It is well-known that a finitely generated group $\Gamma$ has Kazhdan's
property (T) if and only if the Laplacian element $\Delta$ in ${\mathbb
R}[\Gamma]$ has a spectral gap. In this paper, we prove that this phenomenon is
witnessed in ${\mathbb R}[\Gamma]$. Namely, $\Gamma$ has property (T) if and
only if there are a constant $\kappa>0$ and a finite sequence $\xi_1,...,\xi_n$
in ${\mathbb R}[\Gamma]$ such that $\Delta^2-\kappa\Delta = \sum_i
\xi_i^*\xi_i$. This result suggests the possibility of finding new examples of
property (T) groups by solving equations in ${\mathbb R}[\Gamma]$, possibly
with an assist of computers.Comment: 6 pages; a few improvement (v2); update (v3

Ozawa, Narutaka

Journal of the Institute of Mathematics of Jussieu

English

arXiv

Kyoto University Research Information Repository

TitleNONCOMMUTATIVE REAL ALGEBRAIC GEOMETRYOF KAZHDAN’S PROPERTY (T)Author(s)Ozawa, NarutakaCitationJournal of the Institute of Mathematics of Jussieu (2016),15(01): 85-90Issue Date2016-01URL http://hdl.handle.net/2433/204543Right© Cambridge University Press 2014; This is the acceptedmanuscrip of the article is available athttp://dx.doi.org/10.1017/S1474748014000309Type Journal ArticleTextversionauthorKyoto UniversityNONCOMMUTATIVE REAL ALGEBRAIC GEOMETRY OFKAZHDAN’S PROPERTY (T)NARUTAKA OZAWAAbstract. It is well-known that a finitely generated group Γ has Kazhdan’s property(T) if and only if the Laplacian element ∆ in R[Γ] has a spectral gap. In this paper,we prove that this phenomenon is witnessed in R[Γ]. Namely, Γ has property (T) ifand only if there are a constant κ > 0 and a finite sequence ξ1, . . . , ξn in R[Γ] such that∆2 − κ∆ = ∑i ξ∗i ξi. This result suggests the possibility of finding new examples ofproperty (T) groups by solving equations in R[Γ], possibly with an assist of computers.1. IntroductionKazhdan’s property (T) is one of the two most important concepts in analytic grouptheory (the other being amenability). The mere existence of an infinite group withproperty (T) is surprising and has a wide range of applications. Therefore, it is notso surprising that any proof of property (T) for any infinite group is necessarily in-volved. See the monograph [BHV08] for the detailed treatment of property (T). Inthis paper, we investigate the noncommutative real algebraic geometric aspects (cf.[NT13, Oz13, Sc09]) of property (T) and see that property (T) is provable. Let Γ be afinitely generated group and µ be a finitely supported symmetric probability measureon Γ whose support generates Γ. Then the corresponding Laplacian element ∆µ is thepositive element in the real group algebra R[Γ], defined by∆µ =12∑x∈Γµ(x)(1− x)∗(1− x) = 1−∑x∈Γµ(x)x.It is well-known ([Va84, BHV08]) that the group Γ has property (T) if and only ifthe Laplacian element ∆µ has a spectral gap, i.e., there is κ > 0 such that for everyorthogonal Γ-representation pi on a real Hilbert space H the spectrum of pi(∆µ) ∈ B(H)is contained in {0} ∪ [κ,+∞). In turn, this is equivalent to that ∆2µ − κ∆µ ≥ 0 in thefull group C∗-algebra C∗Γ. See Chapter 5 in [BHV08] or Chapter 12 in [BO08] for theDate: 2014/01/08.2010 Mathematics Subject Classification. 16A27; 46L89, 22D10, 20F10.Key words and phrases. Kazhdan’s property (T), Positivstellensa¨tz, spectral gap.The author was partially supported by JSPS (23540233).12 NARUTAKA OZAWAproof of this fact. The main result of this paper is that the inequality ∆2µ− κ∆µ ≥ 0 iswitnessed in R[Γ].Main Theorem. Let Γ, µ, and ∆µ be as above. Then, the group Γ has Kazhdan’sproperty (T) if and only if there are a constant κ > 0 and a finite sequence ξ1, . . . , ξn inR[Γ] such that ∆2µ − κ∆µ =∑i ξ∗i ξi. Moreover, if Γ has property (T), µ takes rationalvalues, and κ > 0 is a rational number such that the