Universal covering space of the noncommutative torus

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by  Petr R. Ivankov

Universal covering space of the noncommutative
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torus
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January 28, 2014
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PetrR.Ivankov*

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e-mail: *[email protected]
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Abstract
v
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4 Gelfand-Na˘imarktheoremsuppliescontravariantfunctorfromacategoryofcom-
7 mutative C∗− algebras to a category of locally compact Hausdorff spaces. Therefore
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any commutative C∗− algebra is an alternative representation of a topological space.
.
1 Similarlyacategoryof(noncommutative) C∗−algebrascanberegardedasacategory
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ofgeneralized(noncommutative)locallycompactHausdorffspaces. Generalizationsof
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topological invariants may be defined by algebraicmethods. Forexample SerreSwan
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: theorem states that complex topological K - theory coincides with K - theory of C∗ -
v
algebras. However the algebraic topology have a rich set of invariants. Some invari-
i
X ants do not have noncommutative generalizations yet. This article contains a sample
of noncommutative universal covering. General theory of noncommutative universal
r
a coverings is being developed by the author of this article. However this sample has
independentinterest,itisveryeasytounderstandanddoesnotrequireknowledgeof
Hopf-Galoisextensions.
Contents
1 Introduction 2
2 Algebraicconstructionofknownuniversalcoveringspaces 3
2.1 Algebraicconstruction oftheR →S1 covering . . . . . . . . . . . . . . . . . . 3
2.2 Algebraicconstruction oftheR2 → S1×S1 covering . . . . . . . . . . . . . . 4
1
3 Universalcoveringofanoncommutativetorus 5
4 Discussion 6
5 Acknowledgment 6
1 Introduction
FollowingGelfand-Na˘imarktheorem[2]statesthatcategoryoflocallycompactHausdorff
topologicalspacesisequivalenttoacategoryofcommutative C∗−algebras.
Theorem1.1. LetHausbeacategoryoflocallycompactHausdorffspaceswithcontinuousproper
mapsas morphisms. And, let C∗Comm be the category of commutativeC -algebras with proper
*-homomorphisms(sendapproximateunitsintoapproximateunits)asmorphisms. Thereisacon-
travariantfunctorC : Haus→ C∗CommwhichsendseachlocallycompactHausdorffspaceXto
thecommutativeC∗ -algebra C (X) (C(X) if X iscompact). Conversely, thereisa contravariant
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functor Ω : C∗Comm → Haus which sends each commutative C∗ -algebra A to the space of
characterson A(withtheGelfandtopology).
ThefunctorsCandΩ areanequivalenceofcategories.
Soany(noncommutative)C∗−algebramayberegardedasgeneralized(noncommutative)
locallycompactHausdorfftopologicalspace. Wemaysummarizeseveralpropertiesofthe
GelfandNa˘imarkcofunctorwiththe followingdictionary.
TOPOLOGY ALGEBRA
Locallycompactspace C∗ -algebra
Compactspace UnitalC∗ -algebra
Continuous map *-homomorpfism
Minimalcompactification Unitization
Maximalcompactification Algebraifmultpicators
Closedsubset Ideal
Disjoint unionoftopological spaces(∐X) Directsumof(pro)- C∗ algebras(⊕A)
ι ι
Principalfibration Hopf-Galoisextension.
Universalcovering ?
Thisarticleassumeselementaryknowledge offollowingsubjects.
1. Algebraictopology [3].
2. C∗−algebrasandoperatortheory[1],[2],
Weusefollowing notation.
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Symbol Meaning
N monoid ofnaturalnumbers
Z ringofintegers
R (resp. C) Fieldofreal(resp. complex)numbers
Q Fieldofrationalnumbers
H Hilbertspace
B(H) AlgebraofboundedoperatorsonHilbertspace H
K(H) or K AlgebraofcompactoperatorsonHilbertspace H
U(H)⊂B(H) Group ofunitaryoperatorsonHilbertspace H
U(A)∈ A Group ofunitaryoperatorsofalgebra A
A+ C∗−algebra A withadjointedidentity
M(A) Amultiplier algebraof C∗-algebra A
C(X) C∗ -algebraofcontinuous complexvalued
functions ontopologicalspace X
C (X) C∗ -algebraofcontinuous complexvalued
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functionsontopologicalspacewhichtendsto0atinfinity
C (X) Algebraofcontinuous functionswithcompactsupport
c
sp(a) Spectrumof elementof C∗-algebraa ∈ A
If X˜ → X is an universal covering then there is a natural ∗-homomorphism f : C (X) →
0
M(C (X˜)). Homomorphism f can be defined by a faithful representations π : C (X) →
0 0
B(H),π∗ :C (X˜) → B(H)suchthat
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π∗(f(x)x˜) =π(x)π∗(x˜), π∗(x˜f(x))= π∗(x˜)π(x), x ∈ C (X), x˜ ∈ C (X˜). (1)
0 0
Wewouldlike constructananalogueof 1foranoncommutative torus.
