On Almost Representations of Property (T) Groups

ON ALMOST REPRESENTATIONS OF PROPERTY (T) GROUPS

arXiv:0708.1290v2 [math.OA] 12 Aug 2007

V. M. MANUILOV AND CHAO YOU

Abstract. Property (T) for groups means a dichotomy: a representation either has an invariant vector or all vectors are far from being invariant. We show that, under a stronger ˙ condition of Zuk, a similar dichotomy holds for almost representations as well.

1. Introduction Let Γ be a group generated by a symmetric finite set S and let π : Γ → U(Hπ ) be a unitary representation of Γ. Suppose that Γ has the property (T) of Kazhdan (i.e. the trivial representation is isolated in the dual space of Γ). We refer to [3] for P basic information 1 about property (T). It is well known [4] that the spectrum of π(x) = |S| s∈Γ π(s) has a gap near 1: Sp(π(x)) ⊂ [−1, 1 − c] ∪ {1},

where c is the Kazhdan constant for Γ (with respect to S). In terms of the group C ∗ -algebra, this means that we can apply function f such that f (1) = 1 and f (t) = 0 for any P a continuous 1 ∗ g ∈ C (Γ) to obtain the canonical projection p = f (x) ∈ C ∗ (Γ) t ∈ [−1, 1−c] to x = |S| g∈S corresponding to the trivial representation [10]. Our aim is to generalize the above property for the case of almost representations of Γ. Recall that, for ε ≥ 0, an ε-almost representation π of Γ (with respect to the given set S of generators) is the map π : S → U(Hπ ) satisfying kπ(s1 s2 ) − π(s1 )π(s2 )k ≤ ε

for any s1 , s2 , s1 s2 ∈ S and π(s−1 ) = π(s)∗ for any s ∈ S. This definition appeared in [1] and then (in a slightly different form) in [7]. If ε = 0 (in the case, when Γ is finitely presented and S is sufficiently big) then a 0-almost representation obviously generates a genuine representation of Γ. It is known that for some applications it suffices for π to be defined on S only instead of the whole Γ. Any small perturbation of a genuine representation is an almost representation, but there exist almost representations that are far from any genuine representation [11]. One should distinguish almost representations from other group ‘almost’ notions, e.g. quasi-representations, almost actions etc. [9, 2], which are completely different. If we have an asymptotic representation (i.e. a continuous family of εt -almost representations (πt )t∈[0,∞) with limt→∞ εt = 0) then it follows from the theory of C ∗ -algebra asymptotic homomorphisms that the spectrum of πt (x) has a gap for t sufficiently great: there is a continuous function α = α(t) > 0 such that limt→∞ α(t) = 0 and Sp(πt (x)) ⊂ [−1, 1 − c + α(t)] ∪ [1 − α(t), 1]. Unfortunately, if we are interested in a single almost representation, it may be impossible to include an almost representation into an asymptotic one [6], and we don’t know how to check existence of a spectral gap because there is no nice formula for the projection p. The first named author was partially supported by RFFI grant 05-01-00923. 1

2

V. M. MANUILOV AND CHAO YOU

Nevertheless, there is a condition, which is only slightly stronger than the property (T) and which provides a gap in Sp(π(x)) for an almost representation π. The importance of ˙ this condition was discovered by A. Zuk [12]. Let us recall his construction. It is supposed that the neutral element doesn’t belong to S. A finite graph L(S) is assigned to the set S of generators as follows: the set of vertices of L(S) is S and the set T of edges of L(S) is the set of all pairs {(s, s′) : s, s′ , s−1 s′ ∈ S}. By including some additional elements in S, one can assume that the graph L(S) is connected [12]. For a vertex s ∈ L(S), let deg(s) denote its degree, i.e. the number of edges adjacent to s. Let ∆ be a discrete Laplace operator acting on functions f defined on vertices of L(S) by 1 X f (s′ ), (1) ∆f (s) = f (s) − deg(s) ′ s ∼s