spectrum of ∆µ in C∗Γ is containedin {0} ∪ (κ,+∞), then∆2µ − κ∆µ ∈ {n∑i=1ξ∗i ξi : n ∈ N, ξ1, . . . , ξn ∈ Q[Γ]}.The present work is motivated by the following two facts. The first is that an explicitformula for ∆2µ − κ∆µ =∑i ξ∗i ξi is known in some cases ([BS´97, Z˙u03], see Example 5below). The second is the following result of Shalom.Corollary (Shalom [Sh00]). Every property (T) group is a quotient of a finitely pre-sented property (T) group.Proof. Let Γ be a property (T) group (which is necessarily finitely generated) and takea symmetric finite generating subset S. Then, by Main Theorem, for the uniformprobability measure µ on S, there a constant κ > 0 and a finite sequence ξ1, . . . , ξn inR[Γ] such that ∆2µ − κ∆µ =∑i ξ∗i ξi. Let FS be the free group over S and Q : FS → Γbe the corresponding quotient map. Let µ˜ be the uniform probability measure on Swhich viewed as a subset of FS and ξ˜i ∈ R[FS] be lifts of ξi through Q. Thus, one hasQ(∆2µ˜ − κ∆µ˜ −∑i ξ˜∗i ξ˜i) = 0. Therefore for the finite subsetR = {x−1y ∈ FS : x, y ∈ supp(∆2µ˜ − κ∆µ˜ −∑i ξ˜∗i ξ˜i) such that Q(x) = Q(y)},the finitely presented group 〈S | R〉 has property (T), by Main Theorem again, and Γis a quotient of 〈S | R〉. Another consequence of Main Theorem is that property (T) is semidecidable, a factwhich is already observed by Silberman [Si11]. At this moment, the author is unableto provide new examples of property (T) groups, but only hopes this result be usefulin proving property (T) for some new cases, possibly with an assist of computers.Acknowledgment. The author has benefited from the conversations with M. Mimura,H. Sako, Y. Suzuki, and R. Willet, during the conference “Metric geometry and analy-sis” held in Kyoto University in December 2013.KAZHDAN’S PROPERTY (T) 32. PreliminaryIn this section, we review basic facts about convex geometry and their applicationto the study of group algebras. See Chapters II and III of [Ba02] or [Oz13] for theformer, and [NT13, Oz13, Sc09] for the latter. Let V be an R-vector space and C ⊂ Vbe a convex subset. An element c ∈ C is called an algebraic interior point of C if forevery v ∈ V there is t ∈ (0, 1] such that (1 − t)c + tv ∈ C. Let int(C) denote theset of algebraic interior points of C. Notice that for every c ∈ int(C) and w ∈ C, onehas tc + (1 − t)w ∈ int(C) for every t ∈ (0, 1]. In particular, int(int(C)) = int(C)for every convex subset C. We equip V with a locally convex topology, called thealgebraic topology (also known as the finest locally convex topology), by declaring thatany convex set C such that C = int(C) is open. Then, every linear functional on V iscontinuous and every linear subspace of V is closed.Now we consider the case where the R-vector space V is equipped with a linearinvolution ∗ and the subspace V h = {v ∈ V : v = v∗} of the hermitian elements hasa distinguished cone V + ⊂ V h. A linear functional φ : V → R is said to be positive ifφ(v∗) = φ(v) and φ(V +) ⊂ R≥0. We note that every linear functional on V h uniquelyextends to an hermitian linear functional on V . An element e ∈ V + is called an orderunit if for every v ∈ V h there is R > 0 such that v + Re ∈ V +. Thus order units arenothing but algebraic interior points of V + in V h. By the Hahn–Banach separationtheorem (also known as the Eidelheit–Kakutani theorem in this context), one has forevery order unit e ∈ V + thatV + = {v ∈ V h : φ(v) ≥ 0 for all positive linear functionals φ on V }= {v ∈ V h : v + e ∈ V + for every  > 0}.Moreover, one has int(V +) = int(V +) whenever int(V +) 6= ∅.Next, we recall Positivstellensa¨tz for group algebras. Let Γ be a discrete groupand R[Γ] be the real group algebra. A typical element in R[Γ] will be denoted byξ =∑x∈Γ ξ(x)x, where ξ : Γ→ R is a finitely supported function. The unit