Fixed point property for universal lattice on Schatten classes
aa r X i v : . [ m a t h . G R ] J un FIXED POINT PROPERTY FOR UNIVERSAL LATTICE ONSCHATTEN CLASSES
MASATO MIMURA
Abstract.
The special linear group G = SL n ( Z [ x , . . . , x k ]) ( n at least 3 and k finite) is called the universal lattice. Let n be at least 4, p be any real numberin (1 , ∞ ). The main result is the following: any finite index subgroup of G hasthe fixed point property with respect to every affine isometric action on thespace of p -Schatten class operators. It is in addition shown that higher ranklattices have the same property. These results are generalization of previoustheorems repsectively of the author and of Bader–Furman–Gelander–Monod,which treated commutative L p -setting. Introduction and main results
In this paper, every group is assumed to locally compact and second countable,and every subgroup of a ( topological ) group is assumed to be closed . The symbol k is used for representing any finite natural (positive) number, and we set A = Z [ x , . . . , x k ]. Hereafter, by “ higher rank lattices ” we mean lattices (, namely,discrete subgroup with finite covolume) in semisimple algebraic groups over localfields (possibly archimedean) with each factor having local rank ≥
2. We use thesymbol h· | ·i for the inner product on a Hilbert space.The special linear group G = SL n ( A )= SL n ( Z [ x , . . . , x k ]) (where n ≥
3) iscalled the universal lattice by Y. Shalom in [Sha1]. It was a long standing problemto determine whether this group satisfies
Kazhdan’s property ( T ) ([Ka]), and finallyShalom and L. Vaserstein has answered this problem in the affirmative. Theorem 1.1. ( Shalom [Sha4] , Vaserstein [Va])
Universal lattices G = SL n ( Z [ x , . . . , x k ])( n ≥ have property (T) . Because property (T) passes to group quotients, this result immediately im-plies groups such as SL n ≥ ( Z [1 /p ]) (here p is a prime number); SL n ≥ ( Z [ √ , √ n ≥ ( F q [ x ]) ( F q is the field of order q and q is a prime power); and SL n ≥ ( Z [ t, t − ])have property (T). Note that in four examples above, all but last one are higherrank (hence arithmetic (or S -arithmetic)) lattices. Kazhdan’s original result [Ka]states that higher rank lattices (recall the notation in this paper from the first para-graph) has property (T). Hence in these cases property (T) is classical. Howeverthe last one in the examples above cannot be realized as a higher rank (hence arith-metic) lattice because it contains an infinite normal subgroup of infinite index, andproperty (T) for this group had not been obtained before the Shalom–Vasersteintheorem. Therefore property (T) for universal lattices can be regarded as a non-arithmetization of extreme rigidity for higher rank lattices. Recall that the cele-brated Delorme–Guichardet theorem, see [BHV], states property (T) is equivalentto property ( F H ), which is defined as follows: a group Λ is said to have property ( F H ) if every (continuous) affine isometric action of Λ on a Hilbert space has aglobal fixed point.
Key words and phrases. fixed point property; Kazhdan’s property (T); Schatten class opera-tors; noncommutative L p -spaces; bounded cohomology.The author is supported by JSPS Research Fellowships for Young Scientists No.20-8313. From this point, we always assume p ∈ (1 , ∞ ). In 2007, Bader, Furman, Ge-lander, and Monod considered fixed point properties in much wider framework, andthey defined the fixed point property, property ( F B ), for a family B of Banach spaces(, or a single Banach space B ). They in particular paid heavy attention to the caseof B = L p , which denotes the family of all L p -spaces, because property (F L p ) turnsout to be much stronger than property (T) ( ⇔ property (FH) = (F L )), providedthat p ≫
2. Indeed, P. Pansu [Pa] shown the group Sp n, , which has property (T)if n ≥
2, fails to have property (F L p ) if p > n + 2. Moreover G. Yu [Yu] shown thatevery hyperbolic group H , including wide range of groups with property (T), has p > l p -space. In comparison, one of the main resultsof Bader–Furman–Gelander–Monod in [BFGM] is the following: Theorem 1.2. ([BFGM])
Higher rank lattices, in the notation of this paper, remainto have property (F L p ) for all p ∈ (1 , ∞ ) . In [Mi], the author extended the Shalom–Vaserstein theorem and obtained thefollowing “non-arithmetization” of Theorem 1.2:
Theorem 1.3. ([Mi])
Let n ≥ . Then for any p ∈ (1 , ∞ ) the universal lattice G = SL n ( Z [ x , . . . , x k ]) has property (F L p ) . Note that property (F L p ) is of high importance also from the view point of groupactions on the circle, see [Na, § n = 3 seems to remain open.The main result of this paper is to extend this result to that for the family C p ofthe spaces C p of p -Schatten class operators on any separable Hilbert space. Herefor a separable Hilbert space H , a bounded linear operator a ∈ B ( H ) is said tobe of p -Schatten class ( a ∈ C p ) if Tr( | a | p ) < ∞ holds, where Tr is the canonical(semifinite) trace: take an orthonomal basis ( ξ i ) i ∈ N for H and for t ∈ B ( H ) + , setTr( t ) := X i ∈ N h tξ i | ξ i i ∈ [0 , ∞ ](, which is independently determined of the choice of an orthonormal basis of H );and | a | := ( a ∗ a ) / is the absolute value of a . It can be regarded as a somegeneralization of Theorem 1.3 to noncommutative L p -setting. Motivating fact onthis study is that fixed point property on C p has potential for some application togroup actions on higher dimensional manifolds.Furthermore, in this paper we consider stronger property than property (F B ),called property ( F F B ). Recall every affine isometric action α : G y B can bewritten as α ( g ) · ξ = ρ ( g ) ξ + b ( g ), where ρ is a isometric linear representation and b : G → B is a ρ - (1 - ) cocycle , namely, for any g, h ∈ G , b ( gh ) = b ( g )+ ρ ( g ) b ( h ) holds.We consider a “quasification” of cocycles, namely we allow uniformly bounded errorfrom being cocycles. Property (FF B ) is the boundedness property for any quasi-cocycle into every isometric representation in B . Property (FF B ) / T is a weakerform of (FF B ) and that asserts the boundedness of quasi-cocycles modulo triviallinear part . Now we state our main result: Theorem 1.4.
