Finitely summable Fredholm modules over higher rank groups and lattices

Finitely summable F redholm mo dules o v er higher rank group s and lattices Mic hael Pusc hnig g 1 In tro du ction In this note w e use recen t progress in rigidit y theory to adress the question of existence of finitely summable F redholm mo dules o v er group C ∗ -algebras. W e begin by giving some motiv a t io n and rec alling previous w ork on this problem. According to Kasparov [Ka1], the K -homology groups of a C ∗ -algebra A may b e defined as the group of homotop y classes of b ounded F redholm mo dules A . In [Co1] Connes constructed a Chern-c hara cter for F redholm mo dules whic h satisfy a strong regularit y condition, called finite summabilit y (with resp ect to a dense subalgebra of A ). This Chern-c haracter is giv en b y a completely e xplicit cyclic co cycle. In particular, the index pairing o f the g iv en mo dule with K -theory can b e calculated b y a simple index formula. It a ppears therefore natural to ask whether ev ery K - homology class can b e repre- sen ted b y a finitely summable mo dule. In the case of the C ∗ -algebra of con tin uous functions on a smo oth compact manifold the answ er is a ffirmat iv e. Ev ery K -homolog y class may b e represen ted by a mo dule whic h is finitely summable ov er the algebra of smo oth functions and Connes’ c har- acter fo rm ula b oils down to the classical Atiy ah-Singer index formula in de Rham cohomology . The situation c hanges if one passes to noncomm utative C ∗ -algebras. Recall that K -homology of C ∗ -algebras can b e describ ed either in terms of b ounded or in terms of un b ounded F redholm mo dules. In [Co3 ] C onnes ha s show n that there are no finitely summable un b ounded F redholm mo dules ov er the reduced C ∗ -algebra of an y nonamenable discrete group at all! F or bo unded modules the story is differen t. In [Co1] Connes constructs bo unded F redholm mo dules o v er the reduced group C ∗ - algebras of free groups and lattices in S U (1 , 1) whic h a r e finitely summable ov er the group ring while represen ting nontriv ial classes in K -homology . (Note that these groups are not amenable.) The main observ ation in this note is that suc h examples do not exist if one passes 1 to lattices in higher rank Lie groups. In fact w e hav e Theorem 1.1. i) L et G b e a pr o duct of simple r e al Lie gr oups of r e al r ank at le ast 2. Then ev ery finitely summable F r e d holm r epr esentation (se e se ction 1.2) of G is homotopic to a multiple of thr trivial r epr e s e ntation. ii) L et Γ b e a lattic e i n a pr o duct of simple r e al Lie gr oups o f r e al r ank at le ast two. Then every F r e dholm m o dule over C ∗ max (Γ) , which is finitely summable over l CΓ , i s homotopic to a finite dimens i o nal virtual r epr esentation of Γ . As a consequence w e obtain in section 2 a complete description of the classes in K K G ( l C , l C ), K K ∗ ( C ∗ max (Γ) , l C ) and K K ∗ ( C ∗ r ed (Γ) , l C) whic h can be realized by finitely summable F redholm represen tations or F redholm mo dules. In particular, no non trivial class in K K ∗ ( C ∗ r ed (Γ) , l C ), Γ a higher rank lattice, c an b e represen ted b y a finitely summable mo dule. This sho ws that for noncomm utative C ∗ -algebras it is in general not p ossible to realise ev ery K -ho mo lo gy class b y a finitely summable F redholm mo dule. The previous results are a simple consequence of recen t deep w ork b y Bader, F urman, Gelander and Monod in rigidit y theory . They conjecture that e ve ry equicon tin uous isometric action of a higher ra nk group or lattice on a unifo rmly conv ex Banac h space admits a global fixed p oin t. They v erify this conjecture for isometric actions on classical L p -spaces. One of their main results reads as follo ws: Theorem 1.2. (Bader, F urman, Gelander, Mono d) [BFGM] L et G b e a higher r ank gr oup or a higher r ank latt ic e as in The or em 1.1. L et ( X , µ ) b e a stand a r d me asur e sp ac e and let ρ b e an isometric li n e ar r epr ese n tation of G on L p ( X , µ ) . Then H 1 cont ( G, ρ ) = 0 for 1 < p < ∞ . Adapted to noncommutativ e ℓ p -spaces their result imm ediately implies our main theorem. The link is giv en b y the w ell known correspondence b etw ee n gro up coho - mology and the Ho chs c hild cohomolog y of group algebras, as w as already observ ed and exploited b y Connes in [Co3]. After dev eloping the notions of finitely summable K -homology and finitely summable F redholm represen tations in section 2 w e are ready to state our results in section 3. Section 4 recalls the parts of the w ork of Bader, F urman, Gelander and Mono d needed fo r our purp ose and in section 5 w e finally giv e the pro of s of the theorems stated in section 3. I w an t to expres s m y gratitude to Thie rry F ack, F uad Kittaneh, Hideki Kosaki and An ton y W assermann for helpful discussions and correspondence concerning eigen- v alue estimates for compact op erators and in particular Hideki Kosaki for bringing Ando’s w ork [A] to m y atten tion. 2 2 The s etup 2.1 Smo oth K -h omology The K -homolo gy groups of a C ∗ -algebra A a re defined, according to Kasparo v [Ka 1], as the groups of homotop y classes of F redholm mo dules o v er A . Recall that an eve n F redholm mo dule E = ( H ± , ρ ± , F ) ov er A is g iven b y a Z Z / 2 Z Z -g raded complex Hilb ert space H = H ± , a pair ρ = ρ ± : A → L ( H ) ± of represen tations of A , and an o dd b ounded linear op erator F : H ± → H ∓ satisfying ρ ( A )( F 2 − 1 ) ⊂ K ( H ) , ρ ( A )( F ∗ − F ) ⊂ K ( H ) , [ F , ρ ( A )] ⊂ K ( H ) . (2 . 1) Here K ( H ) denotes the ideal of compact op erators in L ( H ). The notion o f an o dd F redholm represen tation is obtained by forgetting the Z Z / 2 Z Z -grading. F redholm mo dules should b e view ed as g eneralized elliptic operat o rs ov er A . In fact, if M is a smoo th compact manifold without b oundary , then ev ery linear elliptic differen tial op erator ov er M giv es rise to a F redholm mo dule ov er A = C ( M ), the algebra of con tinuous complex-v alued functions on M . It can b e sho wn that suc h geometric mo dules generate the whole K -homolog y group K K ∗ ( C ( M ) , l C) o f M . The previous example s uggests a notion of smo othness (finitely summabilit y) for F redholm mo dules. Definition 2.1. Let A b e a C ∗ -algebra and let E = ( H ± , ρ ± , F ) b e a F redholm mo dule o v er A . Fix 1 ≤ p < ∞ . Then E is said to b e p-summa ble ov er the dense (in v olutive ) subalgebra A ⊂ A if instead of (2.1) the stronger conditions ρ ( A )( F 2 − 1 ) ⊂ ℓ p ( H ) , ρ ( A )( F 2 − 1 ) ⊂ ℓ p ( H ) , [ F , ρ ( A )] ⊂ ℓ p ( H ) . (2 . 2) hold where ℓ p ( H ) = { T ∈ K ( H ) , T r ace ( T ∗ T ) p 2 < ∞ } (2 . 3) denotes the Sc hatten ideal of p -summable compact o perato rs. According to W eyl, the F redholm mo dule asso ciated to an elliptic op erator D o v er the smo oth compact manifold M is p -summable o ve r C ∞ ( M ) ⊂ C ( M ) for p > dim ( M ) or d ( D ) . Kasparo v’s K -homology groups K K ( A, l C) are defined as the groups of equiv alence classes of ev en (o dd) F redholm mo dules o v er A with resp ect to the equiv alence rela- tion generated by unita ry equiv alence , additio n of degenerate mo dules, and op erator homotop y . F ollo wing Nistor [Ni], we intro duce similar relations for smo oth mo dules. 3 Definition 2.2. Let A b e a C ∗ -algebra and let A ⊂ A b e a dense (in volutiv e) subalgebra. Denote b y E ( p ) ∗ (( A, A ) , l C) the set of ev en(o dd) F redholm mo dules ov er A whic h are p -summable ov er A . Recall that a F redholm mo dule is called degenerate if all the terms in (2.1) are iden tically zero. Eve ry degenerate mo dule o v er A is th us p -summable o v er any dense subalgebra so that the set E ( p ) ∗ (( A, A ) , l C) is stable under addition of degenerate mo dules. There is a lso an obvious notion of unita r y equiv alence a mo ng smo o th F redholm mo dules. One has to b e a bit careful ab out the appropriate notion of op erator homotop y in the smo oth con text. Definition 2.3. A smoo th op erator homotopy b et w een E 0 = ( H , ρ, F 0 ) and E 1 = ( H , ρ, F 1 ), E 0 , E 1 ∈ E ( p ) ∗ (( A, A ) , l C), is given b y a fa mily F t , t ∈ [0 , 1] o f b ounded op erators on H connecting F 0 