2 Algebraic construction of known universal covering spaces
2.1 Algebraic construction of the R → S1 covering
Ourconstruction containstwoingredients:
1. Algebraicanalogueof n -listedcoveringprojection f : S1 →S1 ;
n
2. AlgebraicanalogueofR →S1;
2.1. Constructionofn-listedcoveringprojection. Itiswellknown thatC(S1)isa C∗-algebra
which is generated by a single unitary element u ∈ U(C(S1)). The C(S1) algebra can be
faithfully represented, i.e. there is an inclusion C(S1) → B(H). Let sp(u) ∈ C be the
spectrum of the u (it is known that sp(u) = {z ∈ C | |z| = 1}), φ ∈ B∞(sp(u)) is a
Borel-measurablefunctionsuchthat
(φ(z))n =z (∀z ∈sp(u)). (2)
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According to spectral theorem [2] there exist v = φ(u) ∈ U(B(H)) and vn = u. Let
C(u) → B(H) (resp. C(v) → B(H)) be a C∗- algebra generated by u (resp. v), then we
have an inclusion C(u) ⊂ C(v) which corresponds to an n - listed covering projection
f : S1 →S1.
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2.2. ConstructionofR → S1. A circle S1 can be parameterizedby an angle parameter θ ∈
[−π,π]. Any function g ∈ C([−1,1]) such that g(−1) = g(1) corresponds to ϕ ∈ C(S1)
g
suchthatφ (θ)= g(θ/π). Letusfixasequenceofoperatorsu =u,u ,u ,...∈ B(H)such
g 0 1 2
that u2 = u , wehavea sequence of inclusions C(u) = C(u ) ⊂ C(u ) ⊂ C(u ) ⊂ ...→
n+1 n 0 1 2
B(H). Wewouldliketoprovethatthissequenceandanyrepresentationπ : C(u)→ B(H)
naturallydefinesa representation π∗ : C (R) → B(H). Let C (R) ⊂ C (R) be analgebra
0 c 0
offunctionswithcompactsupport. Thereisanaturalinclusioni : C (R)→ B(H)defined
c
byfollowing way. If f ∈ C (R) thenthereis n ∈ N suchthatsupportof f iscontainedin
c
[−2n,2n]. If f∗ ∈ C([−1,1])issuchthat f(x) = f∗(2nx)then f∗(−1)= f∗(1)=0,andwe
can define φf∗ ∈ C(S1). If πn : C(S1) ≈ C(un) → B(H) then we set i(f) = πn(φf∗). It is
clearthatthisdefinitiondoesnotdependonn. SinceC (R)isdenseinC (R),aninclusion
c 0
i : C (R) → B(H) canbe continuously extendedto a representation π∗ : C (R) → B(H).
c 0
Representationsπ andπ∗ satisfy(1).
2.2 Algebraic construction of the R2 → S1 ×S1 covering
2.3. ConstructionofC∗-algebra. TheC(S1×S1)isgeneratedbytwounitaryelementsu,v∈
U(C(S1×S1)). ThereisafaithfulrepresentationC(S1×S1) → B(H). Thisrepresentation
inducestwofaithfulrepresentationsπ :C(u) → B(H), π : C(v)→ B(H). Construction
u v
fromsection2.1suppliesfollowingtworepresentations:
1. π∗ :C (R) → B(H),
u 0
2. π∗ : C (R) → B(H).
v 0
ItisnaturallytosupposethatC (R2)isisomorphictothenormcompletionofsubalgebra
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of B(H)generatedbyoperatorsoffollowing type:
π∗(f )π∗(f ), π∗(f )π∗(f ); (f , f ∈ C (R)). (3)
u 1 v 2 v 1 u 2 1 2 0
Howeveritisnot alwaystrue. Thisconstruction isnot unique because therearedifferent
Borel-measurable functions which satisfy (2). This algebra is not always commutative,
because one can select element u ∈ B(H) such that u = u2 and u v = −vu . However
1 1 1 1
anyalgebraconstructedby(3)isrepresentativeofanunique Moritaequivalenceclass.
2.4. Morita equivalence. Although constructed above algebra is not always isomorphic to
C (R2)itisstronglyMoritaequivalenttoit[1]. Asitisprovenin[4]aσ-unitalC∗-algebra
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A is strongly Morita equivalent to a σ-unital C∗-algebra B if there is a ∗-isomorphism
A⊗K ≈ B⊗K. I find that good noncommutative theory of universal coverings should
beinvariantwithrespecttoMoritaequivalence. ThistheorycanreplaceC∗-algebraswith
theirstabilizations(recallthatthestabilizationofa C∗ algebra A isa C∗-algebra A⊗K).
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Definition2.5. Let A bea C∗-algebra, A → B(H)isafaithfulrepresentation,u∈U(A+),
v ∈ U(B(H)), is such that vn = u and vi ∈/ U(A+), (i = 1,...,n−1). A generated by v
algebraisaminimal subalgebraof B(H)whichcontainsfollowingoperators:
1. via; (a ∈ A, i =0,...,n−1)
2. avi.
Denoteby A{v} ageneratedby v algebra.