Let n ≥ . Then for any p ∈ (1 , ∞ ) the universal lattice G =SL n ( Z [ x , . . . , x k ]) has property (F C p ) . Equivalently, every affine isometric actionof G on the space C p of p -Schatten class operators ( on any separable Hilbert space ) has a global fixed point. Furthermore, for any p ∈ (1 , ∞ ) , G has property (FF C p ) / T( “property (FF C p ) modulo trivial part” ) . In particular, for any isometric linearrepresentation ρ on C p which satisfies ρ G , every quasi- ρ -cocycle is bounded.Both property (F C p ) and property (FF C p ) / T remain valid by taking group quo-tients and taking finite index subgroups of G above. NIVERSAL LATTICES 3
For precise definitions of property (FF B ) and property (FF B ) / T, see Section 2.In the scope of the author, it seems unknown at the moment, whether quasi-cocycleson universal lattices into the trivial representation must be bounded. Neither doesthe author know whether (FF B ) / T is strictly weaker than (FF B ).Therefore for any commutative and finitely generated ring R (we always assume aring R is unital and associative), the following holds: the elementary group E n ≥ ( R )and finite index subgroups therein have (F C p ), and property (FF C p ) / T. Here the elementary group over R is the multiplicative group in n × n matrix ring M n ( R )generated by elementary matrices . An elementary matrix in M n ( R ) is an n × n matrix whose entries are 1 on diagonal and all but one entries off diagonal are 0.The Suslin stability theorem ([Su]) states for A = Z [ x , . . . , x k ] E n ( A ) coincideswith SL n ( A ) provided n ≥
3, whereas E ( A ) is a proper subgroup of SL ( A ) ([Co]).Property (F B ) and Property (FF B ) have natural interpretation in terms of (ordi-nary and bounded) group cohomology. Thus by Theorem 1.4, we have the followingcorollary. Here bounded cohomology is defined by requesting additional conditionthat every cochain has bounded range, and the map Ψ in below is induced bythe natural inclusion from bounded to ordinary cochain complexes (Ψ is calledthe comparison map in degree 2). For details of bounded cohomology with Banachcoefficients, see [Mo], [BM1], and [BM2]. Corollary 1.5.
Let n ≥ and R is a ( unital, associative, ) commutative and finitelygenerated ring. Then for any p ∈ (1 , ∞ ) , every finite index subgroup Γ in E n ( R ) satisfies the following: • for any isometric linear representation ρ on C p , H (Γ; C p , ρ ) = 0 ; • for any isometric linear representation ρ on C p which satisfies ρ Γ , thenatural map from second bounded cohomology to second cohomology: Ψ : H (Γ; C p , ρ ) → H (Γ; C p , ρ ) is injective. The latter item follows from that the kernel of Ψ above is naturally isomorphicto the following space: { quasi- ρ -cocycles } / ( { ρ -cocycles } + { bounded maps } ) . Our proof of Theorem 1.4 for universal lattices (and group quotients) consistsof two steps: in the first step, in Section 2 we show certain criteria for a family B of Banach spaces with respect to which universal lattices satisfy property (F B )and property (FF B ) / T. To the best of the author’s knowledge, no such criteria hadbeen observed. Since our criteria seems to be of their own interest and importance,we state it here. For the definition of (relative) property (T B )), and of ultraproducts of Banach spaces, we refer to Section 2. Theorem 1.6. ( criteria for fixed point properties for universal lattices ) Set A = Z [ x , . . . , x k ] . Let B be a family of superreflexive Banach spaces. Suppose either ofthe following two conditions is fulfilled: ( i ) The pair E ( A ) ⋉ A D A has relative property (T B ) ; and B is stable underultraproducts. ( ii ) As properties for countable discrete groups, relative property (T) implies rel-ative property (T B ) : that means; for any pair of a countable discrete group Λ and a normal ( not necessarily proper ) subgroup Λ therein whenever the pair Λ D Λ has relative property (T) , it has relative property (T B ) .Then for any n ≥ , the universal lattice SL n ( A ) possesses property (F B ) andfurthermore possesses property (FF B ) / T . Note that “condition ( ii ) implies the conclusion” deeply relies on Theorem 1.1.Also, if condition ( ii ) is satisfied, then relative property (T B ) for E ( A ) ⋉ A D A is automatic because relative property (T) for that pair was shown in [Sha1] muchearlier. Here by Λ := E ( A ) ⋉ A D A =: Λ , we meanΛ = (cid:26) ( Z, ζ ) = (cid:18)
Z ζ (cid:19) : Z ∈ E ( A ) , ζ ∈ A (cid:27) D { ( I , ζ ) : ζ ∈ A } = Λ . For some examples of B which satisfy condition ( i ) or condition ( ii ), see Section 5and the last part of Section 2.In the second step, we verify that the family C p indeed fulfills condition ( ii ): Proposition 1.7.
For any p ∈ (1 , ∞ ) relative property (T) implies relative property (T C p ) among locally compact and second countable groups. This part is inspired by a work of M. Puschnigg [Pu], who extended a method ofBader–Furman–Gelander–Monod for the case of commutative L p -spaces on σ -finitemeasures.For the proof of Theorem 1.4 for finite index subgroups of universal lattices,we need the p -induction theory for (quasi)-1-cocycles of Bader–Furman–Gelander–Monod. We also need to deal with noncommutative L p -space associated with thevon Neumann algebra L ∞ ( D ) ⊗ B ( H ) (with the canonical trace), where D is a finitemeasure space. We examine them in Section 4. The arguments above in additionimplie the following: Theorem 1.8.
