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T T C Property ( ) and strong Property ( ) for unital -algebras ∗ ∗ Chi-Wai Leung and Chi-Keung Ng January 14, 2009 9 0 0 Abstract 2 ∗ In this paper, we will give a thorough study of the notion of Property (T) for C - n algebras (as introduced by M.B. Bekka in [3]) as well as a slight stronger version of it, a J called“strongproperty(T)”(whichis alsoananalogueofthe correspondingconceptinthe 4 case of discrete groups and type II1-factors). More precisely, we will give some interesting 1 equivalent formulations as well as some permanence properties for both property (T) and strong property (T). We will also relate them to certain (T)-type properties of the unitary ] ∗ group of the underlying C -algebra. A O 2000 Mathematics Subject Classification: 46L05, 22D25 ∗ . Keywords: Property (T); unital C -algebras;Hilbert bimodules h t a m [ 1 Introduction 1 v 8 Property (T) for locally compact groups was first defined by D. Kazhdan in [11] and was later 4 extended to Hausdorff topological groups. In [12], Property (T) for a pair of groups H G 9 ⊆ 1 was introduced. This notion was proved to be very useful and was studied by many people (see . e.g. [2], [4], [8], [10], [11], [12] and [16]). 1 0 In [6], A. Connes introduced the notion of property (T) for type II -factors and this notion 9 1 0 was then extended to von Neumann algebras in [7]. A discrete group G has property (T) if and v: only if the von Neumann algebra generated by the left regular representation of G has property i (T) (this was first proved in [7] for ICC groups and was generalized by P. Jolissaint in [9] to X general discrete groups). The notion of property (T) for a pair of von Neumann algebras was r a defined by S. Popa in [15]. This notion was also proved to be very useful in the study of von Neumann algebras. Recently, M.B. Bekka introduced in [3] the interesting notion of property (T) for a pair consisting of a unital C -algebra and a unital C -subalgebra. He showed that a countable ∗ ∗ ∗This work is jointly supported by Hong Kong RGC Research Grant (2160255), Hong Kong RGC Direct Grant (2060319), theNational Natural Science Foundation of China (10771106) and NCET-05-0219 1 discrete group G has property (T) if and only if its full (or equivalently reduced) group C - ∗ algebra has property (T). In [5], N.P. Brown did a study of property (T) for C -algebras and ∗ showed that a nuclear unital C -algebra A has property (T) if and only if A = B C where B ∗ ⊕ is finite dimensional and C admits no tracial state. The aim of this paper is to give a thorough study of property (T) as well as a slightly stronger version called strong property (T) for unital C -algebras. On our way, we will show ∗ that our stronger version is equally good (if not a better) candidate for the notion of property (T) for a pair of unital C -algebras. ∗ The paper is organised as follows. In Section 2, we will give two simple and useful reformulations of both property (T) and strong property (T). In section 3, we consider two Kazhdan constants tA and tA for a C -algebra A which are the analogous of the Kazhdan constant for u c ∗ locally compact groups (see [16]). We will show that A has property (T) (respectively, strong property (T)) if and only if tA > 0 (respectively, tA > 0). Through them, we obtain some c u interesting reformulations of property (T) and strong property (T). In particular, we show that one can check property (T) by looking at just one bimodule. We will also show that one can express the gap between property (T) and strong property (T) by another Kazhdan constant tA. s In section 4, we obtain some permanence properties for property (T) and strong property (T), including quotients, direct sums, tensor products and crossed products. In section 5, we willshowthatfinitedimensionalC -algebrashavestrongproperty(T). Moreover, weshowthat ∗ a corresponding result of Bekka concerning relation between property (T) of discrete groups and their group C -algebras as well as a corresponding result of Brown concerning amenable ∗ property (T)C -algebras also holdsfor strong property(T). InSection 6, we studythe relation ∗ between property (T)(as well as strongproperty (T)) of a unitalC -algebra Aand certain (T)- ∗ type properties of the unitary group of A. Let us first set the following notations that will be used throughout the whole paper. Notation 1.1 (1) A is a unital C -algebra and B A is a C -subalgebra containing the ∗ ∗ ⊆ identity of A. Set ADou := A Aop (where Aop is the “opposite C -algebra” with aopbop = max ∗ ⊗ (ba)op). (2) F(E) is the set of all non-empty finite subsets of a set E and S (X) is the unit sphere of 1 a normed space X. (3) U(A) and S(A) are respectively the unitary group and the state space of A. (4) Bimod (A) is the collection of unitary equivalence classes of unital Hilbert bimodules over ∗ A (or equivalently, non-degenerate representations of ADou). For any H Bimod (A), let ∗ ∈ HB := ξ H :b ξ = ξ b for all b B { ∈ · · ∈ } and PB : H HB be the orthogonal projection. Elements in HB are called central vectors for H → B. Moreover, for any (Q,β) F(A) R , set + ∈ × V (Q,β) := ξ S (H) : x ξ ξ x < β for all x Q . H 1 { ∈ k · − · k ∈ } 2 Elements in V (Q,β) are called (Q,β)-central unitvectors. On the other hand, a net of vectors H (ξ ) in S (H) is called an almost central unit vector for A if a ξ ξ a 0 for any i i I 1 i i ∈ k · − · k → a A. ∈ (5) For any topological group G, we denote by Rep(G) the collection of all unitary equivalence classes of continuous unitary representations of G. If (π,H) Rep(G), we let ∈ HG := ξ H :π(s)ξ = ξ for all s G { ∈ ∈ } and PG :H HG be the orthogonal projection. Furthermore, if F F(G) and ǫ > 0, we set H → ∈ V (F,ǫ) = ξ S (H) : π(t)ξ ξ < ǫ for all t F . π 1 { ∈ k − k ∈ } (6) For any (µ,H),(ν,K) Rep(G), we write (µ,H) (ν,K) if (µ,H) is a subrepresentation ∈ ≤ of (ν,K). 2 Definitions and basic properties Let us first recall Bekka’s notion of property (T) in [3]. The pair (A,B) is said to have property (T) if there exist F F(A) and ǫ > 0 such that for any H Bimod (A), if V (F,ǫ) = , then ∗ H ∈ ∈ 6 ∅ HB = (0). In this case, (F,ǫ) is called a Kazhdan pair for (A,B). Moreover, A is said to have 6 property (T) if the pair (A,A) has property (T). Note that Bekka’s definition comes from the original definition of property (T) for groups (see e.g. [10, Definition 1.1(1)]). We will now give a slightly stronger version which