalgroup. However this group can be reduced if set of finite coverings is factorized by equiv-alence relation. This relation is Morita equivalences induced by bimodules over Hopf-Galois extensions. Defined in 4.7 invariants are invariant with respect to this equivalencerelation.

5.1 General theory

Definition 5.1. Let A and B be algebras. A Morita context connecting A and B is a sextupleM = (A, B, M, N, α, β)

1. M ∈ AMB,

2. N ∈ BMA,

3. α : M⊗B N → A (α is a morphism in AMA) ,

24

4. β : N ⊗A M→ B (β is a morphism in BMB) .

such that

1. α(x⊗ y)x′ = xβ(y⊗ x′);

2. β(y⊗ x)y′ = yα(a⊗ y′); ∀x, x′ ∈ M, ∀y, y′ ∈ N.

Definition 5.2. A Morita context M = (A, B, M, N, α, β) is strict if both α and β are iso-morphisms.

Let M be a right H-comodule, and N a left H-comodule. The cotensor product MH N isthe C - module

MH N = ∑i

mi ⊗ ni ∈ M⊗ N | ∑i

ρ(mi)⊗ ni = ∑i

mi ⊗ λ(ni). (22)

If H is cocommutative, then MH N is also a right (or left) H-comodule. If G is finitegroup and H = C(G) then

MH N = MG N ⊂ M⊗ N,

where MG N is generated by elements a⊗ b ∈ M⊗ N such that

ag⊗ b = a⊗ gb; ∀g ∈ G.

Definition 5.3. [16] Let A and B be right H-comodule algebras. An H-Morita contextconnecting A and B is a Morita context (A, B, M, N, α, β) such that M ∈ AMH

B , N ∈BMH

A , α : M⊗A N → A is a morphism in AMHB and β : N ⊗B M → A is a morphism in

BMHA

A morphism between two H-Morita contexts (A, B, M, N, α, β) and (A′, B′, M′, N′, α′, β′)is defined in the obvious way: it consists of a quadruple (κ, λ, µ, ν), where κ : A → A′

and λ : B → B′ are H-comodule algebra maps, µ : M → M′ is a morphism in AMHB and

ν : N → N′ is a morphism in BMHA such that κ α = α′(µ ⊗ ν) and λ β = β′(ν ⊗ µ)

MoritaH(A, B) will be the subcategory of the category of H-Morita contexts, consisting of

H-Morita contexts connecting A and B, and morphisms with the identity of A and B asthe underlying algebra maps.

Lemma 5.4. Let (A, B, M, N, α, β) be a strict H-Morita context. Then we have a pair of inverseequivalences (M⊗B −, N ⊗A −) between the categories AMH and BMH .

Proof. Let P ∈B MH . Then M⊗B P ∈B MH , with right H-action

ρ(m⊗B p) = m[0] ⊗B p[0] ⊗m[1]p[1].

The rest of the proof is straightforward.

25

Definition 5.5. [16] Assume that A and B are right faithfully flat H-Galois extensions ofAcoH and BcoH . A H-Morita context between AcoH and BcoH is a Morita context (AcoH

and BcoH , M1, N1, α1, β1) such that M1 (resp. N1) is a left AH Bop-module (resp. BH Aop-module) and

α1 : M1 ⊗BcoH N1 → AcoH is left AH Aop − linear,

β1 : N1 ⊗AcoH M1 → BcoH is left BH Bop − linear.

A morphism between two H-Morita contexts connecting AcoH and AcoH is a morphismbetween Morita contexts of the form (AcoH , BcoH , µ1, ν1), where µ1 is left AH Bop-linearand ν1 is left BH Aop-linear. The category of H-Morita contexts connecting AcoH andAcoH will be denoted by MoritaH (AcoH , BcoH).

Theorem 5.6. Let A and B be right faithfully flat H-Galois extensions of AcoH and AcoH . Thenthe categories MoritaH(A, B) and MoritaH (AcoH , BcoH) are equivalent. The equivalence func-tors send strict contexts to strict contexts.

Definition 5.7. Let B and C be right H-comodule algebras such that BcoH = CcoH = A.So M and N are A− A bimodules. Suppose that (B, C, M, N, α, β) is an H - Morita context(see definition 5.3). The M is called an A− H-Morita context if both α : M⊗C N → B andβ : N ⊗C M→ B are morphisms in AMA, i. e. morphisms of A− A bimodules.

If we concern with C∗ algebras then notion of strong Morita equivalence [10],[33] is ratheradequate then Morita equivalence. According to [33] Morita equivalence of C∗ algebras Aand B is equivalent to stable isomorphism, i.e.

A⊗K ≈ B⊗K

Following definition is adoption of 5.7 to noncommutative finite coverings.

Definition 5.8. Covering projections A → B, A → C are Morita equivalent or G - Moritaequivalent if

1. Both covering projections have the same covering transformation group G.

2. There is G equivariant isomorphism B⊗K ≈ C⊗K. This isomorphism is also A− Abimodule isomorphism.

Remark 5.9. From K ⊗ K ≈ K it follows that for for all f : A → B ⊗ K it is *- homo-morphism f : A⊗K → B⊗K. Conversely if p ∈ K is one dimensional projection then(IdA ⊗ p)A ⊗ K(IdA ⊗ p) ≈ A so we have correspondence between *- homomorphismsA ⊗ K → B ⊗ K and A ⊗ K → B ⊗ K. If A → B, A → C are G - Morita equivalentcovering projection then there is G - equivariant *- homomorphism B → C⊗K which isA− A bimodule homomorphism.

Example 5.10. Consider finite coverings Aθ → Aθ′ , Aθ → Aθ′′ from 4.9, θ′ = θ/mn,θ′′ = (θ + k)/mn. Covering transformation group of both coverings is G = Zm ⊗Zn. LetU, V ∈MN=mn(C) be unitary matrixes such that

UV = e2πik/nmVU.

26

There is following G equivariant isomorphism Aθ′ ⊗MN(C) ≈ Aθ′′ ⊗MN(C)

u′ ⊗ 1→ u′′ ⊗U; v′ ⊗ 1→ v′′ ⊗V.

This isomorphism is also Aθ − Aθ bimodule isomorphism. From K ⊗MN(C) ≈ K itfollows that there exist isomorphism Aθ′ ⊗K ≈ Aθ′′ ⊗K such that conditions of definition5.8 are satisfied. So considered coverings are Morita equivalent.

6 Category of finite noncommutative covering projections

We would like construct category such that Morita equivalent coverings are isomorphic.So we replcace any C∗ - ablebra A by A⊗K Objects of this category are finite noncom-mutative finite covering projections. Let A - be a stable C∗ - algebra, f1 : AG1

1 → A1,f2 : AG2

2 → A2 are noncommutative covering projections such that AG11 ≈ AG2

2 ≈ A, k :G2 → G1 is a surjective group homomorphism. Consider *- homomorphisms f : A1 → A2such that

1. f (k(g)a) = g f (a); ∀g ∈ G2, ∀a ∈ A1.

2. f is A− A bimodule homomorphism.

Definition 6.1. Let A → A1 , A → A2 be to finite noncommutative covering projection.An A covering morphism is a *- homomorphisms f : A1 → A2 such that above conditionsare satisfied.

6.2. Let A be a stable C∗ algebra. Let us introduce category Cov(A) of A - covering projec-tions. Objects of the category are noncommutative finite covering projections, morphismsare A covering morphisms.

Definition 6.3. Above category is said to be category of A covering projections.

6.4. Let p : BG → B be noncommutative covering projection, A → BG then G naturallyacts on C = A ⊗BG B. If G′ = G/(ker(G → Aut(C)) then A = CG′ and we have IfCG′ → C.

Definition 6.5. Let us consider situation from 6.4. A * - homomorphism A→ C if said tobe a induced by A→ B noncommutative covering projection if A→ C is a G′ covering projec-tion. A surjective homomorphism G → G′ is said to be a induced by A→ B homomorphismof covering transformation groups

7 Covering projections of noncommutative torus

Noncommutative torus covering projections are already considered in example 4.9. Wewould like to prove this example covers whole set of noncommutative torus covering

27

projections. Let Aθ be a noncommutative torus, generated by unitaries u, v ∈ U(Aθ) Thecommutavive torus T2 = U(1)×U(1) exactly acts on Aθ by following way:

(z1, z2)u = z1u, (z1, z2)v = z2v, (z1, z2) ∈ T2 = U(1)×U(1).

