On Coarse Spectral Geometry in Even Dimension

On Coarse Sp ectral Geometr y in Ev en Dimension Rob ert Y unc k en Marc h 18, 2019 Abstract Let σ b e the involution of the Ro e algebra C ∗ | R | which is indu ced from the reflection R → R ; x 7→ − x . A graded F redh olm mo dule o ver a separable C ∗ -algebra A giv es rise to a homomorphism ˜ ρ : A → C ∗ | R | σ to the fix ed-p oint subalgebra. W e use this observ ation to give an even- dimensional analogue of a result of Ro e. Namely , we sh o w t hat the K - theory of this symmetric Roe algebra is K 0 ( C ∗ | R | σ ) ∼ = Z , K 1 ( C ∗ | R | ) = 0, and that the induced map ˜ ρ ∗ : K 0 ( A ) → Z on K -theory gives the index pairing of K -homology with K -th eory . 1 In tro duc tion In [Ro e9 7], Ro e o bserved that a Dirac op erator D on an o dd-dimensio nal closed manifold M gives rise to a C ∗ -algebra homomor phism ˜ ρ : C ( M ) → C ∗ | R | (1.1) from the con tinuous functions on M to the Ro e alg ebra of the re a l line R . The space R app ear s b ecaus e, up to co arse equiv alence, it is the sp ectrum of the self-adjoint op er ator D . The K -theo ry of C ∗ | R | is K n ( C ∗ | R | ) ∼ = ® 0 , n = 0 , Z , n = 1 , and the map ˜ ρ ∗ : K 1 ( C ( M )) → K 1 ( C ∗ | R | ) ∼ = Z (1.2) agrees with the index pairing o f K -theor y w ith the K -homolo gy class [ D ] ∈ K 1 ( M ). This p oint of vie w was extensively dev elop ed by Luu ([Luu05]), who showed that analytic K - homology can be r eformulated ent irely in the langua g e of coar se sp ectral geometry . Sp ecifica lly , let A b e a separable C ∗ -algebra . Luu defined groups K C n ( A, C ) whose c ycles are ∗ - homomorphisms ρ : A → C ∗ | R n | , 1 and 1 The most natural coarse structure on R n here is the topologically controlled coarse structure asso ciated to the compactification of R n b y a sphere at infinity . (See [Ro e03] for the definition.) If A is separable, it turns out to b e equiv alent to use the standard metric coarse structure on R n , although the construction b ecomes somewhat more tec hnical. The K -theory of C ∗ | R n | i s the same in either case. 1 then prov ed that K C n ( A, C ) ∼ = K K n ( A, C ). In fact, Luu work ed w ith a n arbi- trary ( σ -unital) co efficient algebr a B , to pr o duce gro ups K C n ( A, B ) isomorphic to K K n ( A, B ). W e c ho ose not to work in that g enerality her e. Luu’s picture of K -homology is aesthetica lly very pleas ing . The price of this eleg ance, how ever, is some co mputational co mplexity in even dimensions. The isomorphism of K K and K C in even dimension is ac hieved via a map K K 0 ( A, C ) → K C 2 ( A, C ) whic h r equires a s input a b alanc e d F r edholm mo dule, i.e. a graded F redholm mo dule of the form ( H = H 0 ⊕ H 0 , ρ = ρ 0 ⊕ φ 0 , F =  0 U ∗ U 0  for some Hilbert spa c e H 0 , representation ρ 0 and F r edholm op era tor U : H 0 → H 0 . While every K 0 -class can b e represented by a balanced F redholm mo dule, the pr o cess of “balancing” is quite heavy-handed. F or instance, given a Dirac o pe rator on an ev en dimensional ma nifold, the Hilb ert spa ce o f the asso ciated bala nced F r edholm mo dule is an infinite direct sum o f L 2 -sections of the spinor bundle. (See [HR00, Prop osition 8.3 .12].) The relationship b etw een