Higher rank lattices, in the notation of this paper, have prop-erty (F C p ) for all p ∈ (1 , ∞ ) . More precisely, the following holds true: Let G =Π mi =1 G i ( k i ) , where k i are local field, G i ( k i ) are k i -points of Zariski connected sim-ple k i -algebraic group G i . Assume each simple factor G i ( k i ) has k i -rank at least . Then G and lattices therein have property (F C p ) for any p ∈ (1 , ∞ ) . This result is a noncommutative analogue of Theorem 1.2, and can be seen ageneralization of (a part of) a work of Puschnigg [Pu, Corollary 5.10]. For noncom-mutative L p -spaces associated with semifinite von Neumann algebra other than C p (associated with ( B ( H ) , Tr)), see Remark 5.2.
Organization of this paper : Section 2 is devoted to basic definitions and proof ofTheorem 1.6. In Section 3, we prove Proposition 1.7, and thus prove Theorem 1.4for universal lattices. In Section 4, we consider p -inductions and generalization ofProposition 1.7. By these, we complete the proof of Theorem 1.4 and Theorem 1.8.Corollary 1.5 is immediate from Theorem 1.4 and a mere interpretation (see [BHV],[BFGM], [BM1], [BM2], and [Mo]), so that we will not exhibit a proof of that. InSection 5, we make some concluding remarks on condition ( i ) in Theorem 1.6.2. Property (T B ) , (F B ) , (FF B ) , (FF B ) / T ; and our criteria Hereafter, we assume all Banach spaces B in this paper are superreflexive. Thiscondition is equivalent to that B has a compatible norm to a uniformly convex anduniformly smooth norm. Basic example is any L p -space with p ∈ (1 , ∞ ). We referto [BL, § A] for a comprehensive study of this topic. In this section, Λ is a locallycompact second countable group, ρ is a (continuous) isometric linear representationof Λ in B . Also every subgroup of Λ is assumed to be closed.Here we recall some basic results from [BFGM, § B , there exists a uniformly convex and uniformly smooth norm on B ,compatible to the original norm, with respect to which ρ is still isometric. Hencewe can assume B is uniformly convex and uniformly smooth. Second, then by theuniform smoothness, there is a natural complement B ′ ρ (Λ) of B ρ (Λ) in B which isa ρ (Λ) invariant space. Here B ρ (Λ) is the subspace of B of ρ (Λ)-invariant vectors.Also, B ′ ρ (Λ) is isomorphic to B/B ρ (Λ) as ρ (Λ) representation spaces. Precisely, NIVERSAL LATTICES 5 B ′ ρ (Λ) is the annihilator of the subspace ( B ∗ ) ρ † (Λ) ⊆ B ∗ in B , where ρ † denotesthe contragradient representation: for the duality h· , ·i : B × B ∗ → C and for any g ∈ Λ, any x ∈ B , and any φ ∈ B ∗ , h x, ρ † ( g ) φ i := h ρ ( g − ) x, φ i . Finally, by uniformconvexity of B , for any (continuous) affine isometric action, the existence of a Λ-fixed point is equivalent to boundedness of some (, equivalently, any) Λ-orbit. Byconsidering orbit of the origin 0 ∈ B , this means for any ρ on B , a ρ (1-)-cocycle b is a coboundary if and only if it has bounded range.Now we recall definitions of property (T B ), property (F B ) in [BFGM]; and prop-erty (FF B ), property (FF B ) / T in [Mi]. Note that if Λ E Λ is a normal subgroup,then B = B ρ (Λ ) ⊕ B ′ ρ (Λ ) can be seen as a decomposition of B as Λ-representationspaces. Definition 2.1.
Let Λ, B , ρ be as in the setting of this section. • The representation ρ of Λ on B is said to have almost invariant vectors ,written as ρ ≻ Λ , if the following holds: for any compact subset K ⊂ Λ,there exists a sequence of unit vectors ( ξ n ) n in B such that max s ∈ K k ξ n − ρ ( s ) ξ n k → n → ∞ . • A continuous map b : Λ → B is called a ρ -cocycle if for any g, h ∈ Λ, b ( gh ) = b ( g ) + ρ ( g ) b ( h ) holds. The map b is called a quasi- ρ -cocycle ifsup g,h k b ( gh ) − b ( g ) − ρ ( g ) b ( h ) k < ∞ holds. Definition 2.2. ([BFGM],[Mi]) Let Λ, B , ρ be as in the setting of this section,and fix B .(1) A group pair Λ D Λ is said to have relative property ( T B ) if the following holdstrue: for any ρ of Γ on B , the isometric linear representation ρ ′ , constructedby restricting ρ on B ′ ρ (Λ ) , ρ ′ : Λ → O ( B ′ ρ (Λ ) ) does not have almost invariantvectors. The group Λ is said to have property ( T B ) if Λ D Λ has relativeproperty (T B ).(2) A group pair Λ > Λ is said to have relative property ( F B ) if for any ρ of Λon B , every ρ -cocycle b is a coboundary on the subgroup Λ , namely, thereexists ξ ∈ B such that for any h ∈ Λ , b ( h ) = ξ − ρ ( h ) ξ holds. Equivalently, b is bounded on Λ . The group Λ is said to have property ( F B ) if Λ > Λ hasrelative property (F B ).(3) A group pair Λ > Λ is said to have relative property ( F F B ) if for any ρ of Λon B , every (continuous) quasi- ρ -cocycle b is bounded on Λ . The group Λ issaid to have property ( F F B ) if Λ > Λ has relative property (FF B ).(4) The group Λ is said to have property ( F F B ) /T (, which means property ( F F B ) modulo trivial part ,) if for any ρ of Λ on B and any (continuous) quasi- ρ -cocycle b , b ′ (Λ) is bounded. Here b ′ : Λ → B ′ ρ (Λ) means the projection of b to a quasi- ρ ′ -cocycle which ranges in B ′ ρ (Λ) , associated with the decomposition B = B ρ (Λ) ⊕ B ′ ρ (Λ) .If B = B is a family of Banach spaces, these seven properties are defined as havingcorresponding properties for all Banach spaces E ∈ B .We make two remarks on the definition above: first, if B is a general (possiblynot superreflexive) Banach space, B ′ ρ (Λ) is replaced with B/B ρ (Λ) in the definitionsof (relative) (T B ) and (FF B ) / T. Secondly, if Λ has a compact abelianization, thenproperty (FF B ) / T implies property (F B ). In particular, the following holds true: Lemma 2.3.