where s′ ∼ s means that the vertex s′ is adjacent to the vertex s. Operator ∆ is a nonnegative, self-adjoint operator on the (finitedimensional) Hilbert space l2 (L(S), deg) and zero is a simple eigenvalue of ∆. Let l1 (L(S)) be the smallest non-zero eigenvalue of ∆. We say ˙ that a group Γ with the generating set S satisfies the Zuk’s condition if l1 (L(S)) > 21 . One ˙ of the main results of [12]claims that the Zuk’s condition implies property (T) with the 1 ˙ . We appreciate Zuk’s approach because it allows Kazhdan constant c = √23 2 − λ1 (L(S)) to work with almost representations as well. The main result of this paper is the following statement: ˙ Theorem 1. Let Γ, S satisfy the Zuk’s condition and let c be as above. There is a continuous function α = α(ε) ≥ 0 such that α(0) = 0 and  1 X  Sp π(s) ⊂ [−1, 1 − c/2 + α(ε)] ∪ [1 − α(ε), 1] |S| s∈S for any ε-almost representation π. Corollary 2. For any ε-almost representation π there exists an (ε + 6|S|α(ε))-almost representation π ′ such that kπ ′ (s) − π(s)k ≤ 3|S|α(ε) for any s ∈ S and π ′ = τ ⊕ σ, where τ is a trivial representation and σ is an (ε + 6|S|α(ε))-almost representation satisfying  1 X  Sp σ(s) ⊂ [−1, 1 − c/2 + (1 + 3|S|)α(ε)]. (2) |S| s∈S P 1 Proof. Let H ⊂ Hπ be the range of the spectral projection of |S| s∈S π(s) corresponding to the set [1 − c + α(ε), 1]. Then kπ(s)ξ − ξk ≤ |S|α(ε)kξk for any s ∈ S and for any ξ ∈ H and B ) with respect to the decomposition H ⊕ H ⊥ then kBk ≤ if we write π(s) as a matrix ( CA D |S|α(ε) and kCk ≤ |S|α(ε), hence there exists a unitary D ′ such that kD ′ − Dk ≤ 2|S|α(ε). Put π ′ (s) = ( 10 D0′ ). Then kπ ′ (s) − π(s)k ≤ 3|S|α(ε) and π ′ is obviously an (ε + 6|S|α(ε))⊥ almost representation, for all s ∈ S. Since
P which is trivial ⊥on H. Hence1 H P is π(s)-invariant 1 the restriction of |S| s∈S π(s) onto H satisfies |S| s∈S π(s) |H ⊥ ≤ 1 − c/2 + α(ε), we get (2).  The remaining part of the paper is devoted to the proof of Theorem 1. The proof follows ˙ the proof of Zuk for genuine representations, but has additional argument because relations for almost representations do not hold exactly, but only approximately. It will be seen from the proof that one can take α(ε) = O(ε2/5 ) in Theorem 1.

ON ALMOST REPRESENTATIONS OF PROPERTY (T) GROUPS

3

2. Proof of the theorem The following Hilbert spaces and operators are defined exactly as in [12]: It doesn’t matter that π is not a representation here. Definition 3 ([12]). For r = 0, 1 and 2 let C r be the Hilbert spaces defined as follows: C 0 = {u : u ∈ Hπ }; hu, wiC 0 = hu, wiHπ |T | for u, w ∈ C 0 ;

C 1 = {f : S → Hπ : f (s−1 ) = −π(s−1 )f (s) for all s ∈ S}; hf, giC 1 = X

C 2 = {g : T → Hπ }; hf, giC 2 =

t∈T

X s∈S

hf (s), g(s)iHπ n(s);