the sp ectrum of U and tha t of the original oper ator D is not obvious. In this pap er, we describ e an alter native appro ach to co ntrolled sp ectr al geometry in even dimension which is more conv enient for geometric applicatio ns. Let s : R → R denote the re flec tion through the o rigin. This induces a ∗ - inv o lution σ of the Ro e algebra C ∗ | R | (see Section 3 ). Given a gra ded F r edholm mo dule ( H , ρ, D ) for A , Ro e’s cons truction in fact pro duce s a ∗ -homomor phism ˜ ρ : A → C ∗ | R | σ int o the fixed-p o int algebra of σ . Our ma in res ult is the following. Theorem 1.1 . The K -the ory of the symmetric R o e algebr a is K n ( C ∗ | R | σ ) ∼ = ® Z , n = 0 , 0 , n = 1 , and t he induc e d map ˜ ρ ∗ : K 0 ( A ) → K 0 ( C ∗ | R | σ ) ∼ = Z (1.3) agr e es with the index p airing of [( H , ρ, D )] ∈ K 0 ( A ) with K -the ory. The author would like to thank Vi ˆ et-T rung Luu for stimulating chats. 2 Preliminaries: Th e Ro e algebra C ∗ | R | W e sha ll use | R | to denote the r eal line equipp e d with the top olo gical coa rse structure induced from the t wo-point compactification R := R ∪ { ± ∞} . Thus, a set E ⊆ R × R is c ontr ol le d if for any sequence ( x n , y n ) ∈ E , x n → ∞ (resp. −∞ ) if and only if y n → ∞ (resp. − ∞ ). W e shall r efer to a Hilber t spa ce H equipp ed with a nondeg enerate repre - sentation m : C 0 ( R ) → B ( H ) as a ge ometric R -Hilb ert sp ac e . By the spectr a l theorem, m ex tends natur ally to the algebra of Bor el functions B ( R ). W e sha ll 2 t ypically suppres s mention o f m in the notatio n. W e use χ Y to denote the characteristic function of a subset Y ⊂ R . An ope r ator T ∈ B ( H ) is lo c al ly c omp act if f T , T f ∈ K ( H ) fo r all f ∈ C 0 ( R ). It is c ont ro l le d (for the ab ove to p o logical coar s e str ucture) if for all R ∈ R there exists S ∈ R s uch that χ ( −∞ ,R ] T χ [ S, ∞ ) = 0 , χ [ S, ∞ ) T χ ( −∞ ,R ] = 0 , χ ( − R, ∞ ] T χ ( −∞ , − S ] = 0 , χ ( −∞ , − S ] T χ [ − R, ∞ ) = 0 . One defines C ∗ ( | R | ; H ) as the nor m-closure of the lo c a lly compact and controlled op erator s on H . This C ∗ -algebra is independent of the choice of H as long a s H is ample , i.e. m ( f ) is noncompact for all nonzero f ∈ C 0 ( R ). In that ca se, the algebra is referred to as the R o e algebr a C ∗ | R | . The following standard facts are easy co nsequences of the definitions. The reader familiar with Ro e algebras may pre fer to recognize them as consequences of the coar sely excis ive decomp osition R = ( −∞ , 0] ∪ [0 , ∞ ), wher e we note that the ideal C ∗ | R | ( |{ 0 }| ; H ) a sso ciated to the inclusion of a po int in to R is just the compact op erator s. (See [HR Y93],[HPR97].) Lemma 2.1. L et T ∈ C ∗ ( | R | ; H ) . F or any R 1 , R 2 ∈ R , (i) χ ( −∞ ,R 1 ] T χ [ R 2 , ∞ ) and χ [ R 2 , ∞ ) T χ ( −∞ ,R 1 ] ar e c omp act op er ators. (ii) [ T , χ ( −∞ ,R 1 ] ] and [ T , χ [ R 2 , ∞ ) ] ar e c omp act op er ators. 