For any p ∈ (1 , ∞ ) , property (FF L p ) / T implies property (F L p ) .Proof. Let Λ have (FF L p ) / T. Suppose H := Λ / [Λ , Λ] is noncompact. Then con-sider the ( L p -)left regular representation λ H of H in L p ( H ) and a representation MASATO MIMURA ρ of Λ which is the pull-back of λ H by Λ ։ H . Since H is abelian and noncom-pact, λ H ≻ H but λ H H . Therefore there exists a λ H -cocycle which is not acoboundary (for details, see [BFGM, § ρ -cocycle which is nota coboundary. However this contradicts ρ Λ and (FF L p ) / T for Λ. (cid:3)
By the Delorme–Guichardet theorem, if B is the family H of all Hilbert spaces(= L ), then property (F H ) = (FH) is equivalent to property (T H ) = (T). Wenote that in the sprit of this, property (FF H ) was previously defined and called property ( T T ) by N. Monod [Mo]. In the spirit of this, we also call (FF H ) / T property ( T T ) /T . By Lemma 2.3, (TT) / T implies (T).We now explain one more concept which appears in Theorem 1.6, namely, an ultraproduct of Banach spaces. For precise definition and comprehensive treatment,see [He]. Here we briefly recall the definition: Take a non-principal ultrafilter ω on N and fix it. For a sequence (( B n , k · k n )) n of pairs of a Banach space and thenorm, we define the ultraproduct lim ω B n = ( B ω , k · k ω ) as follows: we define B ω as( L n ∈ N ( B n , k·k n )) ∞ / N . Here ( L n ∈ N ( B n , k·k n )) ∞ means the set of sequences withbounded norms. We define a seminorm k · k there by k ( ξ n ) n k :=lim ω k ξ n k n , and set N as the null subspace with respect to k · k . Finally, define a norm k · k ω on B ω asthe induced norm by k · k above. We say a family B is stable under ultraproducts ifwhenever B n ∈ B for all n ∈ N , lim ω B n ∈ B holds.Before proving of Theorem 1.6, we make a remark that “condition ( i ) implies theconclusion” has sprit of the original Shalom’s strategy [Sha4] to prove Theorem 1.1,and of an observation by M. Gromov. Precisely, for the proof of this part, oneessential part is to use reduced -1-cohomology. The original argument in [Sha2] usesconditionally negative definite functions, but Gromov [Gr] observed that it canbe done in terms of scaling limits on a metric space, which are special cases ofultraproducts for Banach spaces. For details, see [Mi, §
5] for instance.
Proof. ( Theorem 1.6 ) The proof follows from a combination of previous results in[Mi]. With keeping the same notation as in Theorem 1.6, we list necessary results:
Theorem 2.4. ([Mi, Theorem 1.3, Theorem 6.4])
Let B be a superreflexive spaceor a family of them. Suppose E ( A ) ⋉ A D A has relative property (T B ) . Thenfor any m ≥ , SL m ( A ) ⋉ A m > A m has relative property (F B ) . In fact, this pairhas relative property (FF B ) . Theorem 2.5. ([Mi, Theorem 5.5] , Shalom’s machinery ) Let B be a family ofsuperreflexive Banach spaces and n ≥ . Suppose SL n ( A ) > A n − has relativeproperty (F B ) , where A n − sits on a unipotent part, from (1 , n ) -th to ( n − , n ) -thentries. If B is stable under ultraproducts, then SL n ( A ) possesses property (F B ) . Proposition 2.6. ([Mi, Proposition 6.6])
Let B be a superreflexive Banach spaceor a family of them, and let n ≥ . Suppose SL n ( A ) > A n − has relative property (FF B ) , where A n − sits in the same way as Theorem 2.5. If SL n ( A ) moreoversatisfies property (T B ) , then SL n ( A ) possesses property (FF B ) / T . Indeed, for the last two of three, we need a quadruple (Λ , Λ ′ , H , H ) of a count-able discrete group Λ with finite abelianization and subgroups Λ ′ , H , H thereinsatisfying the following three conditions:(1) The group Λ is generated by H and H together.(2) The subgroup Λ ′ normalizes H and H .(3) The group Λ is boundedly generated by Λ ′ , H , and H .Here we say a subset S ⊂
Λ containing the unit e ∈ Λ boundedly generates agroup Λ if there exists N ∈ Z > such that S N = Λ holds (this equality means, any g ∈ Λ can be expressed as a product of N elements in S ). We warn that in some NIVERSAL LATTICES 7 other literature, the terminology bounded generation is used only for the following confined case: S is a finite union of cyclic subgroups of Λ. These properties relateto some forms of Shalom properties , which are used in the proofs of Theorem 2.5and Proposition 2.6. For more details, compare [Mi, Definition 5.4, Definition 6.5].We get these last two results stated above by letting (Λ , Λ ′ , H , H ) in the orig-inal statements be (SL n ( A ) , SL n − ( A ) , A n − , A n − ). Here Λ ′ sits in the left uppercorner of SL n ( A ); H ∼ = A n − sits in SL n ( A ) as a unipotent part, from (1 , n )-thto ( n − , n )-th entries; and H ∼ = A n − sits in SL n ( A ) as a unipotent part, from( n, n, n − i ) is fulfilled. Then by combiningTheorem 2.4 and Theorem 2.5, we obtain that SL n ≥ ( A ) has property (F B ) (notethat SL m ( A ) ⋉ A m naturally injects into SL m +1 ( A )). Since property (F B ) impliesproperty (T B ) (it is a general fact. See [BFGM, Theorem 1.3]), SL n ≥ ( A ) hasproperty (T B ). Then by Proposition 2.6, we obtain property (FF B ) / T as well.Secondly, we consider the case of that condition ( ii ) is satisfied. In general, wemay not apply Theorem 2.5. However in this case, we can first apply Theorem 2.4,and next appeal directly to Proposition 2.6. The point here is that since relative(T) implies relative (T B ), the pairs E ( A ) ⋉ A D A ; and SL n ( A ) D SL n ( A ) ( n ≥