and F 1 suc h that • E t = ( H , ρ, F t ) ∈ E ( p ) ∗ (( A, A ) , l C) , 0 ≤ t ≤ 1, • { t 7→ F t } ∈ C ∞ ([0 , 1] , L ( H )), • the maps a 7→ ρ ( a )( F 2 t − 1 ) , a 7→ ρ ( a )( F ∗ t − F t ) , and a 7→ [ F t , ρ ( a )] lie in L ( A , C ∞ ([0 , 1] , ℓ p ( H )). In the last conditio n A is assumed to b e equipp ed with a b ornolo gy turning the inclusion A ⊂ A in to a b ounded op erator. Th us b ounded linear ma ps from A to b ornological (or F r ´ ec het) spaces make sense. In o ur examples we will only deal with the fine b ornology generated b y finite subsets, so that the b oundedness is automatic. Definition 2.4. Let 1 ≤ p < ∞ and let A b e a dense (in volutiv e) subalgebra of the C ∗ -algebra A . The smo oth ( p -summable) K - homology groups KK ( p ) ∗ (( A, A ) , l C) = E ( p ) ∗ (( A, A ) , l C) / ∼ of the pair ( A, A ) a re defined as the g roups o f equiv alence classes of p -summable F redholm mo dules o v er ( A, A ) with resp ect to t he equiv a lence relation generated b y unitary equiv a lence , addition of degenerate mo dules, and smo oth op erator homo- top y . There is an ob vious homomorphism KK ( p ) ∗ (( A, A ) , l C) − → K K ∗ ( A, l C) (2 . 4) One is mainly in terested in the image of this map. If a K -ho mology class can be represen ted b y a smo oth K - cycle , then its Che rn- Connes c haracter can b e giv en by a simple explicit c haracter for mula. This make s index calculations for smo oth K -cycles mu ch easier than in the general case. 4 Prop osition 2.5. L et A b e a dense (involutive) sub algebr a of the sep ar able C ∗ - algebr a A . L et ˇ ch ( p ) : KK ( p ) ∗ (( A, A ) , l C) − → H P ∗ ( A ) (2 . 5) b e Connes’ Ch e rn char acter on sm o oth K -homolo gy with values in p erio dic cyclic c ohomolo gy [Co1] and let ˇ ch : K K ∗ ( A, l C) − → H C ∗ loc ( A ) (2 . 6) b e the Chern-C onnes char acter on K -homolo gy with va l ues in lo c al cyclic c ohomolo gy [Pu]. Then ther e is a natur al c ommutative diagr am KK ( p ) ∗ (( A, A ) , l C) − → K K ∗ ( A, l C) ˇ ch ( p ) ↓ ↓ ˇ ch H P ∗ ( A ) − → H C ∗ loc ( A ) ← − H C ∗ loc ( A ) (2 . 7) 2.2 Represen tation rings of lo cally compact groups Let G b e a lo cally compac t (second coun table) group. W e denote by R ( G ) the ring of unitary equiv alence classes of finite dimensional unitary represen tations of G . Sum a nd pro duct in R ( G ) are give n b y the direct sum and the tensor pro duct of (equiv alence classes) of represen tations. A r emark able generalization of the notion of finite dimensional unita r y represen ta- tion has b een in tro duced b y G. Kasparov [Ka 2]: Definition 2.6. A F redholm representation E G = ( H , ρ, F ) of G is giv en b y • a Z Z / 2 Z Z -graded Hilb ert space H ± • a pair of unitary r epresen tatio ns ρ ± : G − → U ( H ± ) of G o n the ev en(o dd) part of H . • an o dd b ounded op erator F : H ± − → H ∓ , whic h almost in tertwine s the represen tations ρ + and ρ − , i.e. { g 7→ ρ ( g ) F ρ ( g ) − 1 } ∈ C ( G, L ( H )) , F 2 − 1 ∈ K ( H ) , ( F − F ∗ ) ∈ K ( H ) , ρ ( g ) F ρ ( g ) − 1 − F ∈ K ( H ) , ∀ g ∈ G. W e prop ose the follo wing notion of a smo oth F redholm repre sen tat io n Definition 2.7. A F redholm represen tation E G = ( H , ρ, F ) of a lo cally compact group G is p-summa ble if F 2 − 1 ∈ ℓ p ( H ) , F ∗ − F ∈ ℓ p ( H ) , { g 7→ ρ ( g ) F ρ ( g ) − 1 − F } ∈ C ( G, ℓ p ( H )) . 5 Denote by F r ed ( G ) and F r ed ( p ) ( G ), resp ectiv ely , the set of F redholm represen ta- tions and p -summable F redholm represen tations of G , resp ectiv ely . If G = Γ is a discrete group, there is a tautological bijection F r ed ( p ) (Γ) ≃ − → E ( p ) 0 (( C ∗ max (Γ) , l CΓ) , l C) (2 . 8) b et w een the set of p -summable F redholm represen tations o f Γ a nd the set of ev en F redholm mo dules o v er the env eloping C ∗ -algebra C ∗ max (Γ) of ℓ 1 (Γ), which are p - summable o v er l C Γ. There a r e obvious notions of unitary equiv alence, degeneracy , and op erator homo- top y of F redholm represen tations. F or finitely summable represen tations one puts Definition 2.8. A smo oth op erator homotopy b et we en p -summable F redholm rep- resen tations E 0 = ( H , ρ, F 0 ) and E 1 = ( H , ρ, F 1 ) is giv en b y a family F t , t ∈ [0 , 1] of b ounded op erators on H connecting F 0 and F 1 suc h that • E t = ( H , ρ, F t ) ∈ F r ed ( p ) ( G ) , 0 ≤ t ≤ 1, • { t 7→ F t } ∈ C ∞ ([0 , 1] , L ( H )), • { g 7→ ρ ( g ) F t ρ ( g ) − 1 − F t } ∈ C ( G, C ∞ ([0 , 1] , ℓ p ( H ))). Kasparo v defines the G -equiv arian t K -homology of l C a s K K G ( l C , l C) = F r ed ( G ) / ∼ (2 . 