hf (t), g(t)iHπ ,

where n(s) = #{s′ ∈ S : (s, s′ ) ∈ T }. Since the graph L(S) is connected, n(s) P> 0 for every s ∈ S and n(s) = deg(s). Moreover, −1 it is easy to see that n(s) = n(s ) and s∈S n(s) = |T |. Definition 4 ([12]). Let us define linear operators d1 : C 0 → C 1 and d2 : C 1 → C 2 as follows: d1 u(s) = π(s)u − u, for all u ∈ C 0 ; d2 f ((s, s′ )) = f (s) − f (s′ ) + π(s)f (s−1 s′ ), for all f ∈ C 1 . P Lemma 5 ([12]). One has d∗1 f = −2 s∈S f (s) n(s) for any f ∈ C 1 and kd∗1 kC 1 →C 0 ≤ 2. |T | In [12] it is shown that d2 d1 = 0 for any unitary representation. However, if π is only an almost representation then this doesn’t hold any more. One can only show that this composition is small. Lemma 6. For any u ∈ C 0 and (s, s′ ) ∈ T one has kd2 d1 u((s, s′ ))kHπ ≤ ε kukHπ

(3)

Proof. By the definitions of d1 and d2 , we have

kd2 d1 u((s, s′))kHπ = d1 u(s) − d1 u(s′) + π(s)d1u(s−1 s′ ) Hπ

= (π(s)u − u) − (π(s′ )u − u) + π(s)(π(s−1 s′ )u − u) Hπ

= π(s′ )u − π(s)π(s−1 s′ )u Hπ ≤ ε kukHπ ,

hence we have kd2 d1 u((s, s′ ))kHπ ≤ ε kukHπ . Corollary 7. kd2 d1 kC 2 →C 0 ≤ ε.

That’s why we have to introduce two more (sub)spaces. Definition 8. For any β ≥ 0 set B 0 (β) = {PΩ (d∗1 d1 )(u) : u ∈ C 0 } ⊂ C 0 ,

B 1 (β) = {d1 u : u ∈ B 0 (β)} ⊂ C 1 ,

where PΩ is the spectral projection corresponding to Ω = [β, +∞). It is clear that B 0 (β) and B 1 (β) are invariant subspaces for d∗1 d1 and d1 d∗1 respectively.



4

V. M. MANUILOV AND CHAO YOU 2

δ Proposition 9. If there exists c > 0 and 0 < δ < c/2 such that for every f ∈ B 1 ( |T ) |

hd1 d∗1 f, f iC 1 > chf, f iC 1

(4)

kπ(s)u − ukHπ < δ kukHπ for any s ∈ S

(5)

max kπ(s)u − ukHπ ≥ c/2 kukHπ

(6)

then, for any ε-almost representation π, either there exists u ∈ C 0 such that or s∈S

for every u ∈ C 0 . 2

δ ) = C 0 . If this is Proof. First, we show that if there is no u ∈ C 0 satisfying (5) then B 0 ( |T |

∗

δ2

d d1 u ⊥ 0 < ). Since not true then there exists a non-zero vector u⊥ orthogonal to B 0 ( |T 1

⊥ ∗|



C δ2 ⊥ δ 2 ⊥ 2 ⊥ ⊥ ∗ ⊥ ⊥ ⊥

u C 0 , we have hd1 u , d1 u iC 1 = hu , d1 d1 u iC 0 ≤ u C 0 d1 d1 u C 0 < |T | u C 0 , |T |

which implies that d1 u⊥ C 1 < √δ u⊥ C 0 . By definition of k·kC 1 , it is easy to see that |T | p ⊥





π(s)u⊥ − u⊥ ≤ d1 u⊥ 1 < √δ

u = δ u⊥ for any s ∈ S, which is in |T | Hπ C Hπ Hπ |T |

contradiction with the assumption. δ2 δ2 ) → B 1 ( |T ) has a bounded Next we prove that (4) implies that the operator d1 d∗1 : B 1 ( |T | | 2

2

2

δ δ δ )) is closed in B 1 ( |T ). If d1 d∗1 (B 1 ( |T )) were different from inverse. By (4), d1 d∗1 (B 1 ( |T | | | 2