3) have relative (T B ) (these respectively follow from [Sha1] and Theorem 1.1).Thus for SL n ≥ ( A ), we obtain property (FF B ) / T. Since SL n ≥ ( A ) has the trivialabelinanization, we get (F B ) as well. (cid:3) We mention Bader–Furman–Galander–Monod [BFGM, § L p ( µ ) ) for any σ -finite measure µ . Moreover, Heinrich[He] has proven the family L p satisfies condition ( i ). Thus although we saw property(F L p ) is stronger than property (T) for p large enough (recall this from the intro-duction), for universal lattices with n ≥ L p ) / T ([Mi,Theorem 1.5]). Also in [Mi, Remark 6.7], we have obtained (TT) / T (=(FF L ) / T)for SL n ≥ ( A ). Note that (TT) / T is strictly stronger than (T) because any non-elementary hyperbolic group, including one with (T), is known to admit an un-bounded quasi-cocycle into the left regular representation, see [MMS].In Section 3, we shall see the family C p satisfies condition ( ii ). On a family B having condition ( i ) but not satisfying condition ( ii ), see Section 5.3. Relative property (T) implies relative property (T C p )We refer to [PX] for comprehensive treatments on noncommutative L p -spacesincluding C p . First of all, the Clarkson-type inequality (for instance, see[PX, § C p for any p ( ∈ (1 , ∞ )) is uniformly convex and uniformly smooth.Before proceeding to the proof of Proposition 1.7, we shortly recall the strategyin [BFGM, § § L p ( µ ) ) for σ -finite measure space µ . The keys totheir proof are the following two tools:Tool 1. (The Mazur map : interpolation between L p -spaces, see [BL]) For p, r ∈ (1 , ∞ ) and σ -finite measure, the map M p,r : L p ( µ ) → L r ( µ ); M p,r ( f ) =sign( f ) ·| f | p/r is a (non-linear) map, and this induces a uniformly continuoushomeomorphism between the unit spheres M p,r : S ( L p ( µ )) → S ( L r ( µ )).Tool 2. (The Banach–Lamperti theorem : classification of linear isometries on an L p -space, see [FJ]) For any 1 < p < ∞ with p = 2, any linear isometry V of L p ( X, µ ) has the form
V f ( x ) = f ( F ( x )) h ( x ) (cid:18) dF ∗ µdµ ( x ) (cid:19) p , MASATO MIMURA where F is a measurable, measure class preserving map of a Borel space( X, µ ), and h is a measurable function with | h ( x ) | = 1 almost everywhere.Their proof goes as follows: suppose a group Λ does not have property (T L p ( µ ) ).Then there exists a (continuous) isometric linear representation ρ on B = L p ( µ )such that ρ ′ ≻ Λ (, namely, ρ ′ has almost invariant vectors). Here ρ ′ is therestriction of ρ on the subspace B ′ := B ′ ρ (Λ) , recall the definitions above fromSection 2. Through Tool 1, define π by π ( g ) = M p, ◦ ρ ( g ) ◦ M ,p ( g ∈ Λ). Thenthanks to Tool 2, one can show this π maps each g ∈ Λ to a linear (unitary) operatoron the Hilbert space H := L ( µ ). Thus one obtains the unitary representation π : Λ → U ( H ). Finally, by uniform continuity of the Mazur maps, it is not difficultto see that ρ ′ ≻ Λ implies π ′ ≻ Λ (, where π ′ is the restriction of π on theorthogonal complement of H π (Λ) ). This means Λ does not have property (T).Hence property (T) implies property (T L p ( µ ) ). Also in this argument one can leadthe same conclusion for relative properties. Proof. ( Proposition 1.7 ) Puschnigg [Pu] shown a noncommutative analogue of Tool1, for the case of p -Schatten class C p : Theorem 3.1. ( Puschnigg [Pu, Corollary 5.6])
Let < p, r < ∞ . Then the non-commutative Mazur map M p,r : S ( C p ) → S ( C r ); a u ·| a | p/r is a uniform continu-ous homeomorphims between unit spheres. Here a = u · | a | is a polar decomposition. In [Pu], he considered isometric linear representations on C p which come fromunitary representations, and hence he did not need an analogue of Tool 2. For ourcase, it is needed, and is obtained by J. Arazy [Ar]. Since our Hilbert space H isseparable, by choosing a countable orthonormal basis, we can identify H with asquare integrable sequence space ℓ . Through this identification, we can considerthe transpose map ; a T a on B ( H ) ∼ = B ( ℓ ). Although the transpose map dependson the choices of bases, it is easy to obtain the following facts: Lemma 3.2.