of Γ, as wellas the unit of R[Γ], is simply denoted by 1. The ∗-operation is given by ξ∗(x) = ξ(x−1),and the positive cone of R[Γ]h is given by the sums of hermitian squaresΣ2R[Γ] = {n∑i=1ξ∗i ξi : n ∈ N, ξ1, . . . , ξn ∈ R[Γ]} ⊂ R[Γ]h.Then, 1 is an order unit for R[Γ] and the algebra R[Γ], together with the ∗-positivecone Σ2R[Γ], is a semi-pre-C∗-algebra in the sense of [Oz13], and Positivstellensa¨tz (seeCorollary 2 in [Oz13] or Proposition 15 in [Sc09]) states the following.Theorem 1. For ξ ∈ R[Γ]h, the following are equivalent.4 NARUTAKA OZAWA(1) pi(ξ) ≥ 0 for every orthogonal Γ-representation pi.(2) ξ ∈ Σ2R[Γ], i.e., for every  > 0 one has ξ + 1 ∈ Σ2R[Γ].Here, by pi(ξ) ≥ 0 we mean that the self-adjoint operator pi(ξ) is positive semi-definite.It follows that for ∆µ as in Introduction, the group Γ has property (T) if and only ifthere is a constant κ > 0 such that for every  > 0 one has ∆2µ − κ∆µ + 1 ∈ Σ2R[Γ].However, this characterization is not very satisfactory, because for a given κ > 0 onehas to solve infinitely many equations to verify property (T) of Γ.3. Proof of Main TheoremLet I[Γ] = {ξ ∈ R[Γ] : ∑x ξ(x) = 0} be the augmented ideal of the group algebraR[Γ]. We note that I[Γ] is spanned by {1 − x : x ∈ Γ} and that two natural positivecones of I[Γ]h coincide: I[Γ]∩Σ2R[Γ] = Σ2I[Γ] (cf. Lemma 4.5 in [NT13]). The reasonwe look at I[Γ] is that the Laplacian element ∆µ, defined in Introduction, belongs toΣ2I[Γ]. In fact, it is an order unit.Lemma 2. The Laplacian element ∆µ is an order unit for I[Γ]. In particular,Σ2I[Γ] = {ξ ∈ I[Γ]h : ξ + ∆µ ∈ Σ2I[Γ] for every  > 0}.Proof. For ξ, η ∈ I[Γ]h, we write ξ ≤ η if η − ξ ∈ Σ2I[Γ]. Since I[Γ]h is spanned by2− x− x−1 = (1− x)∗(1− x), x ∈ Γ, it suffices to show{x ∈ Γ : ∃R > 0 such that (1− x)∗(1− x) ≤ R∆µ} = Γ.It is clear that the left hand side contains the support of µ. Also since(1− xy)∗(1− xy) = (1− x+ x(1− y))∗(1− x+ x(1− y))≤ 2(1− x)∗(1− x) + 2(1− y)∗(1− y),it forms a subgroup and hence is equal to Γ. In general for a subspace W ⊂ V , the inclusion W ∩ V + ⊂ W ∩ V + can be strict.Here we prove that the equality holds for the case I[Γ] ⊂ R[Γ].Lemma 3. For every positive linear functional φ : I[Γ] → R, there is a sequence ofpositive linear functionals φn : R[Γ] → R such that φn → φ pointwise on I[Γ]. Inparticular, one has Σ2I[Γ] = I[Γ] ∩ Σ2R[Γ] in R[Γ].Proof. We use the theory of conditionally negative type functions, for which we referthe reader to Appendix C in [BHV08] or Appendix D in [BO08]. Let a positive linearfunctional φ : I[Γ]→ R be given. We claim that ψ(x) = φ(1−x) defines a conditionallynegative type function on Γ. Indeed, for every ξ ∈ I[Γ], one has∑x,y∈Γψ(y−1x)ξ(y)ξ(x) = −φ(ξ∗ξ) ≤ 0.KAZHDAN’S PROPERTY (T) 5(In case Γ has property (T), the conditionally negative type function ψ is bounded andso it is of the form ‖v − pi(x)v‖ for some orthogonal representation pi and a vector v,which implies that φ extends to a positive linear functional on R[Γ].) It follows fromSchoenberg’s theorem that the functions φt(x) := exp(−tψ(x)) are of positive type forall t ∈ R≥0, i.e., they extend to positive linear functionals on R[Γ]. Sincelimt→0t−1φt(1− x) = limt→01− exp(−tφ(1− x))t= φ(1− x)for every x ∈ Γ, one has t−1φt → φ on I[Γ]. The second assertion follows from this. Combining the above lemmas with Theorem 1, we arrive at the following.Proposition 4. For every ξ ∈ I[Γ]h, the following are equivalent.(1) pi(ξ) ≥ 0 for every orthogonal Γ-representation pi.