2

2

δ δ δ B 1 ( |T ), there would exist a non-zero vector f ∈ B 1 ( |T ) orthogonal to d1 d∗1 (B 1 ( |T )). Then | | | we would have, by (4), 0 = hf, d1 d∗1 (f )iC 1 > chf, f iC 1

which is a contradiction. δ2 δ2 δ2 δ2 ) → B 1 ( |T ) has a bounded inverse (d1 d∗1 )−1 : B 1 ( |T ) → B 1 ( |T ) and Thus d1 d∗1 : B 1 ( |T | | | | −1 ∗ −1 k(d1 d1 ) kB1 ( δ2 )→B1 ( δ2 ) ≤ c . |T |

|T |

Now suppose that neither (5) nor (6) holds. Then there is some γ < c/2 and some u ∈ Hπ such that kukHπ = 1 and kπ(s)u − ukHπ ≤ γ for every s ∈ S. Therefore X X X kd1 uk2B1 ( δ2 ) = kd1 u(s)k2Hπ n(s) = kπ(s)u − uk2Hπ n(s) ≤ γ 2 n(s) = γ 2 |T | |T |

s∈S

s∈S

which gives kd1 ukB1 ( δ2 ) ≤ γ |T |

∗

d (d1 d∗ )−1 d1 u 1

1

C0

p

s∈S

|T |. Then

≤ kd∗1 kB1 ( δ2 )→C 0 · (d1 d∗1 )−1 B1 ( δ2 )→B1 ( δ2 ) · kd1 ukB1 ( δ2 ) |T | |T | |T | |T | p p −1 ≤ 2 · c · γ |T | < |T |.

By definition of the norm in C 0 one has then d∗1 (d1 d∗1 )−1 d1 u = u′ , whence ku′kHπ < 1, so the vector u − u′ is non-zero. Finally, d1 (u − u′ ) = d1 u − d1 (d∗1 (d1 d∗1 )−1 )d1 u = d1 u − d1 u,

which means that, for every s ∈ S,

π(s)(u − u′ ) − (u − u′) = 0.

Thus u − u′ is a non-zero invariant vector and (5) holds, which gives a contradiction.



ON ALMOST REPRESENTATIONS OF PROPERTY (T) GROUPS

5

Following [12], define the operator D : C 1 → C 2 by

Df ((s1 , s2 )) = f (s1 ) − f (s2 ),

where f ∈ C 1 and (s1 , s2 ) ∈ T . In [12], the relation between d2 and D was investigated and it was shown that 1 hd2 f, d2 f iC 2 = hDf, Df iC 2 − hf, f iC 1 . Since here an almost representation is engaged, 3 we have to estimate the difference between the left and the right hand side. Lemma 10. For every f ∈ C 1 one has X hf (s−1 s′ ), f (s−1 s′ )iHπ ; hf, f iC 1 =

(7)

(s,s′ )∈T

kd2 f ((s, s′ )) + d2 f ((s′ , s))kHπ ≤ ε f ((s′ )−1 s) Hπ ;

d2 f ((s, s′)) = −π(s)d2 f ((s−1, s−1 s′ )); X X hd2 f ((s, s′ )), −f (s′ )iHπ ; hd2 f ((s−1 , s−1 s′ )), −f (s−1 s′ )iHπ =

(8) (9) (10)

(s,s′ )∈T

(s,s′ )∈T



d2 f ((s, s′)) − π(s′ )d2 f (((s′ )−1 , (s′ )−1 s)) ≤ ε f ((s′ )−1 s) ; Hπ Hπ 5 X 1 hd2 f ((s, s′)), π(s)f (s−1s′ )iHπ − hd2 f, d2 f iC 2 < ε kf k2C 1 3 3 ′

(11) (12)

(s,s )∈T

Proof. The proof of (7) and (9) in [12] doesn’t depend on the property of π being a representation. d2 f ((s, s′)) = = = = hence

f (s) − f (s′ ) + π(s)f (s−1 s′ ) −(f (s′ ) − f (s) + π(s)π(s−1 s′ )f ((s′ )−1 s)) −(f (s′ ) − f (s) + π(s′ )f ((s′ )−1 s) + π(s′ )f ((s′ )−1 s) − π(s)π(s−1 s′ )f ((s′ )−1 s)) −d2 f ((s′ , s)) + (π(s′ )f ((s′ )−1 s) − π(s)π(s−1 s′ )f ((s′ )−1 s)),

kd2 f ((s, s′ )) + d2 f ((s′, s))kHπ = π(s′ )f ((s′ )−1 s) − π(s)π(s−1 s′ )f ((s′ )−1 s) Hπ