Stick to the setting in above. ( i ) The transpose map is a linear isometry on each C p . ( ii ) The transpose map is compatible with the adjoint operation. Namely, for any a ∈ B ( ℓ ) , ( T a ) ∗ = T ( a ∗ ) . ( iii ) For any a, b ∈ B ( ℓ ) , T ( ab ) = T b T a . ( iv ) If u is a unitary, then so is T u . ( v ) If t is positive and α > , then T t is positive and T ( t α ) = ( T t ) α . Now we state the theorem of Arazy:
Theorem 3.3. ( Arazy [Ar])
Let < p < ∞ with p = 2 and C p be the space of p -Schatten class operators on ℓ . Then every linear isometry V on C p is either ofthe following two forms: (1) there exist unitaries w, v ∈ U ( ℓ ) such that V : a wav ; (2) there exist unitaries w, v ∈ U ( ℓ ) such that V : a w T av . Thanks to the two theorems above, by following footsteps of Bader–Furman–Gelander–Monod we accomplish the conclusion. Indeed, for linearity of the com-position, let us take a linear isometry V on C p . By Lemma 3.2, for e V = M p, ◦ V ◦ M ,p : S ( C ) → S ( C ) we have the following (recall that one can take a partialisometry in polar decomposition as a unitary):( i ) in the case of (1), for any x ∈ C with a polar decomposition x = u | x | , apolar decomposition of wu | x | /p v is ( wuv )( v ∗ | x | /p v ). Therefore, we have e V · x = wuv ( v ∗ | x | /p v ) p/ = wuvv ∗ | x | v = wu | x | v = wxv. NIVERSAL LATTICES 9 ( ii ) in the case of (2), for any x ∈ C with a polar decomposition x = u | x | , apolar decomposition of w T ( u | x | /p ) v = w ( T | x | ) /p ( T u ) v is( w T uv ) { v ∗ ( T u ) ∗ ( T | x | ) /p ( T u ) v } . Therefore, we have e V · x = w T uv { v ∗ ( T u ) ∗ ( T | t | ) /p ( T u ) v } p/ = w T uvv ∗ ( T u ) ∗ ( T | x | ) T uv = w T | x | T uv = w T ( u | x | ) v = w ( T x ) v. Hence in both cases e V is linear. Now recall that C , the space of Hilbert–Schmidt operators, is indeed a Hilbert space equipped with the inner product: h x | y i := Tr( y ∗ x ) (this holds for noncommutative L spaces in general setting). (cid:3) Proposition 1.7 together with Theorem 1.6 immediately implies Theorem 1.4 foruniversal lattices and group quotients of them. In Section 4, we deal with the caseof finite index subgroups by utilizing p -inductions.4. Finite index subgroups, Lattices, and L p -inductions The p -induction of (quasi-)cocycles is given by Shalom [Sha3, §
3. III] for p = 2,and later by Bader–Furman–Gelander–Monod [BFGM, §
8] for general p .Let G be a (locally compact second countable) group, Γ G be a lattice, and D be a Borel fundamental domain for Γ (namely, D is a Borel subset of G such that G = F γ ∈ Γ D γ ). For the existence of such a domain, see [BHV, § B]. We let µ be aHaar measure of G with µ ( D ) = 1; and we identify D with G/ Γ and regard D as a(left) G -space. We define a Borel cocycle β : G × D → Γ by the following rule: β ( g, d ) = γ if and only if g − dγ ∈ D . Now for given isometric Γ-representation σ on a Banach space B and a (quasi-) σ -cocycle b : Γ → B , under some integrability condition in below, we can define the p -induction ˜ b of b as follows: • the p -induced Banach space is L p ( D , B ), the space of p -Bochner integrablefunctions; • the p -induced representation ρ = Ind G Γ σ in L p ( D , B ) is defined as follows:for g ∈ G , ξ ∈ L p ( D , B ) and d ∈ D , ρ ( g ) ξ ( d ) := σ ( β ( g, d )) ξ ( g − · d ); • we define the ˜ b : G → L p ( D , B ) as˜ b ( g )( d ) := b ( β ( g, d )) ( g ∈ G, d ∈ D ) , provided that ˜ b ( g ) is p -integrable for all g ∈ G . Then under the condition above, this ˜ b becomes a (quasi-)Ind G Γ σ -cocycle. Remark 4.1.
The p -induction procedure of (quasi-)cocycles requires the integrablecondition above, and in general it is a subtle problem to determine whether thisholds. However, it is known that this condition is satisfied for all p in the followingtwo cases:(1) if Γ is cocompact in G , in particular, if Γ is a finite index subgroup of (acountable) G , then for any D and any b the integrability condition is fulfilled;(2) if G is a semisimple algebraic group with each simple factor having a local rankat most 2, then there exists D such that for any b the integrability condition isfulfilled. For case (1), it is almost trivial to check the integrability. However for case (2), aproof of this fact is considerably involved: one needs a deep result on length func-tions on higher rank lattices [LMR], see [Sha2, §
2] how to deduce the integrability.Now we state the strategy to prove Theorem 1.4 for finite index subgroups ofuniversal lattices; and Theorem 1.8. At the beginning, we consider a general setting.Let G be a (locally compact and second countable) group and Γ be a lattice in G .Let σ be an isometric Γ-representation in C p . Take a Borel fundamental domain( D , µ ) of Γ and consider the p -induction (with the same p as C p ) ρ = Ind G Γ σ of σ on E = L p ( D , C p ). Note that E is identical to the following space: { x : D → C p : k x k pp := Z d ∈D Tr( | x ( d ) | p ) dµ < ∞} . We need to have analogues of Tool 1 and Tool 2 in Section 3. Note that E = L p ( D , C p ) can be seen as the noncommutative L p -space associated with the vonNeumann algebra L ∞ ( D , B ( H )) ∼ = L ∞ ( D ) ⊗ B ( H ) (acting on L ( D ) ⊗ H ) with thecanonical (normal faithful) semifinite trace τ = τ L ∞ ⊗ Tr: for x ∈ L ∞ ( D , B ( H ) + ), τ ( x ) := Z d ∈D Tr( x ( d )) dµ ∈ [0 , ∞ ] . For the definition of noncommutative L p -space L p ( A , τ ) associated with a semifinitevon Neumann algebra ( A , τ ), see [PX]. We use this symbol to distinguish it from L p ( D , C p ) above. For Tool 1, we utilize the following result of Puschnigg: Theorem 4.2. ( Puschnigg, [Pu, Corollary 5.7])
For < p, r < ∞ , the analogue ofthe Mazur map ( Theorem 3.1 )˜ M p,r : L p ( D , C p ) → L r ( D , C r ); x u · | x | p/r is a uniform continuous homeomorhism on unit spehres. Here x = u | x | is a polardecomposition. Note that the unifrom convexity and the uniform smoothness of E follow froma general theory, see [FP].For Tool 2, F. J. Yeadon has shown the following theorem on isometries in thenoncommutative L p -space associated with a semifinite von Neumann algebra: Theorem 4.3. ( Yeadon, [Ye, Theorem2])
Let A be a von Neumann algebra with afaithful semifinite normal trace τ and let L p ( A , τ ) be the associated noncommutative L p -space with p = 2 in (1 , ∞ ) . Suppose V is a linear isometry. Then there exist,uniquely, a partial isometry w ∈ A , a normal Jordan ∗ -monomorphism, and an ( unbounded ) positive self-adjoint operator N affiliated with A ∩ J ( A ) ′ such that • w ∗ w = J (1) = the support projection of N ; • for any x ∈ A + , τ ( x ) = τ ( N p J ( x )) ; • for any x ∈ A ∩ L p ( A , τ ) , V · x = wN J ( x ) . Here A ′ denotes the commutant of A , and a closed and densely defined operator L is said to be affiliated with a von Neumann algebra A (both acting on the sameHilbert space) if every unitary u in A ′ carries the domain of N , onto itself andsatisfies uN u ∗ = N there. A Jordan ∗ -monomorphism of a von Neumann algebra A is an injective linear map J : A → A which satisfies for any x ∈ A , J ( x ) = J ( x ) .With the aid this theorem, we claim the following: Proposition 4.4.