9) as the group of equiv a lence classes of F redholm represen tations of G with respect to the equiv alence relation generated by unitary eq uiv alence, a ddition of degenerate represen tations, a nd o perat o r homotop y . The smo oth v ersion of this is Definition 2.9. Let G b e a lo cally compact group and let 1 ≤ p < ∞ . The p -summable, G -equiv arian t K -homolo gy o f l C KK ( p ) G ( l C , l C) = F r ed ( p ) ( G ) / ∼ (2 . 10) is defined a s the group of equiv alenc e classes of p -summable F redholm represen ta- tions of G with resp ect to the equiv alence relation generated b y unitary equiv alence , addition of degenerate represen tations, and smo oth op erator homotop y . If G = Γ is discrete there are tautological isomorphisms K K Γ ( l C , l C) ≃ − → K K ( C ∗ max (Γ) , l C) (2 . 11) and KK ( p ) Γ ( l C , l C) ≃ − → KK ( p ) (( C ∗ max (Γ) , l C Γ) , l C) (2 . 12) 6 The Kasparov pro duct turns the (smo oth) equiv arian t K -groups (2.9) a nd (2.10) into unital, asso ciativ e rings. The tautological isomorphisms (2.11) and (2.12) b ecome than ring isomorphisms. Finally there is a natural ring homomorphism R ( G ) − → K K G ( l C , l C) (2 . 13) from the repres entation r ing of G to the G -equiv ariant K -homolo g y of l C, whic h assigns to a virtual finite dimensional unitar y represen tation [ ρ + ] − [ ρ − ] the F redholm represen tation E G = (  + ⊕  − , 0). This homomorphism factors thro ugh the smo oth equiv ariant K -homolog y rings K K ( p ) G ( l C , l C) fo r all 1 ≤ p < ∞ . 3 Results In the seque l w e understand b y a higher rank group the group of k -rational p o in ts of a Zariski-connected, simple, isotropic algebraic group of k -rank at least t wo o v er the lo cal field k . Theorem 3.1. L et G b e a finite pr o d uct of higher r ank gr oups. Then KK ( p ) G ( l C , l C) ≃ Z Z (3 . 1) is gener ate d by the trivial r epr esentation of G for 1 < p < ∞ . Theorem 3.2. L et Γ b e a lattic e (i.e. a discr ete sub gr oup of finite c ovolume) in a finite pr o d uct of higher r ank g r oups. Then the tautolo gic al ho momorphisms R (Γ) ≃ − → KK ( p ) Γ ( l C , l C) ≃ − → KK ( p ) 0 (( C ∗ max (Γ) , l CΓ) , l C) (3 . 2) ar e isomo rphisms and KK ( p ) 1 (( C ∗ max (Γ) , l C Γ) , l C) = 0 (3 . 3) for 1 < p < ∞ . Theorem 3.3. L et Γ b e a lattic e in a finite pr o duct of higher r ank gr oups. Then KK ( p ) ∗ (( C ∗ r ed (Γ) , l CΓ) , l C) = 0 (3 . 4) for 1 < p < ∞ , wher e C ∗ r ed (Γ) denotes the r e duc e d C ∗ -algebr a o f Γ , i.e. the envelop- ing C ∗ -algebr a of the r e gular r epr esentation of Γ . Corollary 3.4. Under the assumptions of 3.3 n o nonze r o class in the K -ho molo gy gr oup K K ∗ ( C ∗ r ed (Γ) , l C) c an b e r epr esente d by a F r e dholm mo dule wh ich is finitely summable over l CΓ . 7 4 The work of Bader, F urman, Geland er and Mono d 4.1 Uniformly con v ex B anach spaces Recall that a Banac h space B is stric t ly con v ex if the midp oint of eve ry segmen t joining t w o p oin ts of the unit sphere S ( B ) is in the interior of the unit ball. It is uniformly con v ex if the previous condition holds uniformly in the following sense: for all δ > 0 there exis ts ǫ > 0 suc h that k x − y k ≥ δ ⇒ k x + y 2 k ≤ 1 − ǫ (4 . 1) for all x, y ∈ S ( B ). If B is uniformly con v ex, then for ev ery ξ ∈ S ( B ) there is a unique x ∈ S ( B ∗ ) suc h that h x, ξ i = 1. This a ssignem ent defines a uniformly con tin uous homeomorphism ι : S ( B ) − → S ( B ∗ ) (4 . 2) from the unit sphere of B to that of B ∗ . Uniformly con v ex spaces are reflexiv e. A Banac h space is uniformly smo oth if its dual space is uniformly conv ex . Let ρ : G → O ( B ) b e an orthogonal (i.e. isometric) linear represen tation on a strictly con v ex real Banac h space B . Then the subspace B ρ ( G ) of G - fixed v ectors p ossess es a canonical complemen t B ′ =  ( B ∗ ) ρ ∗ ( G )  ⊖ (4 . 