3 Graded F redholm mo dules and the symmetric Ro e algebra In wha t follows, we shall use the unbounded (‘Baa j-Julg’) picture of K - homology . This is a purely aesthetic choice—see Rema rk 3.3 for the co nstruction using bo unded F re dho lm modules. Let A be a C ∗ -algebra , and let ( H , ρ, D ) b e a g raded unbounded F redholm mo dule for A , i.e. H is a Z / 2 Z -g raded Hilb ert space, ρ is a represe ntation of A by even op erato rs on H , and D is an o dd self-a djo int unbounded o p erator on H such that (1) for a ll a ∈ A , (1 + D 2 ) − 1 2 ρ ( a ) extends to a compact op erato r, (2) for a dense set o f a ∈ A , [ D, ρ ( a )] is densely defined and extends to a bo unded op erato r . Let γ ev , γ od denote the pro jections onto the even and o dd comp onents of H , and γ = γ ev − γ od be the grading op e r ator. Let σ be the inv olution of B ( H ) defined by σ : T 7→ γ T γ . F unctional calculus o n the op era tor D provides H with a g eometric R struc- ture, namely m : B ( R ) → B ( H ); f 7→ f ( D ) . F o r any f ∈ C 0 ( R ), σ ( m ( f )) = f ( γ .D .γ ) = f ( − D ) = m ( f ◦ s ) , 3 where s : R → R is the reflectio n in the orig in. In coar se language, γ is a cov ering isometry for s . It follows that σ restricts to an inv olution of C ∗ ( | R | ; H ). The subalgebra fixed by σ will b e deno ted C ∗ ( | R | ; H ) σ . T aking this symmetry in to account giv es an immedia te strengthening o f Ro e’s construction for ungra de d F redho lm mo dules. Prop ositi on 3.1. The image of ρ lies in C ∗ ( | R | ; H ) σ . Pr o of. The function f ( x ) = x (1+ x 2 ) − 1 2 generates C ( R ), and the idea l g enerated by g ( x ) = (1 + x 2 ) − 1 2 is C 0 ( R ). Using [HR00, Theorem 6 .5.1], Pr o p erties (1) and (2) ab ove imply that ρ ( a ) ∈ C ∗ ( | R | ; H ) for a ny a ∈ A . Since ρ ( a ) is even, σ ( a ) = γ ρ ( a ) γ = ρ ( a ). This geometric R -Hilber t space H is not t ypically ample. Howev er, one can alwa ys embed H into an a mple g e o metric R -Hilb ert space. F or spe c ific ity , let us put H := H ⊕ L 2 ( R ), where L 2 ( R ) has its natural g eometric R -str uctur e. Extension o f op er ators by zero gives an inclusio n ι : C ∗ ( | R | ; H ) ֒ → C ∗ | R | . Put ˜ ρ = ι ◦ ρ : A → C ∗ | R | . The symmetry g 7→ g ◦ s defines a gr a ding op erator on L 2 ( R ). W e sha ll reuse γ to denote the total grading oper ator o n H . Likewise, we use σ to denote conjugation b y γ in B ( H ). Then ˜ ρ has image in C ∗ | R | σ . R emark 3.2 . In the ab ove, w e hav e employ ed a specific choice of symmetry σ ∈ Aut( C ∗ | R | ) asso ciated to the reflection s of R . F o r the ex pe r t co ncerned ab out the uniqueness of this definition, we supply some brief co mmen ts without pro of. They s hall not be nee ded in what follows. Let H be an y ample geo metric R -Hilb ert s pace. By [Luu05, Pro p. 2 .2.11(iii)] (following [HR Y93]), there exists a unitary γ : H → H which cov ers s , in the sense that (1 × s )(Supp( γ )) ⊆ R × R is a controlled set. By carrying out the pro of of this fact in a wa y that maintains the re flective symmetry , one can ensure that γ is in volutiv e, γ 2 = 1 . Then σ : T → γ T γ is an inv olution of C ∗ | R | . If γ ′ is another inv olutive cov ering isometry for s , then there is a con trolled unitary V ∈ B ( H ) such that γ ′ = V γ ([Luu05, Prop. 2.2 .11(iv)] following [HR Y93]). If σ ′ is conjugation by γ ′ , then C ∗ | R | σ ′ = V C ∗ | R | σ V ∗ . Thus the s ymmetric Ro e algebra C ∗ | R | σ is unique up to controlled unitary equiv alence. R emark 3.3 . The b ounded F redholm mo dule corresp onding to ( H , ρ, D ) is ( H, ρ, F := D (1 + D 2 ) − 1 2 ). The map φ : x 7→ x (1 + x 2 ) − 1 2 defines a co arse equiv alence from | R | to the interv al | ( − 1 , 1) | , with to po logical coa rse str ucture asso ciated to its t wo-po int compacification [ − 1 , 1]. Thus, the bo unded picture of K -ho mology provides a morphism ρ : A → C ∗ | (1 , − 1) | ∼ = C ∗ | R | . 