Let Aθ → B be G - Galois extension, Aθ is considerd as subalgebra of B, i. e. Aθ ⊂ B.Let G′ ∈ Aut(B) be the maximal subroup such that there is a natural surjective mapf : G′ → T2 which satisfies following condition:

g′a = f (g′)a, ∀g′ ∈ G′, a ∈ Aθ .

It is clear that G ⊂ G′, and there is following exact sequence of groups:

e → G → G′f−→ T2 → e. (23)

Map f is a covering projection (in topological sense) and G is its covering transformationgroup. Let us consider following special cases of sequense (23):

1. G′ = G×T2

2. G′ is a connected topological space.

7.1. G′ = G×T2

In this case we have following:G′ ≈

⊕g∈G

T2g,

f ((tg1 , ..., tgn)) = tg1 + ... + tgn , (tg1 , ..., tgn) ∈⊕g∈G

T2g.

The⊕

g∈G T2g is a compact Lie group and any its representation is a direct sum of irre-

ducible repersentations. A representation⊕

g∈G T2g → U(1) if given by:

((z1g1 , z2g1), ..., (z1gn , z2gn)) 7→ zig11g1

zjg12g1

... zign1gn

zjgn2gn

, igk , jgk ∈ Z

An element a ∈ B is said to be a homogeneous element of type ((ig1 , jg1), ..., (ign , jgn) if itsatisfies following condition:

((z1g1 , z2g1), ..., (z1gn , z2gn))a = zig11g1

zjg12g1

... zign1gn

zjgn2gn

a.

is said to be of type ((ig1 , jg1), ..., (ign , jgn)) (igk , jgk ))). If a′ (resp. a′′) is a homogeneouselement of type ((i′g1

, j′g1), ..., (i′gn , j′gn)), (resp. ((i′′g1

, j′′g1), ..., (i′′gn , j′′gn))) then the product a′a′′

is a homogeneous element of type ((i′g1+ i′′g1

, j′g1+ j′′g1

), ..., (i′gn + i′′gn , j′gn + j′′gn)). So B is a(Z2)G graded algebra. G naturally acts on

(Z2)G. If x ∈

(Z2)G and a ∈ B is homogeneous

element of type x then ga is a homogeneous element of type gx. Similarly Aθ is a Z2

graded alebra and we for all x ∈ Z2 we can define x homogeneous elements. From

28

exactness of⊕

g∈G T2g action it follows that there is a nonzero homogeneous element ug1 ∈

B of type ((1, 0), (0, 0), ..., (0, 0). Denote by ug a homogeneous element given by:

ug = g′ug1 , g′g1 = g ∈ G.

There is the C - linear map p : B→ Aθ given by:

p(a) = ∑g∈G

ga, ∀a ∈ A0.

It is clear that p(ug1) ∈ Aθ is a (0, 1) homogeneous element. However any (0, 1) homoge-neous element is equal to cu (c ∈ C). If we replace ug1 with c−1ug1 then p(ug1 = u. Fromp(ug1)p(u∗g1) = uu∗ = 1 it follows that

(ug1 + ... + ugn)(u∗g1+ ... + u∗gn) = 1. (24)

Right part of (24) is a ((0, 0), ..., (0, 0)) homogeneous element. If ug1 u∗g26= 0 then left part of

(24) contains a nonzero homogeneous summand of ((1, 0), (0,−1), ..., (0, 0) type. Becauseit is impossible we have ug1 u∗g2

= 0. Similarly we can define elements vg1 , vgn and

vg1 + ... + vgn = v;

(vg1 + ... + vgn)(v∗g1 + ... + v∗gn) = 1.

If ug1 vg2 6= 0 then right part of

uv = (ug1 + ... + ugn)(vg1 + ... + vgn) (25)

contains a nonzero homogeneous summand of ((1, 0), (0, 1), (0, 0), ..., (0, 0)) type. How-ever right part of (25) cannot contain this summand, so ug1 vg2 = 0. Similarly if g′, g′′ ∈ Gand g′ 6= g′′ we have following:

ug′ug′′ = ug′u∗g′′ = u∗g′ug′′ = u∗g′u

∗g′′ = vg′vg′′ = vg′v

∗g′′ = v∗g′vg′′ = v∗g′v

∗g′′ = 0, (26)

ug′vg′′ = ug′v∗g′′ = u∗g′vg′′ = u∗g′v

∗g′′ = vg′ug′′ = vg′u

∗g′′ = v∗g′ug′′ = v∗g′u

∗g′′ = 0. (27)

Element uug1 u∗g1is a sum of homogeneous elements of ((1, 0), (0, 0), ..., (0, 0)), ..., ((0, 0), (1, 0), ..., (0, 0)),...,

((0, 0), (0, 0), ..., (1, 0)) However from (26), (27) it follows that summands of ((0, 0), (1, 0), ..., (0, 0)),...,((0, 0), (0, 0), ..., (1, 0)) types vanish, so we have:

uug1 u∗g1= ug1

orueg1 = ug1

where eg1 = ug1 u∗g1. Siimilarly we can introduce eg for all g ∈ G. From previous equations

it follows that eg is an idempotent and B is a following direct sum of algebras:

B =⊕g∈G

egB,

i.e. B is not a connected algebra. It means that Aθ → B is not a finite covering projection.

29

7.2. G′ is a connected topological space.In this case G′ is a commutative torus T2 = U(1)×U(1). Homomorphism f : G′ → T2 isgiven by:

(z1, z2)→ (zn1 , zm

2 )

where (z1, z2) ∈ U(1)×U(1) ≈ G′, (n, m ∈N). Elemets of G in G′ are given by:(e

2πik1n , e

2πik2m

), (k1, k2 ∈ Z).

Element a ∈ B is said to be a homogeneous of degree (r, s) if it satisfies following condi-tion:

(z1, z2)a = zr1, zs

1.

From exacteness of G′ action it follows that there exist a homogeneous of degree (1, 0)nonzero element u′ ∈ B. Element u′n is G invariant and homogeneous of degree (n, 0). Itis clear that u′n = cu (c ∈ C). Similarly we can prove that there is an element v′ such thatv′m = v. In this case we have finite noncommutative covering projection from example4.9.

8 Relative invariants

Let f : A → B noncommutative covering projection such that covering projection groupG is commutative. Let us select homomorphism fG : Aut(A)→ Aut(B). Let Cov(A, B) ⊂Cov(A) be full subcategory such that if (g : A→ A′) ∈ Cov(A, B) then there is followingdiagram

A′ B⊗K

A

wh

[[[

g

f⊗K

is homotopically commutative and [h] is morphism in Cov(A). Let G′ be covering trans-formation group of A → A′. Then homomorphism fG induces injective homomor-phism G′ → G. Moreover if covering projection A1 → A2 is in Cov(A, B) then thereis injective homomorphism G(A1, A) → G(A2, A). Let A be a C∗ - algebra (F!, F!) ispair of functors Cov(A) → Ab such that condition of definition 4.2 is satisfied. LetI = F! F!( f )(A) ∈ F(A) and M = F(A)/I, then we can define F! F! = F! F! mod I. Letus define pair (π!, pi!) functors on Cov(A, B) such that π!(A′) = π!(A′) = G/G′, whereG′ is covering transformation group of A → A′. If A′ → A′′ is morphism in Cov(A, B)then π! : G/G′ → G/G′′, π! : G/G′′ → G/G′.

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9 Groupoid group

10 Generalization of fundamental group functor

Fundamental group functor[28] is a functor from category of topological sets to categoryof groups. This functor is defined by following way

X 7→ π1(X),

f : X → Y 7→ π1( f ) : π1(X)→ π(Y).

Noncommutative generalization of fundamental group π1(X) is not known yet. Howeverwe know generalization of covering group. So one can construct generalization of π1( f )with respect to covering. First of all we define pi1 with respect to covering in commutativecase.Let f : X → Y be continuous map, and X → X, Y → Y such normal coverings thatfollowing diagram

X Y

X Y

w

u uw

This diagram induces following diagram with surjective vertical arrows.

π1(X) π1(Y)

G(X|X) G(Y|Y)

w

u uw

Definition 10.1. Let us consider above coverings. Homomorphism G(X|X) → G(Y|Y) iscalled Fundamental group homomorphism with respect to coverings X → X, Y → Y.

This definition have noncommutative generalization.

Definition 10.2. Let A, B be C∗ algebras, f : A → B * - homomorphism, A → A, B → Bcoverings. Suppose that it is following commutative diagram:

A B

A B

w

f

w

f

u u

and homomorphism f : G(B|B)→ G(A|A) which satisfies following conditions

f (g · a) = f (g) · f (a).

31

A homomorphism f is called Homomorphism of fundamental groups with respect to A → A,B→ B coverings.

Example 10.3. Homomorphism of fundamental groups of noncommutative torus. Let A = Aθ =C[u, v], B = Aθmn = C[u′, v′] be C∗ algebras and f : A → B *- homomorphism defined byfollowing way:

u 7→ u′m,

v 7→ v′n.

Let A→ A, B→ B coverings defined by following way:

A = Aθ/m′n′ = C[u, v], B = Aθ/mm′nn′ = C[u′, v′],

u 7→ um′ , v 7→ vn′ , u′ 7→ u′mm′

, v′ 7→ vnn′ .

It is clear that G(A, A) ≈ Zm′ ×Zn′ , G(B, B) ≈ Zmm′ ×Znn′ . Homomorphism f : A → Bis defined as:

u 7→ u′m′

, v 7→ v′n′

.

These homomorphisms satisfy conditions of definition 10.2. Direct checking shows thatHomomorphism of fundamental groups with respect to above coverings is natural surjec-tive homomorphism:

f : Zmm′ ×Znn′ → Zm′ ×Zn′ .

11 Abelian covering projections

11.1 Abelian fundamental group

Calculation of Galois groups can be very difficult problem. Often difficult problem isreplaced by simplified one. For example class field theory [37] is a powerful tool calculationof Abelian Galois groups of field extension. Described in ?? construction requires analogueof pointed space. However as it is noted in ?? if all Galois groups are Abelian then one donot need analogue of pointed space.

Definition 11.1. A finite covering is called an Abelian covering if its Galois group is Abeliangroup.

Definition 11.2. Let A be C∗ - algebra. Let us consider category of Abelian coverings of Aand covering morphisms. This category induces diagram of Abelian groups and surjectivehomomorphisms. Inverse limit of this diagram is named an Abelian fundamental group ofA.

32

11.2 Canonical constructions of cyclic covering projection

Definition 11.3. The functiuon u = eiφ ∈ U(C(S1)) = U(C is said to be the unitarygenerator.

11.4. Let U(C) = z ∈ C | |z| = 1 and φ ∈ B∞(U(C)) be a Borel-measurable functionsuch that

(φ(z))n = z (∀z ∈ U(C)). (28)

If u ∈ U(H) then according to spectral theorem there exist v = φ(u) such that vn = u.

Definition 11.5. A Borel-measured function φ ∈ B∞(U(C)) is said to be an n - th root if φsatisfies equation (28). Any n - th root can be represented as unitary operator. Denote byB(φ) ∈ U(B(H)) representation of φ as unitary operator.

11.6. Let A be a C∗ - algebra, A → B(H) a faithful representation, u ∈ U(M(A)) is anunitary, φ is an n -th root, v = φ(u) ∈ U(B(H)) and vk /∈ M(A) (k = 1, .., n− 1). The Acan be extended by new multipliers.

Definition 11.7. Let A be a C∗ - abebra and v ∈ U(B(H)) is such that conditions of 11.6are satisfied. A C∗ - algebra B is said to be the v multiplier extension of A if B containsfollowing operators:

1. via, (∀i ∈ Z, ∀a ∈ A);

2. avi, (∀i ∈ Z, ∀a ∈ A);

Denote by Av = B a v - multiplier extension. If vn ∈ M(A) than A → Av is said tobe a n - th root multiplier extension.

Lemma 11.8. Let A be a C∗ - abebra, u ∈ U(M(A)), φ, ψ are n -th roots, then it is isomorphismAφ(u) ⊗K ≈ Aψ(u) ⊗K

Proof. It is clear that Aφ(u)⊗K ≈ (A⊗K)φ(u)⊗ 1. We have following isomorphism:

(A⊗K)φ(u)⊗ 1 ≈ (A⊗K)ψ(u)⊗ 1,

φ(u)⊗ 1↔ ψ(u)⊗ B(φψ−1).

Some of noncommutative covering projections can be obtained by canonicals constructionwhich is described below.

Lemma 11.9. Let f : A → Av be a n -th root multiplier extension. Then f is a Zn - Galoisextension, and Av is a free finitely generated left and right A - module.

33

Proof. It is clear that u = vn ∈ A is unitary element. Since u and v are unintary elementsthere are following isomorphisms:

C(u) ≈ C(S1); C(v) ≈ C(S1).

Map C(u)→ C(v) corresponds to n listed covering projection p : S1 → S1 of the circle andZn is its group of covering transfiormations. There are functions a1, ..., an, b1, ..., bn suchthat:

n

∑i=1

aibi = 1S1 ;

and if g ∈ Zn is a not trivial element of covering transformatiion group then

n

∑i=1

aigbi = 0.

Functions a1, ..., an, b1, ..., bn can be regarded as elements of A[v]. So according to 3.4 A→A[v] is a Hopf - Galois extension. Also A[v] is a free finitely generated A - module,generated by v, ... vn−1. A generator of Zn acts on A[v] by following way:

v 7→ e2πin v.

It is clear that:Av ≈ A⊕ vA⊕ ...⊕ vn−1 A,

Av ≈ A⊕ Av⊕ ...⊕ Avn−1.

Remark 11.10. Above Galois extension is not always noncommutative covering projectionbecause is it is not always strictly outer.

Remark 11.11. According this lemma some covering projections may be regarded as ex-tension of set of continuous function by adding of some Borel-measured functions.

Example 11.12. Let A be a C∗ - algebra, x ∈ K1(A) a generatior of infinite order, u ∈U((A⊗K)+) is representative of x, i.e. x = [u]. If φ ∈ B∞(U(C)) is an n - th root, thenφ(u) /∈ M(A⊗K). So A⊗K → (A⊗K)φ(u) is a Zn - Galois extension.

Example 11.13. Let u ∈ U(C(S1)) be a representative of a K(C(S1)) group generator,φ ∈ B∞(U(C)) is an n - th root. Then C(S1)→ C(S1)φ(u) is a noncommutative coveringprojection which corresponds to n - listed covering projection of S1.

Example 11.14. Let u ∈ U((C(S3)⊗K)+) be a representative of a K(C(S3)) group genera-tor, φ ∈ B∞(U(C)) is an n - th root. Then C(S3)⊗K → (C(S3)⊗K)φ(u) is a Zn - Galoisextension. However it is not a noncommutative covering projection since automorphismsare not strictly outer.

34

Example 11.15. Let Qs(A) be the stable multiplier algebra of C∗ - algebra A. Let x ∈K1(Qs(A)) be a generator such that nx = 0. Let u ∈ Qs(A) be represenative of x. Thenn[u] = 0 and so un = p(w) where p : M(A) ⊗ K → (M(A) ⊗ K)/A ⊗ K = Qs(A)is the canonical projection. Let φ ∈ B∞(U(C)) is an n - th root, and v = φ(w). Thenvi /∈ M(A⊗K) (i = 1, ..., n− 1). So A⊗K → (A⊗K)v is Zn - Galois extension.

Example 11.16. Let u be a canonical generator, f : S1 → S1 be a n listed covering projectionof the cirle, C( f ) : C(S1) → C(S1) is a corresponding *- homomorphism of C∗ algrebras(u 7→ un), CC( f ) the mapping cone [22]. It corresponds to mapping cone C f of f . CC( f ) isan algebra of continous maps f [0, 1)→ U(C) such that:

f (0) = ∑k∈Z

ukn

where u is the canonical unitary generator. A funtion w = (x 7→ u) (∀x ∈ [0, 1]) is not amultplier of CC( f ). However wn ∈ M(CC( f )). Homomorphism CC( f ) → CC( f )w is a Zn -Galois extension. It a is indeed a noncommutative covering projection which corresopndsto n - listed universal topological covering projection of C f . So π1(C f , x0) = Zn. Other-wise w can be regaraded as representative of K1(Qs(CC( f ))). However K1(Qs(CC( f ))) ≈K0(CC( f )).

11.17. If 0 → J → A → A/J → 0 is an exact sequence of C∗ algebras then there isfollowing sequence (see [22]):

K1(J)→ K1(A)→ K1(A/J) ∂−→ K0(J)→ K0(A)→ K0(A/J).

Where ∂ is defined by following way. Let u ∈ GLn(A/J) and let w ∈ GL2n(A) be a liftof diag(u, u−1). Define ∂([u]) = [wpnw−1]− [pn]. The map ∂ is called the index map. Thereason is the following. Suppose A is a unital C∗ - algebra and u is a unitary in Mn(A/J).If u lifts to a partial isometry v ∈Mn(A), then diag(u, u−1) lifts to the unitary

w =

(v 1− vv∗

1− v∗v v∗

).

so ∂([u]) = [wpnw−1] − [pn] = diag(vv∗, 1− v∗v] = [1− v∗v] − [1− vv∗]. In the specalcase we have following exact sequence

0→ A⊗K → M(A⊗K)→ M(A⊗K)/(A⊗K)→ 0.