3) giv en b y the subspace a nnihilated b y t he space ( B ∗ ) ρ ∗ ( G ) of all b ounded linear func- tionals on B whic h are fixed under the contragredien t represen tation ρ ∗ of G . The r epres entation ρ splits as direct sum of its restriction to an o r thogonal repre- sen tation o n B ′ and the trivial represen tation on B ρ ( G ) . B ≃ B ρ ( G ) ⊕ B ′ (4 . 4) 4.2 Isometric actions on uniformly c onv ex spaces An isometric action of a lo cally group G o n a Banac h space B is called contin uous if the map π : G × B − → B is contin uous. Ev ery isometry of a strictly conv ex space is real affine (it preserv es midp oin ts of segmen ts). Th us π ( g ) v = ρ ( g ) v + ψ ( g ) , ∀ g ∈ G, ∀ v ∈ B (4 . 5) where ρ : G − → O ( B ) is a con tin uous orthogonal (l R-linear isometric) represen tation of G on B and ψ : G − → B is a con tinuous 1-co cycle on G with v alues in the G - mo dule B : ψ ∈ Z 1 cont ( G, ρ ) = { ψ ∈ C ( G, B ) , ψ ( g h ) = ψ ( g ) + ρ ( g ) ψ ( h ) , ∀ g , h ∈ G } . (4 . 6 ) 8 There is th us a bijection b et w een affine isometric G -actions on B with linear part ρ and contin uous 1-co cycles on G with co efficien ts in ρ . The top ology of unifo r m con v ergence on compacta turns Z 1 cont ( G, ρ ) in to a F r´ ec het space. An affine isometric action (with linear part ρ ) p ossesses a fixed p oin t if and only if the corresp onding co cycle b elongs to the sub space B 1 ( G, ρ ) = { µ ∈ Z 1 cont ( G, ρ ) , µ ( g ) = ξ − ρ ( g ) ξ for some ξ ∈ B } . (4 . 7) The v anishing of the c oho mo lo gy group H 1 cont ( G, ρ ) = Z 1 cont ( G, ρ ) /B 1 ( G, ρ ) (4 . 8) is therefore equiv alen t to the a sse rtion that eve ry con tin uous isometric a ction of G on B with linear part ρ p ossess es a fixed p oin t. Lemma 4.1. Supp ose that H 1 cont ( G, ρ ) = 0 and let ψ t , t ∈ [0 , 1 ] b e a smo oth family of c o cycles. Then ther e exists a f a mily ξ t ∈ B , t ∈ [0 , 1] of fixe d p oints of the c orr esp onding affine actions which dep ends smo othly on t . Pro of: There is a canonical exact sequence 0 → B ρ ( G ) → B p → B 1 ( G, ρ ) → 0 (4 . 9) of abstract v ector spaces where p assigns to a v ector ξ ∈ B the co cycle p ( ξ ) : g 7→ ξ − ρ ( g ) ξ . Under our h yp othesis B 1 ( G, ρ ) = Z 1 cont ( G, ρ ) and is thus complete. Therefore (4 .9 ) b ecomes an exact sequence of F r´ ec het spaces whic h splits according to (4.4). In particular, the pro jection p p ossesses a b ounded linear section whic h a ssigns to ev ery co cycle a fixed p oint of the corresp onding affine action. This implies the lemma b ecause b ounded linear maps are smo o th. ✷ The res ult of Ba der, F urman, Gelander and Mono d that w e wish to extend to noncomm utativ e ℓ p -spaces is the Theorem 4.2. [BFGM ] L et G b e a pr o duct of higher r ank gr oups or a lattic e in such a gr oup (se e se ction 3). L et ( X, µ ) b e a me asur e sp ac e a n d let ρ b e any ortho g- onal r epr esentation of G on the Banach sp ac e L p ( X , µ ) wher e 1 < p < ∞ . Then H 1 cont ( G, ρ ) = 0 . It should b e mentioned that Bader, F urman, Gelander a nd Mono d conjecture that an y affine isometric action of a higher rank group or lattice on a uniformly con ve x Banac h space has a fixed p oin t. In fact they deduce the previous theorem from the follo wing result which has a more general fla v or. Theorem 4.3. [BFGM] L et G b e a p r o duct of higher r ank gr oups (se e se ction 2) and let ρ B b e an ortho gonal r epr es e n tation of G on a unif o rmly c onvex an d uniformly smo oth B anach sp ac e B . Supp ose that ther e exists an ortho g onal r epr esentation ρ of G on a Hilb ert sp ac e H and a home omorphism Φ : S ( B ) − → S ( H ) b etwe en the unit sp h er es of B and H such that 9 • Φ intertwines the G -actions induc e d by ρ B and ρ . • Φ and Φ − 1 ar e uniformly c ontinuous. Then H 1 cont ( G, ρ B ) = 0 . W e will sho w in the seq uel that this theorem applies to v ario us affine isometric actions of higher rank groups on op erator ideals. 