≤ ε f ((s′ )−1 s) , Hπ

which proves (8). By (8) and (9),

d2 f ((s, s′)) − π(s′ )d2 f (((s′ )−1 , (s′ )−1 s)) ′

′

Hπ ′

= kd2 f ((s, s )) + d2 f ((s , s)) − d2 f ((s , s)) − π(s′ )d2 f (((s′ )−1 , (s′ )−1 s))kHπ

≤ kd2 f ((s, s′)) + d2 f ((s′ , s))kHπ ≤ ε f ((s′ )−1 s) Hπ ,

which proves (11). Consider the mapping M : T → T , M((s, s′ )) = (s−1 , s−1 s′ ) = (t, t′ ). Then it is easy to see M is a bijection. Hence, X X hdf ((t, t′)), −f (t′ )iHπ , hdf ((s−1, s−1 s′ )), −f (s−1 s′ )iHπ = (s,s′ )∈T

(t,t′ )∈T

which is just a matter of notation. This proves (10).

6

V. M. MANUILOV AND CHAO YOU

X

(s,s′ )∈T

hd2f ((s, s′ )), π(s)f (s−1s′ )iHπ

1 X (hd2 f ((s, s′)), π(s)f (s−1s′ )iHπ + h−π(s)d2 f ((s−1 , s−1 s′ )), π(s)f (s−1s′ )iHπ 3 ′

=

(s,s )∈T

+hπ(s′)d2 f (((s′ )−1 , (s′ )−1 s)), π(s)f (s−1s′ )iHπ ) + D1 1 X (hd2 f ((s, s′)), π(s)f (s−1s′ )iHπ + hd2 f ((s−1 , s−1 s′ )), −f (s−1 s′ )iHπ = 3 ′ (s,s )∈T

+hd2f (((s′ )−1 , (s′ )−1 s)), π((s′ )−1 s)f (s−1 s′ )iHπ ) + D1 + D2 1 X = (hd2 f ((s, s′)), π(s)f (s−1s′ )iHπ + hd2 f ((s, s′)), −f (s′ )iHπ 3 ′ (s,s )∈T

+hd2f (((s′ )−1 , (s′ )−1 s)), −f ((s′ )−1 s)iHπ ) + D1 + D2 1 X (hd2 f ((s, s′)), π(s)f (s−1s′ )iHπ + hd2 f ((s, s′)), −f (s′ ) = 3 ′ (s,s )∈T

+hd2f ((s′ , s)), −f (s)iHπ ) + D1 + D2 1 X = (hd2 f ((s, s′)), π(s)f (s−1s′ )iHπ + hd2 f ((s, s′)), −f (s′ )iHπ 3 ′ (s,s )∈T

+hd2f ((s, s′ )), f (s)iHπ ) + D1 + D2 + D3 1 X (hd2 f ((s, s′)), f (s) − f (s′ ) + π(s)f (s−1 s′ )iHπ + D1 + D2 + D3 = 3 ′ (s,s )∈T