It is shown in [22] that K0(M(A⊗K)) = K1(M(A⊗K)) = 0. So ∂ : K1(M(A⊗K)/(A⊗K)→ K0(A⊗K = Qs(A)) is a isomorphism.

Example 11.18. Let On be the C∗ - algebra generated by n isometries s1, ..., sn with s∗i si = 1,sis∗i = pi, and p1 + .... + pn = 1. The On was sudied by Cuntz [3]. In K0(On) we have[p1] = ...[pn] = [1], so n[1] = [p1] + ... + [pn] = [1], i.e. n[1] = [1], i.e. (n− 1)[1] = 0. It isshown in [4] that K0(On) = Zn−1 with [1] as generator. Let

v =

s1 0 0...0 1 0...

....

∈ M(On ⊗K) = Ms(On).

35

Element v1 is a isometry and

v∗1v− 11 =

p1 − 1On 0 0...0 0 0...

....

∈ On ⊗K.

it is representative of unitary element v ∈ Qs(On) since

v∗1v1 = 1; v∗1v1 − 1 ∈ On ⊗K.

From index map it follows that [v] ∈ K1(Qs(On)) is a generator. From K1(Qs(On)) = Zn itfollows that (n− 1)[v] = [vn−1] = 0. So vn−1 can be lifted to unitatry element w ∈ Ms(On).Let φ ∈ B∞(U(C)) be n− 1 - th root. Then On ⊗K → (On ⊗K)φ(w) is a Zn−1 - Galoisextension.Let us provide explicit expression for Ms(On) representative of [vn

1 ]

Denote by v2, v3, ...vn ∈ Ms(On) following elements

v2 =

1 0 0 ...0 s2 0 ...0 0 1 ...

....

, v3 =

1 0 0 ...0 1 0 ...0 0 s3 ...

....

, ... vn = ....

1 0 0 ... 0 ...0 1 0 ... 0 ...0 0 1 ... 0 ......0 0 0 ... sn ...

.

It is clear that [v1] = [v2] = ... = [vn] ∈ Qs(On). So following patrial isometry

u =

s1 0 0 ... 0 0 ...0 s2 0 ... 0 0 ...0 0 s3 ... 0 0 ......0 0 0 ... sn−1 0 ...0 0 0 ... 0 1 ...

.

is such that [u] = (n − 1)[v1] = 0. So u should be lifted to Ms(On). Following unitaryelement w is the lift of u

w =

s1 s2 s3 ... sn−1 sn 0 ...0 s2 s3 ... sn−1 sn + s1 0 ...0 0 s3 ... sn−1 sn + s1 + s2 0 ......0 0 0 ... sn−1 sn + s1 + ... + sn−2 0 ...0 0 0 ... 0 1 0 ......

.

11.3 Relations with K groups

11.19. Examples 11.13 and 11.16 show that it is correspondence between K - theory andfundamental group. However fundamental group functor is covariant with respect to

36

topological space, but K - functor is contravariant. However K - homology is covariantwith respect to topological spaces. Following diagram represens universal coefficient the-orem.

0→ Ext1Z(K∗(A), K∗(B)) δ−→ KK∗(A, B)

γ−→ Hom(K∗(A), K∗(B))→ 0.

Following diagram represents particulalr case of the universal coefficient theorem.

0 Ext1Z(K0(A), K0(C)) K1(A) Hom(K1(A), Z) 0

0 Rep(K0(A)tors) K1(A) Hom(K1(A)/(K1(A)tors), Z) 0

w w

u

ftors

w

u

f

u

f

w

w w w w

From this diagram if follows that infinite period part of K1 - homology group depends oninfinite part of K1, but finite part of K1 depends on finite part of K0. In 11.13 (resp. 11.16)infinite (resp. finite) K1 homology group is isomorphic to fundamental group. We wouldlike construct a corresondence between fundamental group and K1 homology. It is wellknown that if X is a locally compact Hausdorff space then:

Hodd(X)/(Hev(X))odd ≈ K1(X)/(K1(X))tors; (29)

We wold like define functor K11 from category of C∗ algebras to category of abelian groups,

such that:

1. K11(A) ⊂ K1(A) for any C∗ - algebra A.

2. If X is locally compact Hausdorff topological space then:

H1(X) ≈ K11(C0(X)). (30)

Definition 11.20. [28] Let f : X → Y be a continous map. Let Z f denote the quotinentspace obtained from topological sum of X × I and Y, by identifying (x, 1) ∈ X × I andf (x) ∈ Y. Z f is called the mapping cylinder of f . Incusion map Y ⊂ Z f is a homotopyequivalence. The quotinent C f = Z f /(X× 0) is called the mapping cone of f .

11.21. Let X = S1, Y = S1 and let X → Y n - listed covering projection z 7→ zn. LetA = X× 0 ⊂ Z f . Let x0 ∈ A be a point. There is following exact sequence:

π1(A, x0) π1(Z f , x0) π1(C f , A) π2(A, x0)

Z Z Zn 0u

≈

w

u≈

w

u≈

w

u≈

w×n

w w

37

So π1(C f , A) ≈ Zn. Otherwise there is following Puppe sequence [24] of reduced Kgroups:

K(S(X)) K(S(Y)) K(C f ) K(X) K(Y)

K1(S1) = Z K1(S1) = Z Zn 0 0

Z Z Zn 0 0

u≈

w

u≈

w

u≈

w

u≈

w

u≈

u≈

w×n

u≈

w

u≈

w

u≈

w

u≈

w×n

w w w

So there is an isomorphism K(C f ) ≈ π1(C f , A). Let us explain it. It is well known thatK0(C(C f )) ≈ K(C f ) + Z and K0(C(C f )) ≈ K1(Qs(C(C f ))).

11.4 Further

Example 11.22. Let u ∈ U(A) such that [u] ∈ K(A), n[u] 6= 0 ∀n 6= 0, u 6= nx ∀n ∈ Z, x ∈K1(A). If u = vn then then vi /∈ A (i = 1, ..., n− 1)

Example 11.23. Let u ∈ U(A) such that [u] ∈ K(A) is periodic with simple period p. Ifu = vp then vi /∈ A (i = 1, ..., p− 1)

Example 11.24. Cyclic construction provided by fundamental group Let X, x0 be pointed spaceand f : (S1, s0) → (X, x0) generates cyclic element [ f ] ∈ π1(X, x0) with infinite order.Also let g : X → S1 be such continuous map that g f is homotopic to 1S1 . Map g generatesunitary element u ∈ U(C(X)) because S1 ≈ U(1). Let A be C∗ - algebra and u ∈ U(A)is such unitary that [u] ∈ K1(A) is nontrivial element that n[u] 6= 0&[u] 6= kx ∀n ∈ N

∀m ∈N&m > 1 ∀x ∈ K(A). Element u comply conditions of definition 11.9.

Example 11.25. Cyclic construction provided by K theory Let A be C∗ - algebra and u ∈ U(A)is such unitary that [u] ∈ K1(A) is nontrivial element that n[u] 6= 0&[u] 6= kx ∀n ∈ N

∀m ∈N&m > 1 ∀x ∈ K(A). Element u comply conditions of definition 11.9.

11.5 K theoretic cyclic covering

Let A be C∗ - algebra x ∈ K1(A) such that x 6= iy, (i = 1, ..., n− 1, y ∈ K1(A)) There isu ∈ U(A⊗Mk(C)) such that [u] = x. If A⊗Mk(C)→ B(H) faithful representation thenthere is v ∈ U(H) such that vn = u. Let B = A⊗Mk(C)[v, v∗]. Let P ∈ Mk(C) be onedimensional projection and B = (1⊗ P)B(1⊗ P). From A ≈ (1⊗ P)(A⊗Mk(C))(1⊗ P)it follows that A may be regarded as subalgebra of B and B = A[v, v∗], where v = (1⊗P)v(1⊗ P). It is clear that vn = u, where u = (1⊗ P)v(1⊗ P) ∈ A. Let B ⊂ B be generatedby A ideal. Then A→ B is noncommutative covering.

38

Example 11.26. Canonical cyclic construction of noncommutative torus. Let Aθ be algebra ofnoncommutative torus (See definition 3.6) and u, v ∈ U(Aθ) unitary generators. ThenK1(Aθ) ≈ Z2 and [u], [v] ∈ K1(Aθ) are generators of K1(Aθ). Elements u and v comply todefinition 11.9.