4 K -theory of the symmetric Ro e algebra C ∗ | R | σ Prop ositi on 4.1. The K - t he ory of C ∗ | R | σ is K • ( C ∗ | R | σ ) ∼ = ® Z , • = 0 , 0 , • = 1 . 4 Mor e over, K 0 ( C ∗ | R | ) σ is gener ate d by finite r ank pr oje ctions p ∈ M n ( C ∗ | R | σ ) , and for su ch pr oje ctions, the map to Z is given by [ p ] 7→ dim p H ev − dim p H od . W e use a May er-Vietoris type argument ( cf. [HR Y93 ]). Put Y + := [1 , ∞ ), Y − := ( −∞ , − 1], with their co arse struc tur es inherited from | R | . W e will ab- breviate χ Y ± as χ ± . Since H + := χ + H is an ample geometric Y + -Hilbe r t space, we can define the Ro e algebra C ∗ | Y + | as the co rner algebr a C ∗ ( | Y + | ; H + ) = χ + C ∗ | R | χ + . Likewise for C ∗ | Y − | . Note that σ ( χ ± ) = χ ∓ , so that σ interchanges C ∗ | Y + | and C ∗ | Y − | . Since χ + χ − = 0, the symmetrization map ( I + σ ) : T 7→ T + σ ( T ) is a ∗ -homomo r phism from C ∗ | Y + | into C ∗ | R | σ . W e obtain a morphis m of short- e xact sequences, 0 / / K ( H + ) / / ( I + σ )   C ∗ | Y + | / / ( I + σ )   C ∗ | Y + | / K ( H + ) / / ( I + σ )   0 0 / / K ( H ) σ / / C ∗ | R | σ / / C ∗ | R | σ / K ( H ) σ / / 0 . (4.1) Lemma 4.2 . The right-hand map ( I + σ ) : C ∗ | Y + | / K ( H + ) → C ∗ | R | σ / K ( H ) σ is an isomorphism. Pr o of. Let ψ : C ∗ | R | σ → C ∗ | Y + | denote the cut-down map T 7→ χ + T χ + . By using Lemma 2.1(ii), ψ is a homomorphism mo dulo c ompacts, so it descends to a homomorphism ψ : C ∗ | R | σ / K ( H ) σ → C ∗ | Y + | / K ( H ). By Lemma 2.1(i), for any T ∈ C ∗ | R | σ we hav e T ≡ χ + T χ + + χ − T χ − mo d K ( H ) σ , so that ψ is inv erse to ( I + σ ). Put H ev := γ ev H , H od := γ od H . Lemma 4.3 . We have K ( H ) σ ∼ = K ( H ev ) ⊕ K ( H od ) via T 7→ T γ ev ⊕ T γ od . In p articular, K 0 ( K ( H ) σ ) ∼ = Z ⊕ Z via the map which sends the class of a pr oje ction p to (dim( p H ev ) , dim( p H od )) . Pr o of. Note that a ny T ∈ C ∗ | R | σ commutes w ith γ , so T 7→ T γ ev ⊕ T γ od is indeed a homomorphism. The inv erse homomorphism is T 1 ⊕ T 2 7→ T 1 + T 2 . Lemma 4. 4. Under the identific ations K 0 ( K ( H + )) ∼ = Z a nd K 0 ( K ( H ) σ ) ∼ = Z ⊕ Z , the map ( I + σ ) ∗ is n 7→ ( n, n ) . Pr o of. Let p b e a pro jection in K ( H + ). Then p = χ Y + pχ Y + , so pγ = χ Y + pγ χ Y − , and hence T r( pγ ) = 0. Since γ ev / o d = 1 2 (1 ± γ ), T r( pγ ev ) = T r( pγ od ) = 1 2 T r( p ). Similarly , T r( σ ( p ) γ ev ) = T r( σ ( p ) γ od ) = 1 2 T r( σ ( p )) = 1 2 T r( p ). Hence, T r(( I + σ )( p ) γ ev ) = T r(( I + σ )( p ) γ od ) = T r( p ), a nd the re sult follows from the previous lemma. 