1 X (hd2 f ((s, s′)), d2 f ((s, s′))iHπ + D1 + D2 + D3 3 ′

=

(s,s )∈T

1 X hd2 f, d2 f iC 2 + D1 + D2 + D3 , 3 ′

=

(s,s )∈T

where D1 =

1 X hd2f ((s, s′ )) − π(s′ )d2 f (((s′)−1 , (s′ )−1 s)), π(s)f (s−1s′ )iHπ , 3 ′ (s,s )∈T

D2 =

1 X hd2f (((s′ )−1 , (s′ )−1 s)), (π((s′)−1 )π(s) − π((s′ )−1 s))f (s−1 s′ )iHπ , 3 ′ (s,s )∈T

D3 =

1 X hd2f ((s, s′ )) + d2 f ((s′ , s)), −f (s)iHπ , 3 ′ (s,s )∈T

hence X

(s,s′ )∈T

hd2f ((s, s′ )), π(s)f (s−1s′ )iHπ −

1 X hd2f, d2 f iC 2 = D1 + D2 + D3 . 3 ′ (s,s )∈T

ON ALMOST REPRESENTATIONS OF PROPERTY (T) GROUPS

By Cauchy inequality, definition of k·kC 1 and (7)—(11), we have 1 X ′ ′ ′ −1 ′ −1 −1 ′ hd2 f ((s, s )) − π(s )d2 f (((s ) , (s ) s)), π(s)f (s s )iHπ |D1 | = 3 ′ (s,s )∈T

≤

1 X |hd2 f ((s, s′)) − π(s′ )d2 f (((s′ )−1 , (s′ )−1 s)), π(s)f (s−1s′ )iHπ | 3 ′

≤



1 X

d2 f ((s, s′ )) − π(s′ )d2 f (((s′)−1 , (s′ )−1 s)) π(s)f (s−1 s′ ) Hπ Hπ 3 ′

≤



1 X ε f ((s′)−1 s) Hπ f (s−1 s′ ) Hπ 3 ′

≤

1 ε 3

(s,s )∈T

(s,s )∈T

(s,s )∈T

X




f (s−1 s′ ) 2 1/2 = 1 ε kf k2 1 , C Hπ 3 ′

X




f ((s′ )−1 s) 2 1/2 Hπ

(s,s′ )∈T

(s,s )∈T

1 X ′ −1 ′ −1 ′ −1 ′ −1 −1 ′ hd2 f (((s ) , (s ) s)), (π((s ) )π(s) − π((s ) s))f (s s )iHπ |D2 | = 3 ′ (s,s )∈T

≤



1 X

d2 f (((s′ )−1 , (s′ )−1 s)) (π((s′ )−1 )π(s) − π((s′ )−1 s))f (s−1 s′ ) H Hπ π 3 ′

≤



1 X

f ((s′ )−1 ) − f ((s′)−1 s) + π((s′ )−1 )f (s) ε f (s−1 s′ ) Hπ Hπ 3 ′

≤

X





1 X

f ((s′ )−1 ) f (s−1 s′ ) + 1 ε

f ((s′ )−1 s) f (s−1 s′ ) ε H H H Hπ π π π 3 3 ′ ′

(s,s )∈T

(s,s )∈T

(s,s )∈T

1 + ε 3 ≤

1 ε 3

(s,s′ )∈T

kf (s)kHπ f (s−1 s′ ) Hπ

X




f ((s′ )−1 ) 2 1/2 Hπ

(s,s′ )∈T

1 + ε 3 1 + ε 3 =

(s,s )∈T

X

X




f (s−1s′ ) 2 1/2

(s,s′ )∈T

X




f ((s′ )−1 s) 2 1/2 Hπ

(s,s′ )∈T

X

(s,s′ )∈T

kf (s)k2Hπ


1/2

Hπ

X




f (s−1 s′ ) 2 1/2 Hπ

(s,s′ )∈T

X




f (s−1 s′ ) 2 1/2 Hπ

(s,s′ )∈T

1 1 1 ε kf k2C 1 + ε kf k2C 1 + ε kf k2C 1 = ε kf k2C 1 , 3 3 3

1 X ′ ′ hd2 f ((s, s )) + d2 f ((s , s)), −f (s)iHπ |D3 | = 3 ′ (s,s )∈T

≤

1 X kd2 f ((s, s′)) + d2 f ((s′ , s))kHπ kf (s)kHπ 3 ′ (s,s )∈T

7

8

V. M. MANUILOV AND CHAO YOU

≤

1 X ε f ((s′ )−1 s) Hπ kf (s)kHπ 3 ′ (s,s )∈T

1 ε ≤ 3

X




f ((s′ )−1 s) 2 1/2 Hπ

(s,s′ )∈T

X

(s,s′ )∈T

kf (s)k2Hπ


1/2

=

1 ε kf k2C 1 . 3

So X 1 hd2 f ((s, s′)), π(s)f (s−1 s′ )iHπ − hd2 f, d2 f iC 2 = |D1 + D2 + D3 | 3 ′ (s,s )∈T

≤ |D1 | + |D2 | + |D3 | ≤ which proves (12).