Lemma 11.27. Let A ⊂ B be canonical cyclic covering of degree n. Then A ⊂ B is finitely listedcovering (See definition ??) and its covering group is isomorphic to Zn.

Proof. Let v ∈ B be generator of cyclic covering and vn = u ∈ B. Let us consider *-automorphism α of B which is defined by following rule:

v 7→ ve2πi/n,

v∗ 7→ v∗e−2πi/n.

It is clear that αn = IdA. So α is generator of cyclic group Zn It is clear that A = BZn . LetC(u) ∈ A be C∗ algebra which is generated by u. According to condition 1 of definition11.9 un is not homotopic 1A ∀n ∈ N. If S1 is circle then we have canonical isomorphismC(u) ≈ C(S1). Also we have isomorphism C(u) ≈ C(S1). Inclusion A ⊂ B generatesinjective *-homomorphism C(u)→ C(v) which is defined as

u 7→ vn. (31)

According to direct calculation *-homomorphism (31) induces n listed covering

S1n → S1, (32)

where S1n is a circle. We use S1

n notation instead S1 for avoiding ambiguity. Here and laterS1

n → S1 is n - listed covering of circle S1 by circle S1n. Element α ∈ Zn acts on S1

n as

φ 7→ φ + 2πi/n.

Since S1n is compact it is such finite set Ui ⊂ S1

n (i = 1, ..., N) of open subsets that⋃i=1,...,N

Ui = S1n,

gUi⋂

Ui = ∅, i = 1, ...N, ∀g ∈ Zn (33)

Ui ⊂ S1 that⋃

i Ui = S1, gUi⋂

Ui = ∅ i = 1,...N, ∀g ∈ Zn. Also there are suchnonnegative real functions ei ∈ C(S1) that

1. ei(x) = 0 ∀x /∈ Ui;

2. ∑i=1,...,N ei = 1C(S1n)

According to (33) we have eigei = 0; i = 1, ...N; ∀g ∈ Z2. Now we if one set ai = bi =√

eithen one have

∑i=1,...,N

aibi = 0;

∑i=1,...,N

aigbi = 0; ∀g ∈ Zn& g 6= e.

To above equations are in fact necessary conditions of definition ??

39

Remark 11.28. Noncommutative coverings of commutative torus (See example ??) can beconstructed as well as in example ??.

Example 11.29. Coverings of noncommutative 3D Sphere Algebra of complex functions of 3Dsphere could be generated by four real valued functions x1,...,x4 those satisfy to followingequations:

x21 + x2

2 + x23 + x2

4 = 1. (34)

If we introduce complex valued functions α = x1 + ix2, β = x3 + ix4 then when we canreplace (34) by the following equation:

αα∗ + ββ∗ = 1. (35)

Very interesting involutive noncommutative algebra is considered in [38]. It is generatedby two elements α, β and satisfies to following relations.

α∗α + β∗β = 1; αα∗ + q2ββ∗ = 1; αβ− qβα = 0; αβ∗ − qβ∗α = 0; β∗β = ββ∗, (36)

where q is a real number and 0 < q ≤ 1.By C(SUq(2)) denote C∗ which satisfy above equation. It is clear that if we suppose thatq = 1 then this algebra is commutative and it satisfies to relations (34). If q ≈ 1 then alge-bra C(SUq(2)) could be considered as noncommutative approximation of algebra C(S3)of continuous complex valued functions on 3D sphere. C(SUq(2)) admits the structureof spectral triple[36]. It is well known that 3D sphere is simply connected. So if q = 1then C(SUq(2)) no nontrivial finite coverings. However if q = 1 then it is such unitaryelement u ∈ U(C(SUq(2)) than [u] ∈ K1(C(SUq(2)) is not trivial and has infinite period.According to example 11.25 element u comply conditions of definition 11.9. So one canconstruct construct cyclic covering C(SUq(2)) → B where B is generated over a by suchelement v that vn = u.

Remark 11.30. If q ≈ 1 then algebraic properties of C(SUq(2)) are very close to algebraicproperties of commutative algebra C(S3). However these algebras are principally differ-ent. First one does not have nontrivial coverings but second one has them. Perhaps thisfact is relevant to structure of the Universe. In some models space of the Universe is C(S3).Since C(SUq(2)) is close C(S3) it is reasonably suppose that Universe space correspond toalgebra C(SUq(2)). Since former algebra has nontrivial coverings this fact can occur newcosmological properties.

11.6 Analogy with Kummer extensions

Here analogue with algebraic field extensions is considered. Let K be a field K (resp.Ksep)is its algebraic (resp. separable) closure and G = G(K/K) is Galois group. Cyclicextension [?] L of K (K ⊂ L ⊂ K) have such Galois group H ⊂ G that G/H is finitecyclic group. If number of elements of G/H is equal to n ∈ N then G/H is isomorphic togroup of n-th roots of unity in C. This isomorphism may be regarded as character χ of G

40

with the kernel H; such a character which if order of n, will be said attached to L. If α isrepresentative in G of generator of G/H, there is one and only one character χ of G thatχ(α) = e2πi/n. Conversely if χ any homomorphism from G into C×; it is a character oforder n; its kernel is open subgroup H ∈ G with cyclic subgroup of order n and subfieldL ⊂ Ksep corresponding to H is cyclic of degree n over K; we will than say that L is attachedto L. If K contains distinct n roots of 1; then these make up a cyclic group E of order n,if K is of characteristic p > 1, or assumption implies that n is prime to p. Let ψ be anisomorphism of E onto group of n-th roots of 1 in C; this will be determined uniquely ifwe choose a generator ε1 of E and prescribe ψ(ε1) = e2πi/n. Take any ξ ∈ K×sep, and let xbe any one of roots of equation Xn = ξ in K; when x ∈ K×sep, and equation Xn = ξ has ndistinct roots εx with ε ∈ E. In particular, for each σ ∈ G xσ must be one of the roots, sothat xσx−1 is in E. Now put

χn,ξ(σ) = ψ(xσx−1); (37)

as E ∈ K, the right-hand side does not change if we replace x by εx with ε ∈ E and istherefore independent of choice of a root x for Xn = ξ. For similar reason, we have, forall, ρ, σ ∈ G;

xρσx−1 = (xρx−1)σ(xσx−1) = (xρx−1)(xσx−1),

and thereforeχn,ξ(ρσ) = χn,ξ(ρ)χn,ξ(σ),

and therefore shows that χn,ξ is a character on G. Take now any η ∈ K×, and call y a rootof Xn = η; then xy is root of Xn = ξη, and we have for all σ ∈ G):

(xy)σ(xy)−1 = (xσx−1)(yσy−1)

end thereforeχn,ζη = χn,ζχn,η ,

which shows that ξ 7→ χn,ξ is a morphism of K× into group of characters of G. It isobvious that χn,ξ is trivial if Xn = ξ has one root, hence all its roots, in K, i.e. ifξ ∈ (K×)n; in other words, (K×)n is kernel of ξ 7→ χn,ξ . It would be easy to showthat the image of K× under that morphism consists of all the characters of G whose or-der divides n, but this will not be needed. Let us generalize this construction. Let Abe C∗ - algebra, U(A) group of its unitary elements, U0(A) ∈ U(A) subgroup homo-topic to unity elements, [U(A)] = U(A)/U0(A) factorgroup. By [[U(A)], [U(A)]] denotecommutator of [U(A)], by [U(A)]ab denote factorgroup [U(A)]/[[U(A)], [U(A)]] . LetTors([U(A)]ab) ∈ [U(A)]ab be subgroup of elements which have finite period. Factorgroup[U(A)]ab f ree = [U(A)]ab/Tors([U(A)]ab) is free Abelian group. Let u1, ..., up ∈ U(A) suchunitary elements that:

1. ui satisfy conditions of definition 11.9 for i = 1, ..., p.

2. Classes ui ∈ [U(A)]ab f ree are linearly independent.

41

According to example ?? and/or example ?? ∀ ∈ N one can Abelian construct finitecovering A → B that there are such unitary elements v1, ..., vk ∈ B that vn

k = uk. By G

denote Abelian Galois group of this covering. This group can be represented as followingdirect sum.

G =⊕

j=1,...,k

Gj; Gj ≈ Zn∀j (1 ≤ j ≤ k);

And summand Gj is generated by automorphism σ ∈ G which acts on B by followingway:

vj 7→ e2πi/nvj;

vl 7→ vl ; l 6= j.