5 Pro ofs 5.1 Geometry of Sc hatten ideals W e fix from now on a real n um b er p, 1 < p < ∞ . The Sc hatten-von Neumann ideal ℓ p ( H ) ⊂ L ( H ) is the ideal o f compact op erators whose sequenc e of singular v alues is p -summable. It is a Banac h space with resp ect to the norm k T k p = T r ace (( T ∗ T ) p 2 ) (5 . 1) The Sc hatten spaces are symmetrically normed op erator ideals, i.e. one has k AT B k p ≤ k A k L ( H ) k T k p k B k L ( H ) (5 . 2) for A, B ∈ L ( H ) , T ∈ ℓ p ( H ). In particular, if ρ : G − → U ( H ) is a unitary represen- tation of a lo cally compact group G on H , then ρ p : G − → U ( ℓ p ( H )) , ρ p ( g )( T ) = ρ ( g ) T ρ ( g ) − 1 (5 . 3) is a strongly con tin uous, isometric, l C-linear represen tation of G on ℓ p ( H ). W e will need rather precise information ab out the geometry of the Banac h spaces ℓ p ( H ) for 1 < p < ∞ . W e b egin with Theorem 5.1. (Clarkson-MacCarth y),[Si ] The B anach sp ac es ℓ p ( H ) , 1 < p < ∞ ar e uniformly c onvex. This generalizes the correp onding result for the classical Banach spaces L p ( X , µ ). F rom [L T], 1.e.9.(i) o ne derive s further Corollary 5.2. L et ( X, µ ) b e a Bor el sp ac e. Then for 1 < p < ∞ the Banach sp ac e L p ( X , ℓ p ( H )) (5 . 4) 10 obtaine d f r om the sp ac e C c ( X , ℓ p ( H )) of c ontinuous c omp actly supp orte d functions on X w ith values in ℓ p ( H ) by c ompletion with r esp e c t to the norm k f k p p = Z X T r ace (( f ∗ ( t ) f ( t )) p 2 ) dt (5 . 5) is uniformly c onvex. Lemma 5.3. L et 1 < p, q < ∞ b e such that 1 p + 1 q = 1 . Then ℓ p ( H ) ∗ ≃ ℓ q ( H ) , L p ( X , ℓ p ( H )) ∗ ≃ L q ( X , ℓ q ( H )) (5 . 6) in the notations of 5.2. In particular, the Sc hatten ideals ℓ p ( H ) , 1 < p < ∞ are uniformly con v ex and uniformly smo oth Banac h spaces. No w w e will study the p olar decomposition in ℓ p ( H ). The basic res ult in this direc- tion is Theorem 5.4. (Po w ers-Sto ermer inequalit y),[PS] L et A, B b e p ositive tr ac e clas s op er ators on H . Then k A 1 2 − B 1 2 k 2 2 ≤ k A − B k 1 . (5 . 7) This inequality has b een generalized by v arious authors. In particular, Ando pro v ed the follo wing Theorem 5.5. (Generalized Po w ers-Sto ermer inequalit y),[A] L et 0 < α < 1 an d let A, B b e p ositive o p er ators in ℓ p ( H ) , 1 ≤ p < ∞ . Then k A α − B α k p α p α ≤ k A − B k p p . (5 . 8) The generalized P ow ers-Stoermer inequalities are needed to establish the Corollary 5.6. L et 1 < p, q < ∞ . Then the Mazur map M p,q : S ( ℓ p ( H )) − → S ( ℓ q ( H )) T 7→ sig n ( T ) | T | p q (5 . 9) is a uniform l y c ontinuous home omorphism. Pro of: Note that M p,p = I d and M q ,r ◦ M p,q = M p,r for 1 < p, q , r < ∞ . Therefore it suffices to ve rify t he claim in the follow ing tw o cases: 1 < q < p < 3 q , p ≥ 2 , and p < q , 1 p + 1 q = 1. In the second case it is eas ily se en that M p,q = ∗ ◦ ι 11 where ι is the canonical ho meomorphism (4.2) and ∗ : ℓ q ( H ) → ℓ q ( H ) maps an op erator to it s adjo in t. Th us M p,q is oviously unif o rmly con tinuous in t his case. If 1 < q < p < 3 q , p ≥ 2 , w e find with 0 < r = 1 2 ( p q − 1 ) < 1 that M p,q ( T ) = sg n ( T ) | T | p q = T | T | 2 r = T ( T ∗ T ) r The Hoelder and the g eneralize d P ow ers-Stoermer inequalities yield then for S, T ∈ S ( ℓ p ( H )) k M p,q ( S ) − M p,q ( T ) k q ≤ ≤ k ( S − T )( S ∗ S ) r k q + k T (( S ∗ S ) r − ( T ∗ T ) r ) k q ≤ k S − T k p k ( S ∗ S ) r k p 2 r + k T k p k ( S ∗ S ) r − ( T ∗ T ) r k p 2 r ≤ k S − T k p ( k S ∗ S k p 2 ) r + k T k p ( k S ∗ S − T ∗ T k p 2 ) r ≤ k S − T k p ( k S ∗ k p k S k p ) r + k T k p ( k S ∗ − T ∗ k p k S k p + k T ∗ k p k S − T k p ) r ≤ k S − T k p + (2 k S − T k p ) r whic h sho ws tha t M p,q is uniformly con tin uous. ✷ Similar estimates lead to Corollary 5.7. L et ( X , µ ) b e a Bor el sp ac e and let 1 < p, q < ∞ . Then the Mazur map M p,q : S ( L p ( X , ℓ p ( H ))) − → S ( L q ( X , ℓ q ( H ))) f 7→ sig n ( f ) | f | p q (5 . 10) is a uniform l y c ontinuous home omorphism. 