Universal covering space of the noncommutative 4 torus 1 0 2 n a January 28, 2014 J 7 2 PetrR.Ivankov* ] e-mail: *[email protected] A O . h t a m [ 1 Abstract v 8 4 Gelfand-Na˘imarktheoremsuppliescontravariantfunctorfromacategoryofcom- 7 mutative C∗− algebras to a category of locally compact Hausdorff spaces. Therefore 6 any commutative C∗− algebra is an alternative representation of a topological space. . 1 Similarlyacategoryof(noncommutative) C∗−algebrascanberegardedasacategory 0 ofgeneralized(noncommutative)locallycompactHausdorffspaces. Generalizationsof 4 topological invariants may be defined by algebraicmethods. Forexample SerreSwan 1 : 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 0 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. 2 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 0 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 0 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) 3 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. n 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 0 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 0 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). 4 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 5 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. 6 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. 7