Now we can define character ξn,uj on defined as

ξn,uj(σ) = σ(vj)v−1j ;

This equation can be regarded as analogue of equation 37.

12 Generalization of infinite covering

According to section 1 C∗ it is following mapping:

TOPOLOGY ALGEBRALocally compact space C∗ - algebra

Compact space Unital C∗ - algebraContinuous map *-homomorpfism

This map excludes generalization of infinitely listed coverings by following reasons. LetX be compact Hausdorff space and p : Y → X be infinitely listed covering, then Y is notcompact, C0(X) is unital, but C0(Y) is not unital. Homomorphism C(p) : C0(X)→ C0(Y)which correspond to p does not exist.So one should generalize notion of *- homomor-phism for generalization of infinitely listed coverings .

12.1 Noncommutative generalization of R→ S1 covering

12.1.1 R→ S1 covering

R → S1 covering. Let p : R → S1 well known infinitely listed covering [28]. No functionf ∈ C0(R), f 6= 0 can be obtained from function g ∈ C0(S1). However it is anotheralternative construction which is modification universal object. Let us consider categoryof all coverings of circle.

42

R

... S12n ... S1

21 S1u

hhhhhhhhhhhhhj

w w w w

×2

We assume that all coverings of above coverings are two listed coverings. Symbol S12n ≈ S1

means that that covering degree of initial circle S1 is equal to 2n. It is natural bijectionbetween C(S1) and f :∈ C[−2nπ, 2nπ], f (−2nπ) = f (2nπ)]. Any f ∈ C0(R) can beregarded as limit of functions supported on intervals [−2nπ, 2nπ] , n→ ∞. More preciselyfor f ∈ C0(R) it is following sequence f2n ∈ C0(R):

f2n(x) =

f (x)− f (2nπ) + (x− 2nπ) f (x+2nπ)− f (x−2nπ)

2n+1π; x ∈ [−2nπ, 2nπ]

0; x /∈ [−2nπ, 2nπ](38)

It is evident that sequence f2n is norm convergent to f . Now let us note that

L2(R) =⊕k∈Z

L2([2πk, 2π(k + 1)]), (39)

where ⊕means Hilbert direct sum. Otherwise f2n can be naturally identified with elementf 2n ∈ S1

2n It is clear that there are following natural isomorphisms of Hilbert spaces

L2([2πk, 2π(k + 1)]) ≈ L2([2πn, 2π(n + 1)]) ∀k, n ∈ Z.

Let us also define action f 2n ∈ S12n on L2(R) as action of f2n . It is clear that f 2n trivially

acts on L2([−2πk, 2π(k + 1)]) (n < k∨

n > k + 1).

12.1.2 Generalization of R→ S1 covering

Algebra C(S1) satisfies conditions of definition 11.9. The R→ S1 covering can be general-ized on any C∗ algebra which satisfies conditions of definition 11.9. Let A and u ∈ U(A)be such algebra and its unitary element which satisfy to conditions of definition 11.9. Al-gebra A faithfully acts on Hilbert space H. From lemma 11.27 it follows that there existsfollowing sequence of finitely coverings:

, ... A[v2n , v∗2n ] ... A[v2, v∗2 ] Au u u u

where v2n = u.Above diagram is noncommittal analogue of diagram considered at 12.1.1. Indeed C(S1) ≈C(u) and C(S1

2n) ≈ C(v2n). Let us define action of C0(R) on Hilbert sum H =⊕

n∈Z Hn.First of all note that C(v2k ) acts on

⊕−2k≤n<2k−1 Hn. Suppose that C(v2k ) acts triv-

ially on Hn if n < −2k ∨ n ≥ 2k. This action can be continued whole sum H. Letf ∈ C(R) any function and sequence fn is defined by equation 38. On can define func-tions f 2n ∈ C(S1

2n) = C(v2n). So l f2n ∀n ∈ N defines bounded operator B( f2n) ∈ B(H).

43

Sequence B( f2n) is norm convergent. By B( f ) denote its limit. Denote B(a) ∈ B(H) a ∈ Abounded operator which acts on every component of Hilbert sum as well as a acts on H.

Definition 12.1. Let B ∈ B(H) be norm completion of algebra generated by elementsB(a)B( f ) and B( f )B(a), where a ∈ A and f ∈ C(R). This algebra is called Noncommutativegeneralization of R→ S1 covering.

Remark 12.2. This construction can be generalized. Suppose that there are two elementsu1, u2 ∈ A which satisfy conditions 11.9. We can define two actions of C(R) on H. Letus distinguish these actions for clarity. Action of C(R)1, C(R)2 is constructed by usage ofelements u1 and u2 respectively. Norm completion of algebra generated by B( f1)B( f2)B(a),B( f2)B( f1)B(a), B( f1)B(a)B( f2), B( f2)B(a)B( f1), B(a)B( f1)B( f2), B(a)B( f2)B( f1), wherea ∈ A, f1 ∈ C(R)1 f2 ∈ C(R)2, can be regarded as generalization of covering of torus beplane. Similarly covering of n torus by Rn can be generalized.

Example 12.3. Infinite covering of noncommutative torus. Algebra Aθ of noncommittal torushas two unitary elements u, v which satisfy conditions of definition 11.9. So infinitegeneralization of covering by plane can be constructed. Since v satisfies condition 11.9 onecan construct such sequence v2, v∗2 , ..., v2n , v∗2n that v2n

2n = v, v∗2n2n = v∗; ∀n ∈ N. Elements

of this sequence satisfy following conditions:

uv2n = e2πi(θ+k)/2nv2n u,

where k ∈ Z is arbitrary integer number. Here we set k = 0. In this case uv2n =e2πiθ/2n

v2n u. Sequence v2n induces sequence B( f2n) ∈ B(H) for all f ∈ C0(R). B( f ) ∈B(H) is norm limit of B( f2n) (See 12.1.2). Operators B(u) and B( f2n) satisfy followingcondition.

B(u)B( f2n) = e2πiθ/2nB( f2n)B(u).

Since B( f ) is norm limit of B( f2n) we have.

B(u)B( f ) = B( f )B(u).

From previous equation it follows that algebra generated by elements B(u) and B( f ) ∀ f ∈C0(R) is commutative. So its norm completion is also commutative. One can check thatthis algebra is isomorphic to C0(S1×R). Generalization of infinite covering of C0(S1×R)is C0(R

2). This generalization coincides with commutative covering. So infinite coveringof noncommutative torus is commutative plane.

12.2 Generalization of arbitrary infinite covering

Here we would like construct generalization of arbitrary infinite covering. This construc-tion is analogical to commutative infinite covering. So first of all algebraic construction ofcommutative infinite covering will be constructed.

44

12.2.1 Commutative infinite covering from algebraic viewpoint

Let (X, x0) be pointed topological space space, π(X, x0) → (X, x0) is infinitely listed cov-ering, G = G(X, X) is covering group. According to GNS Construction [23] C∗ - algebraC(X) has a faithful representation, i.e. C(X) is isometrically isomorphic to C∗-algebra ofoperators on a Hilbert space H. Here full representation of C(X) on Hilbert sum

H = ⊕g∈G(X,X)Hg; Hg ≈ H (∀g ∈ G(X, X)) (40)

will be constructed. Let U ∈ X be connected fundamental domain i.e. U is open, limitationπ|U is injective map, and π(U) ∈ X is dense subset. Suppose that x0 ∈ U. GroupG(X, X) acts on X. Group G acts on X and gU is fundamental domain ∀g ∈ G. Denoteby A′′ bicommutant of C∗ - algebra A [23]. Any faithful action of A on Hilbert spaceinduces faithful action of A′′ on same Hilbert space. Since π(U) is dense in X we haveC(π(U))′′ = C(X)′′. Set U = π−1(π(U)) =

⋃g∈G gU is dense open subset of X, C(X)′′ ≈

CU′′ ≈ ⊕g∈G C(gU)′′. Otherwise

⊕g∈G C(gU)′′ acts on Hilbertian sum H =

⊕g∈G Hg,

where Hg ≈ H ∀g ∈ G. So C(X) have faithful representation on H =⊕

g∈G Hg. Actionof a ∈ C(X is defined by following way. Element a is continuous function on X. Itslimitation a|gU , (g ∈ G) is element of C(gU), a ∈ C(gU)′′. So a|gU acts on Hg. Actionof a on H =

⊕g∈G Hg is componentwise action of a|gU on Hg ∀g ∈ G. Let us consider

approximation of this action by actions obtained by finite coverings. Suppose that G canbe included into following diagram of surjective group homomorphism:

G

... Gn ... G1 eu

hhhhhhhhhhhhhj

w w w w

where G is finite ∀n ∈ N. Suppose that⋂

n∈N ker(G → Gn) = e. This diagram inducesfollowing of coverings.