comes from an equivalent form of property (T) for groups (see [10, Theorem 1.2(b2)]). Note that the corresponding stronger version of property (T) for type II -factor is also equivalent to property 1 (T) (see e.g. [7, Proposition 1]) but we do not know if it is the case for C -algebras. ∗ Definition 2.1 The pair (A,B) is said to have strong property (T) if for any α > 0, there exist Q F(A) and β > 0 such that for any H Bimod (A) and any ξ V (Q,β), we have ∗ H ∈ ∈ ∈ ξ PB(ξ) < α. In this case, (Q,β) is called a strong Kazhdan pair for (A,B,α). We say − H (cid:13)that A has s(cid:13)trong property (T) if (A,A) has such property. (cid:13) (cid:13) It is clear that if A has property (T) (respectively, strong property (T)) then so has the pair (A,B). Moreover, by taking α < 1/2, we see that strong property (T) implies property (T). We will see later that strong property (T) is an equally good (if not a better) candidate for the notion of property (T) for a pair of C -algebras. ∗ Let us now give the following simple reformulation of property (T) and strong property (T) which will be useful in Section 6. Lemma 2.2 For any (Q,β) F(A) R , there exists (Q,β ) F(U(A)) R such that + ′ ′ + ∈ × ∈ × V (Q,β ) V (Q,β) for any H Bimod (A). Consequently, one can replace F(A) by H ′ ′ H ∗ ⊆ ∈ F(U(A)) in the definitions of both property (T) and strong property (T). 3 Proof: ThislemmaisclearifQ = 0 . LetQ 0 = x ,...,x andM = max x ,..., x . 1 n 1 n { } \{ } { } {k k k k} Foreachk 1,...,n , consideru ,v U(A)suchthat2x = x ((u +u )+i(v +v )). Ifwe ∈ { } k k ∈ k k kk k ∗k k k∗ take Q to be the set u ,u ,v ,v ,...,u ,u ,v ,v and β = β , then V (Q,β ) V (Q,β) ′ { 1 ∗1 1 1∗ n ∗n n n∗} ′ 2M H ′ ′ ⊆ H for any H Bimod (A). (cid:3) ∗ ∈ Before we give a second simple reformulation, we need to set some notations. Let S(D) and S (D) be respectively the sets of all states and the set of all tracial states on a C -algebra D. t ∗ For any τ S(D) and any cardinal α, we denote by M the GNS construction for τ and by τ ∈ M the α-times direct sum, M , of M (we use the convention that M = 0 ). τ,α α τ τ 0 τ { } L L Definition 2.3 Let H := M and K := M . τ τ τ SM(ADou) τ MSt(A) ∈ ∈ We called H and K the universal and the standard bimodules (over A) respectively. Moreover, a bimodule of the form M is called a quasi-standard bimodule. τ St(A) τ,ατ ∈ L Proposition 2.4 (a) (A,B) has property (T) if and only if for any H Bimod (A), the ∗ ∈ existence of an almost central unit vector for A in H will imply that HB = 0 . 6 { } (b) The following statement are equivalent. (i) (A,B) has strong property (T). (ii) For any almost central unit vector (ξ ) for A in any bimodule H Bimod (A), we have i i I ∗ ξ PB(ξ ) 0. ∈ ∈ i− H i → (cid:13) (cid:13) (iii) F(cid:13)or any almo(cid:13)st central unit vector (ξ ) for A in H and any n N, there exists i I i i I n with ξ PB(ξ ) < 1/n. ∈ ∈ ∈ in − H in (cid:13) (cid:13) (cid:13) (cid:13) Proof: (a) This part is well known. (b) It is clear that (i) (ii) and (ii) (iii). To obtain (iii) (i), we suppose, on the contrary, ⇒ ⇒ ⇒ that (A,B) does not have strong property (T). Then one can find α > 0 such that for 0 any i = (Q,β) I := F(A) R , there exist H Bimod (A) and ξ V (Q,β) with ∈ × + i ∈ ∗ i ∈ Hi ξ PB(ξ ) α . If K = A ξ A, then H = K K and HB = KB (K )B. As k i − Hi i k ≥ 0 i · i· i i ⊕ i⊥ i i ⊕ i⊥ ξ K , we have i i ∈ ξ PB(ξ ) = ξ PB(ξ ) α . k i− Ki i k k i− Hi i k ≥ 0 We set X := K : i I Bimod (A) and K := K. Since all bimodules in X are { i ∈ } ⊆ ∗ 0 K X cyclic (as representations of ADou) and any two elemenLts in∈ X are inequivalent, K0 is a Hilbert sub-bimodule of H. Moreover, each K is equivalent to a unique element in X and this gives i a canonical Hilbert bimodule embedding Ψ : K K . It is easy to check that (Ψ (ξ )) is i i 0 i i i I an almost central unit vector for A in H with Ψ→(ξ ) PB(Ψ (ξ )) = ξ PB(ξ ) α∈for k i i − H i i k k i− Ki i k ≥ 0 4 every i I (since Ψ (K ) is a direct summand of H). This contradicts Statement (iii). (cid:3) i i ∈ SinceProposition2.4issofundamentaltoourdiscussions,wemayuseitwithoutmentioning it explicitly throughout the whole paper. Remark 2.5 (a) Note that if A is separable, then in the above proposition, one can replace almost central unit vector by a sequence of unit vectors that is “almost central” for A. (b) In Proposition 2.4(b)(iii), we only need to check one bimodule (namely, the universal one) in order to verify strong property (T). (c) One may wonder if it is possible to check whether a C -algebra has property (T) by looking ∗ at its universal bimodule alone. However, this cannot be done using the original formulation of property (T) because there exists a unital C -algebra A which does not have property (T) but ∗ HA = (0) (i.e. A has a tracial state). Nevertheless, we will show in Theorem 3.4 below that it 6 is possible to do so using an equivalent formulation of property (T). 3 Kazhdan constants In this section, we will define and study some Kazhdan constants in the case when B = A. Let us start with the following lemma. Lemma 3.1 Let H Bimod∗(A). If HC is the sub-bimodule generated by HA (called the ∈ centrally generated part of H), then HC is a quasi-standard bimodule (Definition 2.3) and HC⊥ contains no non-zero central vector for A. Proof: Without loss of generality, we may assume that C := S (HA) is non-empty. Let 1 S := A ξ : ξ C and { · ∈ } M := M S :K L for any K,L M . { ⊆ ⊥ ∈ } By the Zorn’s lemma, there exists a maximal element M in M and we put H := K. 0 1 K M0 ∈ Then clearly H1 ⊆ HC and H1⊥ contains no non-zero central vector for A. TogetheLr with the fact HA = H1A⊕(H1⊥)A, this shows that HA = H1A ⊆ H1 and hence H1 = HC. Finally, for any A ξ M with ξ C, the functional defined by τ(a) := aξ,ξ (a A) is a tracial state and 0 · ∈ ∈ h i ∈ A ξ = M . This completes the proof. (cid:3) · ∼ τ Suppose that H Bimod (A) and K is a Hilbert subspace of H. For any Q F(A), we set ∗ ∈ ∈ 1/2 tA(Q;H,K) := inf  x ξ ξ x 2 :ξ S1(H K)  xXQk · − · k ∈ ⊖   ∈      (we use the convention that the infimum over the empty set is + ). ∞ 5 Lemma 3.2 Let Q F(A), H Bimod (A) and K be a Hilbert subspace of H. Suppose that ∗ ∈ ∈ H = H such that K = K where K := H K. λ Λ λ λ Λ λ λ λ∩ ∈ ∈ L L (a) If α is a cardinal for any λ Λ, and if we set H := H and K := λ ∈ 0 λ∈Λ(cid:16) αλ λ(cid:17) 0 L L K , then λ∈Λ(cid:16) αλ λ(cid:17) L L tA(Q;H,K)2 ζ 2 x ζ ζ x 2 (ζ H K ). (3.1) 0 0 k k ≤ k · − · k ∈ ⊖ xXQ ∈ (b) tA(Q;H,K) = inf tA(Q;H ,K ). λ Λ λ λ ∈ (c) (tA(Q;H,K)) is an increasing net and lim tA(Q;H,K) =0 if and only if there Q F(A) Q F(A) ∈ ∈ exists an almost central unit vector for A in H K. In this case, one can choose an almost ⊖ central unit vector (ξ ) for A such that for any i I, there exists λ Λ with ξ H K . i i∈I ∈ i ∈ i ∈ λi⊖ λi Proof: (a) For any ζ S (H K ), we have ζ = (ζ ) with ζ H K H K and ζ ∈2 =11. T0⊖hus,0 i,λ λ∈Λ;i∈αλ i,λ ∈ λ⊖ λ ⊆ ⊖ λ Λ i αλk i,λk ∈ ∈ P P ζ ζ 2 tA(Q;H,K)2 ζ 2 x i,λ i,λ x = x ζ ζ x 2. i,λ ≤ k k (cid:13) · ζ − ζ · (cid:13) k · − · k λX∈ΛiX∈αλ xX∈Q(cid:13)(cid:13) k i,λk k i,λk (cid:13)(cid:13) xX∈Q (cid:13) (cid:13) (b) Note that tA(Q;H,K) tA(Q;H ,K ) for all λ Λ (as H K H K). For any λ λ λ λ ≤ ∈ ⊖ ⊆ ⊖ ǫ > 0, there exists ξ S (H K) such that x ξ ξ x 2 tA(Q;H,K)+ǫ. Now, ∈ 1 ⊖ x Qk · − · k ≤ ξ = (ξ ) with ξ H K and ξ P2 =∈ 1. A similar argument as part (a) implies that thλerλe∈Λexists λ0λ ∈Λ sλu⊖ch thλat Pλ∈Λk λk ∈ 2 ξ ξ x λ0 λ0 x tA(Q;H,K)+ǫ. (cid:13) · ξ − ξ · (cid:13) ≤ xXQ(cid:13) k λ0k k λ0k (cid:13) ∈ (cid:13) (cid:13) (cid:13) (cid:13) (c)Itisclear that(tA(Q;H,K)) isincreasingandthattA(Q;H,K) = 0foranyQ F(A) Q F(A) ∈ ∈ if there exists an almost central unit vector for A in H K. Now, suppose that ⊖ sup inf tA(Q;H ,K ) = lim tA(Q;H,K) = 0. λ λ Q F(A)λ Λ Q F(A) ∈ ∈ ∈ Then for any Q F(A) and ǫ > 0, there exists λ Λ and ξ S (H K ) such that ∈ Q,ǫ ∈ Q,ǫ ∈ 1 λQ,ǫ⊖ λQ,ǫ uPnxit∈Qvekcxto·rξQfo,rǫ −A.ξQ,ǫ ·xk2 < ǫ2. It is easy to see that (ξQ,ǫ)(Q,ǫ)∈F(A)×R+ is an almost centra(cid:3)l Now, we define three Kazhdan constants: for any Q F(A), set ∈ tAu(Q) := tA(Q;H,HA), tAc (Q) := tA(Q;H,HC), tAs(Q) := tA(Q;K,KA) (where H and K are the universal bimodule and the standard bimodule respectively) and tA := sup tA(Q), tA := sup tA(Q) as well as tA := sup tA(Q). u u c c s s Q F(A) Q F(A) Q F(A) ∈ ∈ ∈ 6 Lemma 3.3 (a) For any H Bimod (A), we have tA(Q) tA(Q;H,HA) and tA(Q) ∈ ∗ u ≤ c ≤ tA(Q;H,HC). If, in addition, H is quasi-standard, then tAs(Q) ≤ tA(Q;H,HA). (b) tA(Q) min tA(Q),tA(Q) . u ≤ { c s } Proof: (a) There are cardinals α (τ S(ADou)) such that H = M . For any τ ∈ ∼ τ S(ADou) ατ,τ ξ S (H HA), we have ξ = (ξ ) where ξ M MA . ByLIneq∈uality (3.1), we have ∈ 1 ⊖ τ τ ∈ ατ,τ ⊖ ατ,τ tA(Q)2 ξ 2 x ξ ξ x 2 u k τk ≤ k · τ − τ · k xXQ ∈ and so tA(Q)2 x ξ ξ x 2 = x ξ ξ x 2 u ≤ k · τ − τ · k k · − · k τ SX(ADou)xXQ xXQ ∈ ∈ ∈ (as ξ 2 = ξ 2 = 1). Thus, we have tA(Q) tA(Q;H,HA). The arguments for τ S(ADou)k τk k k u ≤ thePoth∈er two inequalities are similar. (b) tAu(Q) ≤ tAc (Q) because HA ⊆ HC and tAu(Q) ≤ tAs(Q) because of part (a). (cid:3) Theorem 3.4 (a) The following statements are equivalent. i. tA > 0. u ii. A has strong property (T). iii. There exists (Q,δ) F(A) R+ such that for any ξ VH(Q,δ), we have PHA(ξ)= 0. ∈ × ∈ 6 (b) The following statements are equivalent. i. tA > 0. s ii. For any ǫ > 0, there exists (Q,δ) F(A) R such that for any quasi-standard bimodule + ∈ × H and any ξ V (Q,δ), we have ξ PA(ξ) < ǫ. ∈ H k − H k iii. There exists (Q,δ) F(A) R+ such that for any ξ VK(Q,δ), we have PKA(ξ) = 0. ∈ × ∈ 6 (c) The following statements are equivalent. i. tA > 0. c ii. A has property (T). iii. There is (Q,δ) F(A) R such that V (Q,δ) H = for any H Bimod (A). + H C⊥ ∗ ∈ × ∩ ∅ ∈ iv. There exists (Q,δ) F(A) R+ such