5.2 Rigidit y of actions on S c hatten ideals W e a r e ready t o apply the results of Bader, F urman, Gelander and Mono d in our framew ork. Theorem 5.8. L et G b e a pr o d uct of higher r ank gr oups as i n se ction 3 and le t ρ b e a unitary r epr esentation of G on the Hilb ert sp ac e H . L e t ρ p b e the c orr esp ond i ng isometric r epr esentation on the Sc hatten ide als ℓ p ( H ) . Then H 1 cont ( G, ρ p ) = 0 for 1 < p < ∞ . (5 . 11) Consider no w a lattice Γ ⊂ G in a higher rank group. The homogeneous space G/ Γ is then of finite v olume. Acc ording to 5.2 and 5.3 the Bana ch spaces L p ( G/ Γ , ℓ p ( H )) are uniformly con v ex and uniformly smo oth fo r 1 < p < ∞ . Supp ose that in addition a unitary represe ntation ρ of Γ on a Hilb ert space H is g iv en a nd let ρ p b e the corresp onding isometric represen tation on ℓ p ( H ). Then L p ( G/ Γ , ℓ p ( H )) can b e iden tified with the completion of the space L [ p ] ( G, ℓ p ( H )) Γ of Γ-equiv arian t maps from G to ℓ p ( H ) with r espect to the norm k f k p = R D k f ( g ) k p dg ( D ⊂ G a Borel fundamen tal domain fo r Γ). The latter space carries a canonical isometric linear G -action coming from left translation. 12 Theorem 5.9. L et G b e a higher r ank gr oup and let Γ b e a lattic e in G . L et ρ b e a unitary r epr e sentation of Γ . Then in the pr evious notations H 1 cont ( G, L [ p ] ( G, ℓ p ( H )) Γ ) = 0 for 1 < p < ∞ . (5 . 12) Corollary 5.10. L et Γ b e a lattic e in a pr o duct of hig h er r ank gr oups as in se ction 2 and let ρ b e a unitary r e pr esen tation of Γ on the Hilb ert sp ac e H . L et ρ p b e the c orr esp onding isometric r epr esentation on the Sc h atten ide als ℓ p ( H ) . Then H 1 (Γ , ρ p ) = 0 for 1 < p < ∞ . (5 . 13) Pro of: (of t he theorem) W e b egin with a pro of of ( 5 .11). W e w ant to apply theorem 4.3 to the isometric represen tations ρ p on the Sc hatten ideals ℓ p ( H ). F or 1 < p < ∞ these ideals are uniformly conv ex and un ifo rmly smo oth 5.3. Note t ha t ℓ 2 ( H ) is the Hilb ert space of Hilb ert-Sc hmidt o p erator s on H . By construction the Mazur map M p, 2 : S ( ℓ p ( H )) → S ( ℓ 2 ( H )) (5.10 ) intert wines the isometric represen tations ρ p and ρ 2 . Moreov er M p, 2 as w ell a s its in v erse M 2 ,p are uniformly contin uous by 5.6. Thus Theorem 4.3 of Bader, F urman, Gelander and Mono d applies and yields (5.11). The demonstration o f (5.1 2 ) is similar a nd uses 5.2 and 5.7 . ✷ Pro of: (of the corollary) The w ell k nown idea is to use induction of represen tations and affine isometric actions from Γ to G in o r der to reduce the statemen t to (5.12). The arg uments in [BFGM], sec tio n 8.1, 8.2 apply v erbatim to o ur situation and yield H 1 (Γ , ρ p ) ≃ H 1 cont ( G, L [ p ] ( G, ℓ p ( H )) Γ ) The claim follo ws then from 5.9. ✷ 5.3 Finitely summable mo du les o v er higher rank groups and lattices The follo wing conse quence of the preceding rigidity theorems will immediately lead to our main results Prop osition 5.11. L et G b e a pr o duct of higher r ank gr oups o r a lattic e i n such a gr oup. L et 1 < p < ∞ . i) Every p -summab l e F r e dholm r epr esentation E = ( H , ρ, F ) is sm o othly op er ator homotopic (among p -summable r epr e sentations) to a F r e dholm r epr esentation E ′ = ( H , ρ, F ′ ) such that [ F ′ , ρ ( G )] = 0 . ii) L et E i = ( H , ρ, F i ) , i = 0 , 1 , b e p -summ a ble F r e dholm r epr esentations which ar e smo othly op er ator homotopic and satisfy [ F i , ρ ( G )] = 0 . Then ther e exists a smo oth op er ator homo topy E t = ( H , ρ, F t ) , t ∈ [0 , 1] , c onne c ting E 0 and E 1 and sa tisfying [ F t , ρ ( G )] = 0 , ∀ t ∈ [0 , 1] . Pro of: L et E = ( H , ρ, F ) b e a p -summable F redholm represen tation. Then the map ψ : G → ℓ p ( H ) , g 7→ ρ ( g ) F ρ ( g ) − 1 − F defines a contin uous one-co cycle 13 ψ ∈ Z 1 cont ( G, ρ p ). According to theorems 5.8 , 5 .9, this co cycle is a cob oundary , i.e. ψ ( g ) = ρ ( g ) F ρ ( g ) − 1 − F = ρ ( g ) T ρ ( g ) − 1 − T fo r some op erator T ∈ ℓ p ( H ) and all g ∈ G . Th us E s = ( H , ρ, F − sT ) is a smo oth op erator homotop y b et w een E and the desired F redholm represen tation E ′ = ( H , ρ, F − T ) satisfying [ F − T , ρ ( G )] = 0. If E t = ( H , ρ, F t ) , t ∈ [0 , 1] is a smo oth op erator homotopy then the corresp