X

... Xn ... X1 Xu

fn

hhhhhhhhhhhhj

π

f1

w w w w

where G(Xn|X) ≈ Gn, maps pn : Xn → X ∀n ∈N are finite coverings.Let g1, ..., gk be all elements of G. Gn is factorgroup of G. Let us select for all gi ∈ Gn suchrepresentative gi that set

⋃i=1,...,m giU is connected. Set p(Un) =

⋃i=1,...,m giU is dense

open subset of Xn. Slight modification of previous speculations shows that C(Xn) havefull representation on direct sum

⊕Hgi

(i = 1, ..., m). Since⊕

Hgi(i = 1, ..., m) ⊂ H C(Xn)

acts on H. Let us select such fundamental domains Ui ∈ X, i ∈ N which correspond to

45

spaces Xi and Ui ⊂ Uj, (i < j). This selection define actions of C(Xi) i ∈ N and all theseactions are compatible with homomorphisms C(Xi) → C(Xj). Let A ∈ B(H) be normcompletion of algebra generated by C(Xi) ∈ B(H). C(X) ∈ A is such subalgebra that ifa ∈ C(X) that for all ε > 0 number of such spaces Hg that ‖a|Hg‖ > ε is finite.

12.2.2 Noncommutative algebraic generalization of infinite covering

Let A be C∗ algebra and

... An ... A1 Awu u u

sequence of finite coverings, and corresponding sequence of covering groups can be in-cluded into following diagram:

G

... Gn ... G1 eu

hhhhhhhhhhhhhj

w w w w

Also suppose that⋂

n∈N ker(G → Gn) = e. Milnor’s construction [24] provides suchinfinite covering space BG that π1(BG) ≈ G. Universal covering of this space is usuallydenoted by EG → BG, G acts on EG and EG/G ≈ BG. This covering induces followingdiagram: This diagram induces following of coverings:

EG

... Xn ... X1 BG

ufn

hhhhhhhhhhhhj

f1

w w w w

where G(Xn, BG) ≈ Gn, ∀n ∈ N. Let U ∈ EG fundamental domain. For all elementsg1, ..., gm ∈ Gn we will select such representatives gi ∈ G that set

⋃i=1,..,m giUn is connected.

In this case Un =⋃

g∈Gn gU is fundamental domain of Xn → BG covering.Let A → B(H) be GNS representation. Let H =

⊕g∈G Hg is Hilbertian sum. Constructed

algebra of infinite coverings subalgebra of B(H). GNS representation is Hilbertian sumof irreducible representations. Irreducible representations of A will be indexed by setΛ i.e. rλ : A → B(Hλ), (λ ∈ Λ). Hilbert space of GNS representation is followingHilbert sum H =

⊕λ∈Λ Hλ. According to [23] . Let r : An → B(H′) any irreducible

representation. According to [27] there exist irreducible limitation r′ : A → B(K), whereK ⊂ H′. A is hereditary subalgebra of An because An is finitely generated projectivemodule. According to [27] K = H′ or it is such unique λ ∈ Lambda that H′ = Hλ For anyrepresentation rλ : A → H unique extension r′λ : An → H will be fixed. Let φλ : An → C

46

be positive functional which defines representation r′λ. By φλg denote following positivefunctional

φλg(a) = φλ(ga); ∀a ∈ An, g ∈ Gn.

Representation defined by φλg has same limitation on A as φλ one. By Hg denote Hilber-tian sum of spaces of representations φλg, ∀λ ∈ Λ. So Hilbert space of GNS represen-tation of An is Hilbertian is a direct sum Hn =

⊕g∈Gn Hg. For all covering group Gn =

gn,1, ..., gn,m we will select such representatives gn,1, ..., gn,m ∈ G that Un =⋃

i=1,..,m gn,iUis connected fundamental domain. Fundamental domains are selected by such way thatif i < j then Ui ⊂ Uj. Selection of these representatives enable us define action of An onHilbertian sum H =

⊕g∈G Hg. Actions of algebras An are compatible with finite cover-

ings Ai → Aj. Let B ∈ B(H) be norm completion of algebra generated by all elementsa ∈ An, n ∈N. Let A ⊂ B be such subalgebra that for all ε > 0 number of elements g ∈ Gwhich satisfy condition ‖a|Hg‖ > ε is finite.

Definition 12.4. In this situation algebra A is named generalization of infinite covering.

13 Generalization of Hurewicz homomorphism

Notion of Hurewicz homomorphism was initially appeared in algebraic topology andthen generalized in several directions. This chapter is devoted to generalization related totheory of C∗ - algebras. First of all let us remind some notions of algebraic topology [28].Let X be topological space, x0 ∈ X is base point, πn(X, x0), Hn(X) are n-th (n ∈ N) aren-th homotopy group and singular homology group respectively. Then ∀n ∈ N there isnatural homomorphism φn : πn(X, x0)→ Hn(X). This homomorphism is named Hurewiczhomomorphism. Pair (X, x0) is named pointed space. if n = 1 then homomorphism isdefined by following way: Let S1 be a circle then H1(S1) ≡ Z. Let c ∈ H1(S1) be generatorof H1(S1). Then Hurewicz homomorphism is defined by following expression: Herewe consider generalization of φ1 only and we shall replace φ1 by φ for simplicity. Forgeneralization of Hurewicz homomorphism we need answer following questions:

1. What is analogue of H1(X)?

2. What is analogue of pointed space (X, x0)?

3. What is analogue of π1(X, x0)?

4. What is analogue of Hurewicz homomorphism?

There is a set of versions of answers which depend on context. Analogue of H1(X) forHurewicz theorem can be different from analogue of H1(X) for other problems. There arethree approaches for noncommutative generalization of classical (commutative) geometri-cal results.

1. Direct (Deductive) From analogues of definitions to analogues of theorems;

47

2. Inverse From analogues of theorems to analogues of definitions;

3. Combined Simultaneous development of analogues of definitions and theorems.

These approaches are schematically represented below:

Generalized (noncommutative) notions

Generalized (noncommutative) theorems

6

?

6

?

Direct approach Inverse approach Combined approach

For example classical notion of Hurewicz homomorphism is based on notion of funda-mental group. However noncommutative generalization of fundamental group can bebased on noncommutative generalization of Hurewicz homomorphism. Combined pointof view implies simultaneous generalization of both fundamental group and Hurewiczhomomorphism.

13.1 Generalization of homology group H1

Equation ?? can be used as definition of Hurewicz homomorphism. This equation canbe generalized for any covariant homotopy invariant functor P which satisfies followingcondition P(S1) ≈ Z. This generalization is defined by following way:

φ([ f ]) = P( f )(c) ∀([ f ] ∈ [S1, s0, X, x0] = π1(X, x0). (41)

This observation provides following requirement for noncommutative generalization ofH1. Generalization of H1 should be such contravariant homotopy invariant functor Pfrom category of C∗- algebras to category of Abelian groups that P(C(S1)) ≈ Z.

Example 13.1. Let K1 be functor of K - homology. Then K1(C(S1)) = Z. In this article theK1 as generalization of H1 is being considered.

13.2 Generalization of pointed space

Let us remind some facts from commutative topology.

1. If X = äi Xi is disjoint union and all Xi are connected then all algebras C(Xi) aresimple and C(X) =

⊕i C(Xi)

2. If C∗ - algebra A =⊕

i Ai then i - th connected component is associated to Ai (Ai issimple algebra.

3. If A is unital then 1A = ∑i 1Ai and 1Ai is selfadjoint idempotent of A.

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4. Any point x0 ∈ X defines homomorphism H0(x0) → H0(X) and generator ofh ∈ H0(X). If X = äi Xi then H0(X) =

⊕i H0(Xi) and H0(X) =

⊕i H0(Xi),

H0(X) ∼ H0(Xi) ∼ Z. Generator h defines path-connected component of x0 satisfiesfollowing conditions:

(a) h has infinite period

(b) h is not divisible

(c) If hi0 generator of H0(Xi0) and hi generator of H0(Xi0) then hi0 _ hi0 = hi0 andhi _ hi0 = 0 (i0 6= i).

Since we consider K1 as analogue of H1 then it is reasonable consider K0 as analogue ofH0.