that VH(Q,δ) HC⊥ = . ∈ × ∩ ∅ (d) tA > 0 if and only if both tA > 0 and tA > 0. u c s 7 Proof: (a) (i) (ii). There exists Q F(A) with tA(Q) > 0. Let m be the number of ⇒ ∈ u elements in Q and δ = tAu(Q)ǫ. For any H Bimod (A) and τ S(ADou), there is a cardinal ∗ √m ∈ ∈ α such that H = M (α can be zero). Pick any ξ V (Q,δ) and consider τ τ S(ADou) ατ,τ τ ∈ H ξ = ξ PA(ξ) (HLA)∈ . Since ξ = (ζ ) where ζ (MA ) , we have, by Inequality ′′ − H ∈ ⊥ ′′ τ τ∈S(A) τ ∈ ατ,τ ⊥ (3.1), ξ′′ 2 tA(Q) 2 x ξ′′ ξ′′ x 2 = tA(Q) 2 x ξ ξ x 2 < ǫ2. k k ≤ u − k · − · k u − k · − · k xXQ xXQ ∈ ∈ (ii) (iii). By taking ǫ = 1/2, we see that Statement (iii) holds. ⇒ (iii) (i). Suppose on the contrary that tA(Q) = 0. Then there exists ξ S ((HA) ) with ⇒ u ∈ 1 ⊥ x Qkx·ξ −ξ ·xk2 < δ2. Hence, ξ ∈ VH(Q,δ) and so PHA(ξ) 6= 0 which contradicts the fact tPhat∈ξ (HA) . ⊥ ∈ (b) The proof of this part is essentially the same as that of part (a) with H and tA being u replaced by K and tA respectively. s (c) (i) (ii). Let Q F(A) such that tA(Q) > 0. Suppose that A does not have property ⇒ ∈ c (T). There exists H Bimod (A) that contains an almost central unit vector (ξ ) for A but ∗ i ∈ HA = 0 . Hence, H = H, and C⊥ { } 1/2 tA(Q;H,HC) = inf  x ξ ξ x 2 : ξ S1(H) = 0.  xXQk · − · k ∈   ∈      Now Lemma 3.3(a) gives the contradiction that tA(Q)= 0. c (ii) (i). Suppose on the contrary that tA(Q) = 0 for any Q F(A). There exists, by Lemma ⇒ c ∈ 3.2(c), an almost central unitvector for Ain H whichcontradicts thefactthatAhas property C⊥ (T) (because of Lemma 3.1). (i) (iii). Let Q F(A) such that tA(Q) > 0 and 0 < δ < tAc(Q) where m is the number of ⇒ ∈ c √m elements in Q. By Lemma 3.3(a), we have tAc (Q)≤ tAc (Q;H,HC). Thus, x ζ ζ x 2 tA(Q) > mδ2 k · − · k ≥ c xXQ ∈ for any ζ S (H ). Suppose that there exists ξ V (Q,δ) H . Then mδ2 < x ξ ∈ 1 C⊥ ∈ H ∩ C⊥ x Qk · − ξ x 2 < mδ2 which is absurd. P ∈ · k (iii) (iv). This is obvious. ⇒ (iv) (i). Suppose on the contrary that tA(Q) = 0. Then there exists ξ S (H ) with ⇒ c ∈ 1 C⊥ x Qkx·ξ−ξ·xk2 < δ2 which gives the contradiction that ξ ∈ VH(Q,δ)∩HC⊥. ∈ (Pd) If tA > 0, then tA > 0 and tA > 0 (by Lemma 3.3(b)). Conversely, suppose that tA = 0. u s c u Then by Lemma 3.2(c), there exists an almost central unit vector (ξ ) for A in i i I ∈ H HA = (H HC) (HC HA). ⊖ ⊖ ⊕ ⊖ 8 Let ηi H HC and ζi HC HA be the corresponding components of ξi. Then either ηi 9 0 ∈ ⊖ ∈ ⊖ or ζ 9 0. Therefore, by rescaling, there exists an almost central unit vector for A in either i H⊖HC or HC⊖HA = HC⊖HCA. In the first case, we have tAc = 0 (by Lemma 3.2(c)). In the Lseecmonmdac3a.s2e(,c)w).e have tAs ≤ supQ∈F(A)tA(Q;HC,HCA) = 0 (by Lemma 3.3(a), Lemma 3.1 an(cid:3)d Part (a) of the above theorem tells us that in order to show that A has strong property (T), it suffices to verify a weaker condition than that of Definition 2.1 for just the universal bimodule H. Remark 3.5 (a) The argument of Theorem 3.4(a), together with Lemma 3.3(a), tell us that for any Q F(A), δ > 0 and H Bimod (A), if ξ V (Q,δ), then ∗ H ∈ ∈ ∈ tA(Q) ξ PA(ξ) < δ√m u k − H k (where m is the number of elements in Q). (b) The argument of Theorem 3.4(c), together with Lemma 3.3(a), tell us that if tA(Q) > 0, c for any Q F(A), H Bimod (A) and δ (0, tAc(Q)), we