Lemma 2.6. Let A be a C∗-algebra, A → B(H) is a faithful representation, u ∈ U(A+) is an
unitary element such that sp(u) = {z ∈ C | |z| = 1}, ξ,η ∈ B∞(sp(u)) are Borel measured
functionssuchthatξ(z)n =η(z)n =z(∀z ∈sp(u)). Thenthereisanisomorphism
A{ξ(u)}⊗K → A{η(u)}⊗K (4)
whichisalsoaleft A-moduleisomorphism. Theisomorphismisgivenby
ξ(u)⊗x 7→ η(u)⊗ξη−1(u)x; (x ∈ K). (5)
Proof. Follows fromthe equalityξ(u) =ξη−1(η(u)).
Let u ∈ U(C(S1×S1)) be an unitary such that u 6= vn for any n > 1,v ∈ U(C(S1×S1)).
One can construct different sequences x = u,x ,x ,... ∈ B(H), y = u,y ,y ,... ∈ B(H),
0 1 2 0 1 2
suchthat x = x2, y = y2 butC∗ -algebras
n+1 n n+1 n
A{x } ⊂ A{x } ⊂...
1 2
arenotisomorphic to C∗ -algebras
A{y }⊂ A{y } ⊂....
1 2
Howeverfollowingsequences
A{x }⊗K ⊂ A{x }⊗K ⊂...
1 2
A{y }⊗K ⊂ A{y }⊗K ⊂...
1 2
contain isomorphic algebras. If A is the norm completion of an algebra generated by (3)
then A⊗K ≈ C (R2)⊗K, i.e. A isstronglyMoritaequivalenttoC (R2).
0 0
3 Universal covering of a noncommutative torus
A noncommmutative torus [5] A is a C∗-algebra generated by two unitary elements
θ
(u,v∈U(A ))suchthat
θ
uv =e2πiθvu, (θ ∈R).
If θ ∈ Q then noncommutative torus is strongly Morita equivalent to commutative one,
i.e. A ⊗K ≈ C(S1×S1)⊗K, and our construction is the same as in the section 2.2. A
θ
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case θ ∈/ Q ismore interesting. Howeveraconstruction universalcoveringfullycoincides
with the considered in section 2.2 one. Let A → B(H) be a faithful representation. This
θ
representationinduces two representations π : C(u) → B(H), π : C(v) → B(H). These
u v
representations induce representations π∗(C (R)) → B(H), π∗(C (R)) → B(H). The
u 0 v 0
universalalgebraofnoncommutative torusisanorm completionof analgebragenerated
byoperatorsoffollowing type
π∗(f )π∗(f ), π∗(f )π∗(f ); (f , f ∈ C (R)).
u 1 v 2 v 1 u 2 1 2 0
ThisalgebraisnotuniquebutitisarepresentativeoftheuniquestrongMoritaequivalence
class.
4 Discussion
It is known the Maxwell’sequations of classical electrodynamic aremore important than
Maxwell’sproof. NowIamoccupiedbygeneraltheoryofnoncommutativeuniversalcov-
erings, which uses theory of Hopf C∗-algebras and Hopf-Galois extensions [6]. However
I obtained a new algebra which can be regarded as a locally compact spectral triple [5].
Maybe this result is more interesting than a general theory. This algebra is also interest-
ing because it contains almost commutative sector, i.e. noncommutativity parameter θ is
infinitesimal. Itmeansthatthereisasequenceof algebras
A → A →...
θ θ/2
such that noncommutativity parameter tends to 0. Maybe this algebra has a physical
sense. I found an analogy of this algebra with [7] Kaluza-Klein theory. In Kaluza-Klein
theorywedonotobservecompactdimensions ofthe Universebecausetheyarecompact.
Weobserveflatquotientspace,whichcorrespondstoasubalgebraoftheUniverse. Maybe
weobservealmostcommutativesubalgebraoftheUniverse,becausewecannotobservea
noncommutative algebra.
5 Acknowledgment
Iwouldliketoacknowledgea"Non-commutativegeometryandtopology"seminar,orga-
nizedby:
1. Prof. AlexanderMishchenko,
2. Prof. IvanBabenko,
3. Prof. EvgenijTroitsky,
4. Prof. VladimirManuilov,
5. Dr. AnvarIrmatov
fordiscussionofmywork.
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References
[1] B. Blackadar. K-theory for Operator Algebras, Second edition. Cambridge University
Press1998.
[2] G.J.Murhpy.C∗-AlgebrasandOperatorTheory.AcademicPress1990.
[3] E.H.Spanier.AlgebraicTopology.McGraw-Hill.NewYork 1966.
[4] LawrenceG. Brown, Philip Green, and Marc A. Rieffel. Stable isomorphismand strong
Morita equivalence of C -algebras. Source: Pacific J. Math. Volume 71, Number 2 , 349-
363,1977
[5] J.C.Várilly.AnIntroductiontoNoncommutativeGeometry.EMS2006.
[6] Lecturenotesonnoncommutativegeometryandquantumgroups,EditedbyPiotrM.Hajac
[7] J.M.Overduin,P.S.Wesson, Kaluza-KleinGravity,arXiv:gr-qc/9805018,1998.
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