For a finite measure space D and p = 2 in (1 , ∞ ) , let V be alinear isometry on L p ( D , C p ) . Then the following composition ˜ V = ˜ M p, ◦ V ◦ ˜ M ,p is a linear isometry map on L ( D , C ) . NIVERSAL LATTICES 11
Note that L ( D , C ) is a Hilbert space (compare with Section 3). Proof. ( Proposition 4.4 ) For the proof, we need the following theorem of Stømeron structures of Jordan monomorphisms (note that e and f below are orthogonal: ef = f e = 0): Theorem 4.5. ( Stømer, [St, Lemma 3.2])
Let A , M be von Neumann algebras and J be a normal Jordan ∗ -monomorphism from A into M such that the von Neumannalgebra generated by J ( A ) equals M . Then there exist two central projections e and f in M with e + f = 1 M such that the map J : x J ( x ) e is a ∗ -homomorphismand that the map J : x J ( x ) f is an anti- ∗ -homomorphism. We apply Theorem 4.3 and Theorem 4.5 to our case: A = L ∞ ( D ) ⊗ B ( H ) withthe canonical trace τ . Thus for V , we obtain w , J , and N in the statement ofTheorem 4.3 and let q be the projection J (1 A ) ∈ A . Set M be the von Neumannalgebra generated by J ( A ) with putting 1 M = q , and through Theorem 4.5 weobtain e, f central in M . Take an arbitrary element x in L ( A , τ ) ∩ A , namely, x is an element in L ( D , C ) which satisfies ess . sup d ∈D k x ( d ) k < ∞ . Let x = u | x | bea polar decomposition of x with u ∈ A being a unitary. Then( V ◦ ˜ M ,p )( x ) = V · ( u | x | /p ) = wN J ( u | x | /p ) e + wN J ( u | x | /p ) f = wN J ( u ) J ( | x | ) /p e + wN J ( | x | ) /p J ( u ) f. Hence the following is a polar decomposition of ( V ◦ ˜ M ,p )( x ) with wJ ( u ) being apartial isometry:( V ◦ ˜ M ,p )( x ) = ( wJ ( u ) e + wJ ( u ) f )( N J ( | x | ) /p e + J ( u ) ∗ N J ( | x | ) /p J ( u ) f )= ( wJ ( u ))( N J ( | x | ) /p e + J ( u ) ∗ N J ( | x | ) /p J ( u ) f )Here we use the facts that J ( u ) is a partial isometry with J ( u ) ∗ J ( u ) = J ( u ) J ( u ) ∗ = q ; e, f are projections with e + f = q which are central in M ; and that N is affiliatedwith A ∩ M ′ = A ∩ J ( A ) ′ . Therefore, one has the following:( ˜ M p, ◦ V ◦ ˜ M ,p )( x ) = ( wJ ( u ))( N J ( | x | ) /p e + J ( u ) ∗ N J ( | x | ) /p J ( u ) f ) p/ = ( wJ ( u ) e + wJ ( u ) f )( N p/ J ( | x | ) e + J ( u ) ∗ N p/ J ( | x | ) J ( u ) f )= wN p/ J ( u ) J ( | x | ) e + wN p/ J ( | x | ) J ( u ) f = wN p/ J ( u | x | ) e + wN p/ J ( u | x | ) f = wN p/ J ( u | x | )= wN p/ J ( x ) . This means that ˜ M p, ◦ V ◦ ˜ M ,p is a linear 2-isomery at least from L ( A , τ ) ∩ A to L ( A , τ ). Since in our case L ( A , τ ) ∩ A is (2-)dense in L ( A , τ ), this map extendsto a linear isometry on L ( A , τ ), as desired. (cid:3) Theorem 4.2 together with Proposition 4.4 lead us to the following corollary(compare with Section 3):
Corollary 4.6.
Let D be a finite measure space and p = 2 in (1 , ∞ ) . Then amonglocally compact and second countable groups, relative property (T) implies relativeproperty (T L p ( D ,C p ) ) .Proof. ( Theorem 1.4 for finite index subgroups; and
Theorem 1.8 )Frist we prove Theorem 1.8. Recall the argument in [BFGM, §
5] of deducing(F L p ) from (strong) relative (T L p ) for higher rank algebraic groups (they utilizes thegeneralized Howe–Moore property, see [BFGM, § C p ) and (F L p ( D ,C p ) ) for higher rank algebraic groups. By p -induction process (it is possible, see (2) of Remark 4.1), thelater property implies (F C p ) for lattices therein, see [BFGM, § G = SL n ≥ ( A ) be the universal lattice and Γ G bea finite index subgroup. Suppose that Γ does not have (FF C p ) / T. Then thereexists an isometric Γ-representation σ on C p and a quasi- σ -cocycle b such that b ′ : Γ → ( C p ) ′ σ (Γ) is unbounded. Take a p -induction of b (it is possible in view ofRemark 4.1 (1)), and get the induced representation ρ = Ind G Γ σ and the inducedquasi- ρ -cocycle c = ˜ b in E = ℓ p ( D , C p ).Now observe that Theorem 1.6 and Corollary 4.6 prove that G has (FF E ) / T.This in particular implies the map c ′ : G → E ′ ρ ( G ) must be bounded. On the otherhand, by the construction of c , the restriction c ′ | Γ can be identified with the quasi- σ ′ -cocycle b ′ : Γ → ( C p ) ′ σ (Γ) and hence is unbounded. Since the measure space D consists of atoms, this forces c ′ to be unbounded. It is a contradiction. ThereforeΓ must have (FF C p ) / T, and also have (F C p ) because Γ has finite abelianization (forinstance, this follows from Theorem 1.1). (cid:3) Remark 4.7.