5 ε kf k2C 1 , 3 

Proposition 11. For every f ∈ C 1 one has hDf, Df iC 2 − 1 hd2f, d2 f iC 2 − hf, f iC 1 ≤ 10 εhf, f iC 1 3 3 Proof. By definition of operator D, we have X hd2f ((s, s′ )) − π(s)f (s−1s′ ), d2 f ((s, s′ )) − π(s)f (s−1 s′ )iHπ hDf, Df iC 2 = (s,s′ )∈T

=

X

(s,s′ )∈T

−2

hd2f ((s, s′ )), d2 f ((s, s′))iHπ +

X

(s,s′ )∈T

X

(s,s′ )∈T

hf (s−1 s′ ), f (s−1 s′ )iHπ

hd2 f ((s, s′ )), π(s)f (s−1s′ )iHπ

2 = hd2 f, d2 f iC 2 + hf, f iC 1 − hd2 f, d2 f iC 2 3 X 2 + hd2 f, d2 f iC 2 − 2 hd2 f ((s, s′ )), π(s)f (s−1s′ )iHπ 3 ′ (s,s )∈T

=

1 hd2 f, d2 f iC 2 + hf, f iC 1 3 X
1 hd2f ((s, s′ )), π(s)f (s−1s′ )iHπ , +2 hd2 f, d2 f iC 2 − 3 ′ (s,s )∈T

hence hDf, Df iC 2 − 1 hd2 f, d2 f iC 2 − hf, f iC 1 3 1 X 10 hd2 f ((s, s′ )), π(s)f (s−1s′ )iHπ ≤ εhf, f iC 1 , = 2 hd2 f, d2 f iC 2 − 3 3 ′ (s,s )∈T

which ends the proof of Proposition 11.



Note that every f ∈ C 1 can be considered as a function on L(S). It was shown in [12] (and the proof doesn’t depend on the property of π to be a representation) that hf, f iC 1 = hf, f iL(S) , X hf (s) − f (s′ ), f (s) − f (s′ )iHπ = 2h∆f, f iL(S) hDf, Df iC 2 = (s,s′ )∈T

ON ALMOST REPRESENTATIONS OF PROPERTY (T) GROUPS

9

λ1 (L(S)) ∗ hd1f, d∗1 f iC 0 . 4

(13)

and h∆f, f iL(S) ≥ λ1 (L(S))hf, f iC 1 −

From now on, for shortness’ sake, we denote λ1 (L(S)) by λ1 . Lemma 12. For every f ∈ C 1 one has 1 10
1 hd2 f, d2 f iC 2 + λ1 hd∗1 f, d∗1 f iC 0 ≥ 2λ1 − 1 − ε hf, f iC 1 . 3 2 3

(14)

Proof. By (13), we have 2λ1 hf, f iC 1 − hf, f iC 1 ≤ 2h∆f, f iL(S) − hf, f iC 1 +

λ1 ∗ hd1 f, d∗1 f iC 0 . 2

Since hDf, Df iC 2 = 2h∆f, f iL(S) , λ1 ∗ hd f, d∗1 f iC 0 2 1 10 λ1 1 hd2 f, d2 f iC 2 + εhf, f iC 1 + hd∗1 f, d∗1 f iC 0 , ≤ 3 3 2

2λ1 hf, f iC 1 − hf, f iC 1 ≤ hDf, Df iC 2 − hf, f iC 1 +

which proves (14).