5 By [Ro e96, Pr op osition 9.4], C ∗ | Y + | has trivial K -theory . The b oundary maps in K -theory induced from the diagram (4.1) give K 1 ( C ∗ | Y + | / K ( H + )) ∂ ∼ = / / ( I + σ ) ∗ ∼ =   K 0 ( K ( H + )) ∼ = Z ( I + σ ) ∗   n _   K 1 ( C ∗ | R | σ / K ( H ) σ ) ∂ / / K 0 ( K ( H ) σ ) ∼ = Z ⊕ Z ( n, n ) . (4.2) W e see that K 1 ( C ∗ | R | σ / K ( H ) σ ) ∼ = Z , a nd the image of its b oundary map into K 0 ( K ( H ) σ ) is { ( n, n ) | n ∈ Z } . The corr esp onding diagram in the other degree gives K 0 ( C ∗ | R | σ / K ( H ) σ ) ∼ = 0. Now the six-ter m exact sequence a sso ciated to the b o ttom row of (4.1 ) b e- comes ( n, n ) Z ⊕ Z / / K 0 ( C ∗ | R | σ ) / / 0   n _ O O Z O O K 1 ( C ∗ | R | σ ) o o 0 o o Thu s, K 0 ( C ∗ | R | σ ) ∼ = Z and K 1 ( C ∗ | R | σ ) ∼ = 0. With an appropria te choice of sign, top-left ho rizontal map is given by ( m, n ) 7→ m − n . Applying Lemma 4.3, this completes the pro of of 4.1. 5 The index pairing Let θ ∈ K 0 ( A ) b e the K -homolog y class of a gra ded unbo unded F r edholm mo dule ( H , ρ, D ), and put F := D (1 + D 2 ) − 1 2 . Let p b e a pro jection in M n ( A ). The index pairing K 0 ( A ) × K 0 ( A ) → Z is given b y ( θ, [ p ]) := Index  ρ ( p )( F ⊗ I n ) ρ ( p ) : ρ ( p ) H n ev → ρ ( p ) H n od  , (where I n denotes the identit y in M n ( C ).) Let P = ˜ ρ ( p ) ∈ M n ( C ∗ | R | σ ), and let f denote the function f ( x ) = x (1 + x 2 ) − 1 2 , as represe nted on the geo metric | R | -Hilb e r t space H . Then ( θ, [ p ]) = Index( P ( f ⊗ I n ) P : P H n ev → P H n od ) . The r ight-hand side here dep ends only on the cla ss of P in K 0 ( C ∗ | R | σ ). By Prop os itio n 4 .1, we ma y therefore repla ce P by a finite rank pro jection Q , and the index is ( θ, [ p ]) = Index( Q ( f ⊗ I n ) Q : Q H n ev → Q H n od ) = dim( Q H n ev ) − dim( Q H n od ) = [ Q ] = ˜ ρ ∗ [ p ] . This completes the pro o f of Theorem 1.1. 6 R emark 5.1 . Given the a b ov e r esults, it is na tur al to ex pe c t a reformulation of K K 0 ( A, C ) in the s pir it of L uu. Indeed, one can define a g roup K C 0 σ ( A, C ) as follows. Cyc les are morphisms from A in to the symmetric Ro e alg ebra C ∗ | R | σ . Equiv a lence of cycles is genera ted by controlled unitary equiv alences (preserving the in volution γ ) and w eak homotopies (resp ecting the symmetry σ ). Then K C 0 σ ( A, C ) ∼ = K K 0 ( A, C ). W e s hall not develop this in detail here, as the results follow [Luu05] clo sely . References [HPR97] Nigel Higson, Erik Kjær Pedersen, and John Ro e. C ∗ -algebra s and controlled top olog y . K -The ory , 1 1(3):209 – 239, 19 97. [HR00] Nigel Higso n and John Ro e. Anal ytic K -homolo gy . Oxford Mathe- matical Mo nogra phs. Oxford University Press, O xford, 20 00. O xford Science Publications . [HR Y93] Nigel Higson, J ohn Ro e, and Guo Liang Y u. A co arse Ma yer-Vietoris principle. Math. Pr o c. Cambridge Philos. So c. , 114(1):85 – 97, 199 3. [Luu05] Viˆ et-T rung Luu. A lar ge sc ale appr o ach to K -homolo gy . PhD thesis, Penn State Univerisity , 2 005. [Ro e96] John Ro e. In dex the ory, c o arse ge ometry, and top olo gy of manifold s , volume 90 of CBMS R e gional Confer enc e Series in Mathematics . Pub- lished for the Co nference Boar d of the Mathematical Sciences, W ash- ington, DC, 199 6. [Ro e97] John Ro e. An example of dual c o ntrol. R o cky Mountain J. Math. , 27(4):121 5–12 21, 1997. [Ro e03] John Ro e. L e ctu r es on c o arse ge ometry , volume 3 1 of University L e c- tur e Series . American Mathematical So ciety , Providence, RI, 20 03. 7