ond- ing family of co cycles is smo oth. According to 4.1 there exists a smo oth family T t ∈ ℓ p ( H ) , t ∈ [0 , 1] satisfying [ F t , ρ ( g )] = [ T t , ρ ( g )] , ∀ g ∈ G, t ∈ [0 , 1 ]. Th us E ′ t = ( H , ρ, F t − T t ) , t ∈ [0 , 1] defines the desired op erator homo t o p y consisting of op erators comm uting strictly with G . ✷ F or the pro of of 3.1 w e still need the w ell kno wn Theorem 5.12. [BFGM] (Mo ore Er godicity Theorem) L et G ( k ) b e the gr oup of k -r ational p oints of a s i m ply c onne cte d, sem i s imple algebr aic gr oup over the lo c a l field k . L et ρ b e a unitary r epr esentation of G o n a Hilb ert sp ac e H . Then either ther e a r e nonzer o ve ctors fixe d under ρ ( G ( k )) or al l matrix c o effic i e nts of ρ vanish at in fi nity, i.e. { g 7→ h ξ , ρ ( g ) η i} ∈ C 0 ( G ) , ∀ ξ , η ∈ H . Pro of of Theorem 3.1: Let E = ( H , ρ, F ) b e a p -summable F redholm represen- tation o f G . After a smo oth op erator homotop y w e may suppo se b y 5.10.i) that F and ρ ( G ) strictly commute . Moreo v er w e may supp ose tha t F is selfadjoin t. The subspace K er ( F ) ⊂ H as w ell as its orthogonal comple men t H ′ are then stable under ρ ( G ). The triple E ′ = ( H ′ , ρ |H ′ , F ′ = F |H ′ ) is t hen smo othly o p erato r homo- topic t o the degenerate triple D ′ = ( H ′ , ρ |H ′ , F ′ | F ′ | ). Thus our original repre sen tatio n is equ iv alen t t o the finite dimensional F redholm represen tation ( K er F , ρ K erF , 0). It remains to sho w that eve ry finite dimensional unitary represen tation of G is trivial. If not, there is a finite dimensional unitary represen tation ρ ′ of G without fixed v ectors. By Mo ore’s ergo dicit y theorem the function g 7→ det ( ρ ′ ( g )) , g ∈ G , whic h is a homogeneous p olynomial in the matr ix co efficien ts of ρ , has to v anish at infinit y . This con tradicts the fact that this function is of absolute v alue 1 ev erywhere b ecause ρ ′ is unitary . ✷ Pro of of T heorem 3.2: The same reasoning as in the pro of of theorem 3.1 sho ws that the canonical homomorphism R (Γ) → K K ( p ) Γ ( l C , l C) is surjectiv e a nd tha t KK ( p ) 1 (( C ∗ max (Γ) , l CΓ) , l C) = 0. W e wan t to sho w that the canonical map is injectiv e as w ell. So let [ ρ ] , [ ρ ′ ] ∈ R (Γ) b e a virtual finite dimensional unitar y repres entations of Γ whic h define the same class in K K ( p ) 0 (( C ∗ max (Γ) , l C Γ) , l C). According to 5.10 ii) one actually ma y find a homot op y of virtual unitary represen tations on a fixed finite dimensional Z Z / 2 Z Z - graded Hilb ert space whic h connects [ ρ ] and [ ρ ′ ]. Ev ery higher r a nk group as considered in section 2 has Kazhdan’s pro perty T [HV], i.e. the trivial represen tation is a n isolated p oint in the unitary dual with resp ect to the F ell top ology . Prop erty T is stable under taking finite pro ducts and pa ssage to lattices [HV]. Moreo ver it is easily seen that under the presence of prop erty T not only the trivial represen tation but in fact ev ery finite dimensional unita ry represen tation is an isolated p oint in the unitary dual. Thus a n y homotopy of finite dimensional virtual unitary represen tations of a prop ert y T g roup is necessarily 14 constan t which pro v es our claim. ✷ Pro of of Theorem 3.3: Using 5.10 again w e see (after using a smo oth op er- ator homotop y and the subtraction o f a degenerate module) that ev ery clas s in KK ( p ) ∗ (( C ∗ r ed (Γ) , l C Γ) , l C) may b e represen ted by a F redholm mo dule E = ( H ′ , ρ, 0) o v er C ∗ r ed (Γ) with finite dimensional underlying Hilb ert space H ′ . According to [Di], 18.9.5 ev ery finite dimensional represen tation o f the r educe d group C ∗ -algebra of a nonamenable lo cally compact g roup is iden tically zero. This applies in part icular to the lat tices under consid eratio n which p ossess property T and are not compact a nd therefore nonamenable. Th us the considered F redholm mo dule E is iden tically zero and the theorem follo ws. ✷ References [A] T . Ando, Comparison of norms || f ( A ) − f ( B ) || and || f ( | A − B | ) || , Math. 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