 2

δ Lemma 13. For any f ∈ B 1 ( |T ) one has |

kd2 f k2C 2 ≤

4|T |2 ε2 kf kC 1 . δ4

2

δ ) we have Proof. By Lemma 6, for any u ∈ B 0 ( |T |

kd2 d1 uk2C 2 =

X

(s,s′ )∈T

2

kd2 d1 u((s, s′ ))kHπ ≤ ε2

2

δ Since u ∈ B 0 ( |T ), it follows that |

4|T |2 ε2 δ4

2

δ2 |T |

X

(s,s′ )∈T

kuk2Hπ = ε2 |T | kuk2Hπ = ε2 kuk2C 0 .

kukC 0 ≤ kd∗1 d1 ukC 0 ≤ 2 kd1 ukC 1 . So kd2 d1 uk2C 2 ≤ 2

2

δ δ δ kd1 ukC 1 . Since d1 (B 0 ( |T )) is dense in B 1 ( |T ), for any f ∈ B 1 ( |T ) it also holds that | | |

kd2 f k2C 2 ≤

4|T |2 ε2 δ4

kf kC 1 .



Corollary 14. For every f ∈ C 1 one has  2 20ε 8|T |2ε2  hd∗1 f, d∗1 f iC 0 ≥ 4 − hf, f iC 1 . − − λ1 3λ1 3λ1 δ 4 Corollary 14 provides the constant c for Proposition 9. Now the function α(ε) from  10ε 8|T |2 ε2 Theorem 1 should satisfy α(ε) = max 3λ + 3λ1 δ4 , δ . In order to get a continuous function 1 2/5 with α(0) = 0 one may take δ = ε . Then α(ε) = O(ε2/5 ) and Theorem 1 directly follows from Corollary 14 and Proposition 9.  The authors are grateful to N. Monod for useful remarks.

10

V. M. MANUILOV AND CHAO YOU

References [1] A. Connes, M. Gromov, H. Moscovici: Conjecture de Novikov et fibr´es presque plats. C. R. Acad. Sci. Paris, s´erie I, 310 (1990), 273–277. [2] D. Fisher, G. Margulis: Almost isometric actions, property (T) and local rigidity. Invent. Math. 162 (2005), 19-80. [3] P. de la Harpe, A. Valette: La propri´et´e (T) de Kazhdan pour les groupes localment compacts. Ast´erisque, 175, (1989). [4] P. de la Harpe, A. G. Robertson, A. Valette: On the spectrum of the sum of generators for a finitely generated group. Israel J. Math. 81 (1993), 65–96. [5] D. Kazhdan: Connection of the dual space of a group with the structure of its closed subgroups. Funts. Anal. Prilozh. 1, (1967), No. 1, 71–74 (in Russian); Funct. Anal. Appl. 1 (1967), 63–65. [6] V. M. Manuilov: Almost representations and asymptotic representations of discrete groups. Izv. Ross. Akad. Nauk Ser. Mat. 63 (1999), No 5, 159–178 (in Russian). English translation: Russian Acad. Sci. Izv. Math. 63 (1999), 995–1014. [7] V. M. Manuilov, A. S. Mishchenko: Almost, asymptotic and Fredholm representations of discrete groups. Acta Appl. Math. 68 (2001), 159–210. [8] M. Reed, B. Simon: Methods of Modern Mathematical Physics Vol. I: Functional Analysis. Academic Press, (1980). [9] A. I. Shtern: Quasirepresentations and pseudorepresentations. Funts. Anal. Prilozh. 25, (1991), No. 2, 70–73 (in Russian); English translation: Funt. Anal. Appl. 25 (1991), 140–143. [10] A. Valette: Minimal projections, integrable representations and property (T). Arch. Math. 43 (1984), 397–406. [11] D. Voiculescu: Asymptotically commuting finite rank unitary operators without commuting approximants. Acta Sci. Math. (Szeged) 45 (1983), 429–431. ˙ [12] A. Zuk: Property (T) and Kazhdan constants for discrete groups. Geom. Funct. Anal. 13 (2003), 643–670.

V. M. Manuilov, Dept. of Mechanics and Mathematics, Moscow State University, Leninskie Gory, Moscow, 119992 Russia, and, Dept. of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R.C. E-mail address: [email protected] Chao You, Dept. of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R.C. E-mail address: [email protected]