Induction of quasi-cocycles from a lattice to a locally compact (topo-logical) group is delicate in general. This is because for non-atomic finite measurespace, the unboundedness of some value does not necessarily imply the unbound-edness of L p -norm. However, Burger and Monod have overcame this difficulity byshowing the induced map H (Γ; B, σ ) → H ( G ; L p ( D , B ) , Ind G Γ σ ) is injective (here H • cb denotes the continuous bounded cohomology). For details, see [BM1], [BM2].5. Concluding remarksRemark 5.1.
We will explain an example of a family B of Banach spaces whichsatisfies condition ( i ) but fails to fulfill condition ( ii ) in Theorem 1.6. This exampleis employed for establishing property (F [ H ] ) for universal lattices in [Mi], where [ H ]denotes the family of Banach spaces Y which is isomorphic to a Hilbert space;namely, on which there exists a Hilbert norm k · k Hilb such that there exists C ≥ C − k · k Hilb ≤ k · k Y ≤ C k · k Hilb (the infimum of such C is called the normratio ). The family [ H ] is not stable, but for any M ≥
1, the family [ H ] M of any Y isomorphic to a Hilbert space with norm ratio ≤ M , is stable under ultraproducts.Theorem 1.3 in [Mi] states that E ( A ) ⋉ A D A has relative property (T [ H ] ), andthe family [ H ] M satisfies condition ( i ) of Theorem 1.6. Therefore SL n ≥ ( A ) hasproperty (F [ H ] M ) for any M . Thus property (F [ H ] ) for SL n ≥ ( A ) is also verified.Note that, by considering k · k Hilb , one can interpret property (F [ H ] ) as the fixedpoint property with respect to all affine uniformly bi-Lipschitz action on Hilbertspaces (similarly, property (T [ H ] ) is interpreted as a property in terms of uniformlybounded linear representations). We note that in [BFGM], property (F [ H ] ) and(T [ H ] ) are respectively called property ( F H ) and ( T H ).We also mention that B = [ H ] and B = [ H ] M for sufficiently large M do not satisfy condition ( ii ) in Theorem 1.6. This is due to an unpublished result of Shalomthat Sp n, and lattices therein fails to have property (T [ H ] ). Remark 5.2.
In the view of Theorem 4.3, our fixed point theorems (and (FF) / T-type property) can be extended, with some effort, to the cases of noncommutative L p -spaces associated with semifinite von Neumann algebras. For uniform continuityof the associated Mazur maps on unit spehres, one can utilize an inequality of Kosaki[Ko, Proposition 7]. Compare with the proof of [Pu, Corollary 5.6].We mention that even from type III von Neumann algebras (, namely, beyondsemifinite cases), one can construct the associated noncommutative L p -spaces. For NIVERSAL LATTICES 13 details, see [PX]. For classification of linear isometries on a general noncommutative L p -space, see [JRS] and [She].Finally, we warn that the class C p is not stable under ultraproducts. To havestability, we need to consider noncommutative L p -spaces associated with type III von Neumann algebras as well. For details, see [Ra].
Remark 5.3.
It is a problem of high interest to determine whether higher ranklattices and universal lattices have property (F B uc ) for B uc being the family of alluniformly convex Banach spaces. One of the main motivations for studying thisproblem is this relates to uniform (non-)embeddability of expander graphs, and thatrelates the coarse geometric Novikov conjecture [KY] and (a possible direction toconstruct a counterexample of the surjectivity-side of) the Baum–Connes conjecture[HLS]. There is a breakthrough by V. Lafforgue [La1], [La2], and his results implySL n ≥ ( F ) ( F is a non-archimedean local field) and cocompact lattices therein have(F B uc ). For archimedean local field cases or noncocompact lattice cases, it does notseem any result is known for this problem.In [BFGM], Bader–Furman–Gelander–Monod observed that for higher rank groupsand lattices, in order to verify property (F B uc ) it suffices to show that SL ( R ) ⋉R D R and symplectic version of this pair have relative property (T B uc ). Thanks toTheorem 1.6, we have an analogue of this observation for universal lattices; namely,the following:“ if E ( A ) ⋉ A D A has relative property (T B uc ) , then SL n ≥ ( A ) hasproperty (F B uc ) . ” The key fact is that the family of uniformly convex Banach spaces with uniform lower bounds for modulus of convexity is stable under ultraproducts[AK], and hence a similar argument to one in Remark 5.1 applies. acknowledgments The author thanks his supervisor Narutaka Ozawa, and Nicolas Monod for com-ments and conversations. He also thanks Takeshi Katsura and Hiroki Sako, whofirst raised a question on p -Schatten classes to the author. Finally, this work wascarried out during a long stay (from February, 2010 to January, 2011) of the au-thor at EPFL (´Ecole Polytechnique F´ed´erale de Lausanne), on the Excellent YoungResearcher Overseas Visiting Program by the Japan Society for the Promotion ofScience. The author is grateful to Professor Nicolas Monod and his secretary Mrs.Marcia Gouffon for their warmhearted hospitality to his stay at EPFL. References [AK] A.G. Aksoy, and M.A. Khamsi,
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