Definition 13.2. Let A be C∗ algebra and h ∈ K0(A). A pair (A, h) is noncommutativegeneralization of pointed space if following conditions are hold:

1. h has infinite period

2. h is not divisible

3. if A =⊕

i Ai and 1A = ∑i 1Ai then there exits such single index i0 that h · [1Ai0] =

1KK0(C,C) and h · [1Ai ] = 0.

13.3 Hurewicz homomorphism with respect to covering

Hurewicz homomorphism is in general homomorphism from noncommutative group tocommutative one. So it can be decomposed by following way:

π1(X)→ πab(X)→ H1(X),

where πab(X) is Abelian group defined as πab(X) = π1(X)/[π1(X), π1(X)]Algebraic topology has good notion of fundamental group. However good noncommuta-tive generalization of fundamental group is not known. But every covering X → X definescovering group G(X, X) which is factorgroup of fundamental group. If this group has nat-ural structure of subgroup then one can define natural homomorphism G(X, X)→ H1(X).Since H1(X) is Abelian we can take into account Abelian covering projections only (seesection 11) i.e. coverings with Abelian covering transformation group. Abelian group issimultaneously subgroup and factorgroup if is direct summand. So if G(X, X) is directsummand of πab(X) then it is natural homomorphism G(X, X)→ H1(X).

Definition 13.3. Let π : X → X be Abelian covering and G(X, X) is direct summandof πab(X). Natural homomorphism G(X, X) → H1(X) is a Hurewicz homomorphism withrespect to π.

Let us generalize this definition. Fundamental group is not defined for noncommutativeC∗ - algebras. However if G(X, X) is direct summand of πab(X) is also direct summandfor all intermediate subgroup G i.e. G(X, X) ⊂ G ⊂ πab(X).

49

This observation enable us define generalization of Hurewicz homomorphism with re-spect to covering by following way.

Definition 13.4. Let π : A → B be such Abelian covering of C∗ - algebras that for allAbelian coverings B → C group G(B, A) is direct summand of G(C, A).Hurewicz homo-morphism with respect to π A→ B is natural homomorphism from G(B, A) to K1(A).

Let us generalize this definition. Fundamental group is not defined for noncommutativeC∗ - algebras. However if G(X, X) is direct summand of πab(X) is also direct summandfor all intermediate subgroup G i.e. G(X, X) ⊂ G ⊂ πab(X).This observation enable us define generalization of Hurewicz homomorphism with re-spect to covering by following way.

13.4 Noncommutative Hurewicz homomorphism

Noncommutative generalization of Hurewicz homomorphism is not group homomor-phism. It is a set of homomorphism’s conditions. In particular cases this conditionsdefine unique group homomorphism. In general this homomorphism does not exist andis not unique. Let G, H be finitely generated Abelian groups and f : G → H is grouphomomorphism. Let Gtors (resp. Htors be torsion of G (resp. H). Then it is followingcommutative diagram with exact rows

0 Gtors G G/Gtors 0

0 Htors H H/Htors 0

w w

u

ftors

w

u

f

u

f

w

w w w w

Homomorphism f uniquely defines both ftors and f , but not vice versa. So ftors and f canbe regarded as properties of f . If one of following conditions is satisfied

1. Gtors ≈ 0;

2. G/Gtors ≈ 0

3. Htors ≈ 0

4. H/Htors ≈ 0then ftors and f uniquely define f . Otherwise if A is C∗ - algebra then it is followingsequence:

0 Ext1Z(K0(A), K0(C)) KK1(A, C) Hom(K1(A), K0(C)) 0

0 Rep(K0(A)tors) K1(A) Hom(K1(A), Z) 0

w w

u≈

w

u≈

u≈

w

w w w w

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Construction of Hurewicz homomorphism generalization has properties of following di-agram:

0 Gtors G Hom(G/Gtors, Z) 0

0 Rep(K0(A)tors) K1(A) Hom(K1(A), Z) 0

w w

u

ftors

w

u

f

u

f

w

w w w w

where G = G(B|A) is Abelian covering group and Rep(G) means representation group∀G (G is finite Abelian group) .Rather we would like construct properties of ftors and f . First of all note that K∗(A) ≈KK∗G(C, A). There are canonical pairings

Gtors × Rep(Gtors)→ A,

Rep(K0(A)tors)× K0(Ators)→ A,

where A is finite Abelian group.So following pairing

Rep(Gtors)× K0(A)tors → A (42)

can be regarded as analog of isomorphism Gtors ≈ Rep(K0(A)tors). Kasparov intersectionproduct KK∗G(C, C)⊗ KK∗G(C, B)→ KK∗G(C, B). From

KK∗G(C, C) ≈ Rep(Gtors),

KK0G(C, B) ≈ KK0(C, A) ≈ K0(A).

it follows that it is following pairing.

Rep(Gtors)× K0(A)→ K0(A).

Since Rep(Gtors) is finite above pairing does not depend on infinite part of K0(A) i.e.

Rep(Gtors)× K0(A)tors → K0(A)tors.

Above formula is in fact pairing (42).There are following natural pairings.

G/Gtors ×Hom(G/Gtors, Z)→ Z.

K1(A)×Hom(K1(A), Z)→ Z.

So following pairingK1(A)×Hom(K1(A), Z)→ A.

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can be regarded as analogue of isomorphism G/Gtors ≈ K0(A) From Kasparov intersec-tion product it follows next pairing

KK1G(C, C)× KK1

G(C, B)→ KK0G(C, B) (43)

If G is finitely generated Abelian group then

KK1G(C, C) ≈ Hom(G, Z) ≈ Hom(G/Gtors, Z);

KK1G(B, C) ≈ KK1(A, C) ≈ K1(A).

So pairing (43) can be regarded as isomorphism

13.5 Construction Hurewicz homomorphism generalization

Now we have all ingredients for construction of Hurewicz homomorphism generalization.Let (A, h) be noncommutative generalization of pointed space (see definition 13.2), π :A → B be Abelian covering which satisfies conditions of definition 13.4, and coveringgroup G = G(B, A) is finitely generated (Abelian) group.Construction of Hurewicz homomorphism generalization with respect to π includes fol-lowing steps.

1. It is natural isomorphism: K∗G(B)→ K∗(A);

2. It is natural isomorphism G ∼ KK1G(C, C);

3. KK1G(C, C) acts on KK0

G(B, C) ∼ K0G(B) ∼ K0(A), So G acts on K0(A), it is pairing

G× K0(A)→ K1(A);

4. Hurewicz homomorphism generalization with respect to π is defined as

G 3 g 7→ (gh− h) ∈ K1(A). (44)

Definition 13.5. Let (A, h) be noncommutative generalization of pointed space and π :A → B be Abelian covering which satisfies conditions of definition 13.4, and coveringgroup G = G(B, A) is finitely generated (Abelian) group. An Abelian group homomor-phism φ : G → K1(A) is called Hurewicz homomorphism generalization with respect to π if φis defined by equation (44).

Remark 13.6. Functionality of Hurewicz homomorphism. Let f : A→ B be *- homomorphism,f is homomorphism of fundamental groups with respect to A → A, B → B coverings,(B, h), (A, K1( f )(h)) generalizations of pointed spaces. Then following natural diagram

G(B|B) G(A|A)

K1(B) K1(B)

w

f

u uw

K1( f )

is commutative. Vertical arrows of above diagram are generalizations of Hurewicz homo-morphism defined by pairs (B, h), (A, K1( f )(h)) .

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Let us remind universal coefficient theorem of KK theory

Theorem 13.7. [35] Let A and B be separable C∗ algebras with A ∈ N. Then there is a shortexact sequence

0→ Ext1Z(K∗(A), K∗(B)) δ−→ KK∗(A, B)

γ−→ Hom(K∗(A), K∗(B))→ 0. (45)

The map γ has degree 0 and δ has degree 1. The sequence is natural and splits unnaturally. So ifK∗(A) is divisible or K∗(B) is divisible, then γ is isomorphism.

Particular case of theorem45 is following sequence:

0→ Ext1Z(K0(A), Z)

δ−→ K1(A)γ−→ Hom(K1(A), Z)→ 0.

Similarly if G is finitely generated Abelian group then it is following exact sequence:

0→ Ext1Z(KK0

G(C, C), Z)α−→ KK1

G(C, C)β−→ Hom(KK1

G(C, C), Z)→ 0.

Generalization of Hurewicz homomorphism induces following natural homomorphismsbetween above exact sequences.

Ext1Z(KK0

G(C, C), Z)) KK1G(C, C) Hom(KK1

G(C, C), Z)

Ext1Z(K0(A), Z) K1(A) Hom(K1(A), Z)

wα

u

w

β

u uw

αw

β

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