(2) ξ ∈ Σ2I[Γ], i.e., ξ + ∆µ ∈ Σ2I[Γ] for every  > 0.Proof of Main Theorem. As it is noted in the introduction, the group Γ has property(T) if and only if there is κ′ > 0 such that ∆2µ − κ′∆µ ∈ Σ2R[Γ]. Thus, the ‘if’ part ofthe assertion follows. For the other direction, let us assume that Γ has property (T).Then, Lemmas 3 and 2 imply that ∆2µ − (κ′ − )∆µ ∈ Σ2I[Γ] for every  > 0. Taking < κ′ and letting κ = κ′ − , we are done. Indeed, this shows that ∆2µ − κ∆µ is aninterior point of Σ2I[Γ] (inside I[Γ]h). Hence, it also belongs to the sub-cone{n∑i=1riξ∗i ξi : n ∈ N, ri ∈ R+, ξ1, . . . , ξn ∈ Q[Γ] ∩ I[Γ]}which is dense and has non-empty interior (see the proof of Lemma 2). Now, if µ andκ are rational, then for a given finite sequence ξ1, . . . , ξn ∈ Q[Γ], the linear equation∆2µ − κ∆µ =∑ni=1 riξ∗i ξi in (ri)ni=1 has a positive real solution if and only if it has apositive rational solution. Examples for which an explicit formula ∆2µ − κ∆µ =∑i ξ∗i ξi is known are providedby [BS´97, Z˙u03] (see also Theorem 5.6.1 in [BHV08] and Theorem 12.1.15 in [BO08]).Example 5. Let Γ be a group generated by a finite symmetric set S such that 1 /∈ S.The link is the graph with the vertex set S and the edge set E := {(x, y) : x−1y ∈ S}.We put on S the probability measure µ(x) = |{y ∈ S : (x, y) ∈ E}|/|E|, and on E theuniform probability measure. Define d : L2(S, µ)→ L2(E) by (dξ)(x, y) = ξ(y)− ξ(x),and let Λ = d∗d/2 be the Laplacian operator on L2(S, µ). If the link graph is connectedand the spectrum of Λ is contained in {0}∪ [λ,+∞), then there are ξi, i ∈ S, such that∆µ − (2 − λ−1)∆µ =∑i ξ∗i ξi. Hence, Γ has property (T) if λ > 1/2. Indeed, for theorthogonal projection P from L2(S, µ) onto the one dimensional subspace of constant6 NARUTAKA OZAWAfunctions, one has λ−1Λ + P − I ≥ 0 and so there is an operator T on L2(S, µ) suchthat λ−1Λ + P − I = T ∗T . Thus, by writing Ax,y = 〈Aδy, δx〉 for each operator A onL2(S, µ), one has∑x,y∈S2(λ−1Λx,y + Px,y − Ix,y)x−1y =∑x,y∈S2∑i∈Sµ(i)−1Ti,xTi,yx−1y =∑i∈Sξ∗i ξi,where ξi = µ(i)−1/2∑x Ti,xx ∈ R[Γ]. But since Λx,x = µ(x), Λx,y = |E|−1 for (x, y) ∈ E,and Λx,y = 0 otherwise; Px,y = µ(x)µ(y); and Ix,y = δx,yµ(x); the left hand side isλ−1(1− 1|E|∑(x,y)∈Ex−1y)+(∑x,y∈Sµ(x)µ(y)x−1y)− 1= λ−1(1−∑xµ(x)x)+(∑xµ(x)x)2 − 1= ∆2µ − (2− λ−1)∆µ.References[BS´97] W. Ballmann and J. S´wia‘tkowski; On L2-cohomology and property (T) for automorphismgroups of polyhedral cell complexes. Geom. Funct. Anal. 7 (1997), 615–645.[Ba02] A. Barvinok; A course in convexity. Graduate Studies in Mathematics, 54. American Math-ematical Society, Providence, RI, 2002. x+366 pp.[BHV08] B. Bekka, P. de la Harpe, and A. Valette; Kazhdan’s property (T). New Mathematical Mono-graphs, 11. Cambridge University Press, Cambridge, 2008. xiv+472 pp.[BO08] N. Brown and N. Ozawa; C∗-algebras and Finite-Dimensional Approximations. GraduateStudies in Mathematics, 88. American Mathematical Society, Providence, RI, 2008.[NT13] T. Netzer and A. Thom; Real closed separation theorems and applications to group algebras.Pacific J. Math. 263 (2013), 435–452.[Oz13] N. Ozawa; About the Connes embedding conjecture: algebraic approaches. Jpn. J. Math. 8(2013), 147–183.[Sc09] K. Schmu¨dgen; Noncommutative real algebraic geometry - some basic concepts and first ideas.Emerging applications of algebraic geometry, 325–350, IMA Vol. Math. Appl., 149, Springer,New York, 2009.[Sh00] Y. Shalom; Rigidity of commensurators and irreducible lattices. Invent. Math. 141 (2000),1–54.[Si11] L. Silberman; Lectures at “Metric geometry, algorithms and groups” at IHP in 2011.http://metric2011.wordpress.com/tag/lior-silbermans-lectures/[Va84] A. Valette; Minimal projections, integrable representations and property (T). Arch. Math.(Basel) 43 (1984), 397–406.[Z˙u03] A. Z˙uk; Property (T) and Kazhdan constants for discrete groups. Geom. Funct. Anal. 13(2003), 643–670.RIMS, Kyoto University, 606-8502 JapanE-mail address: narutaka@kurims.kyoto-u.ac.jp

Noncommutative real algebraic geometry of Kazhdan's property (T)

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