have V (Q,δ) H = . ∈ ∈ ∗ ∈ √m H ∩ C⊥ ∅ (c) The gap between property (T) and strong property (T) is represented by the gap between tA c and tAu or equivalently between HA and HC. Note that in the case of a locally compact group G, such a gap does not exist because the set of G-invariant vectors defines a subrepresentation. By Theorem 3.4 and Lemma 3.2(c), we have the following corollary. Corollary 3.6 (a) A has property (T) (respectively, strong property (T)) if and only if there is no almost central unit vector for A in H (respectively, in (HA) ). C⊥ ⊥ (b) A has strong property (T) if and only if A has property (T) and tA > 0. s Note that one can also obtain part (b) of the above corollary by using a similar argument as that of [7, Proposition 1]. 4 Some permanence properties In this section, we study the permanence properties for property (T) and strong property (T). First of all, we have the following lemma which implies that the quotient of any pair having property (T) (respectively, strong property (T)) will have the same property. Since the proof is direct, we will omit it. Lemma 4.1 Let A and A be two unital C -algebras and let B A and B A be C - 1 2 ∗ 1 1 2 1 ∗ ⊆ ⊆ subalgebras containing the identities of A and A respectively. Suppose that ϕ : A A is 1 2 1 2 → a unital -homomorphism such that B ϕ(B ). If (A ,B ) has property (T) (respectively, 2 1 1 1 ∗ ⊆ strong property (T)), then so does (A ,B ). 2 2 9 Lemma 4.2 Let A , A , B and B be the same as Lemma 4.1. If both (A ,B ) and (A ,B ) 1 2 1 2 1 1 2 2 have property (T) (respectively, strong property (T)), then so does (A A ,B B ). 1 2 1 2 ⊕ ⊕ Proof: The statement for property (T) is well known and we will only show the case for strong property (T). Suppose that H Bimod (A A ) and e = (1 ,0) A A . Then H = 2 H where H is a non-de∈generate∗Hil1be⊕rt A2 -A -bimoduAle1. Sup∈pos1e⊕tha2t (ξ ) k,l=1 kl kl k l i i I ∈ is an Lalmost central unit vector in H for A A and ξ = 2 ξkl where ξkl H . Then 1⊕ 2 i k,l=1 i i ∈ kl P ξ12 2+ ξ21 2 = e ξ ξ e 2 0. i i k · i− i· k → (cid:13) (cid:13) (cid:13) (cid:13) If ξ22 0, then we ca(cid:13)n as(cid:13)sume(cid:13) tha(cid:13)t ξ11 > 1/2 (i I), and ξi11 is an almost cen(cid:13)trial(cid:13)un→it vector for A in H . In thkisicakse, for any∈ǫ > 0, the(cid:16)rekξi1i1sk(cid:17)ii∈I I such that (cid:13) (cid:13) 1 11 0 ∈ ξ11 PB1 ξ11 < ǫ (i i ), which implies that (cid:13) i − H11 i (cid:13) 2 ≥ 0 (cid:13) (cid:0) (cid:1)(cid:13) (cid:13) (cid:13) 2 (cid:13)ξj −PHB1⊕B2(ξj)(cid:13) ≤ r(cid:13)ξi11−PHB111(ξi11)(cid:13) + ξi12 2+ ξi21 2+ ξi22 2 < ǫ (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) when j is large enough. The same conclusion holds if ξ11 0. We consider now the case i → when ξ11 9 0 and ξ22 9 0. There exist a constant(cid:13)κ>(cid:13)0 as well as subnets ξ11 and i i (cid:13) (cid:13) ik k J1 ξ22 (cid:13) s(cid:13)uch that (cid:13)ξ11 (cid:13), ξ22 κ for every k J and l J . One can sho(cid:0)w ea(cid:1)si∈ly that il l(cid:13)J2 (cid:13) (cid:13)ik (cid:13) il ≥ ∈ 1 ∈ 2 (cid:0) ξ1(cid:1)1∈ (cid:13)ξ22 (cid:13) (cid:13) (cid:13) ik and (cid:13)il (cid:13) (cid:13) a(cid:13)re almost central unit vectors for A and A in H and H (cid:18)kξi1k1k(cid:19)k J1 (cid:18)kξi2l2k(cid:19)l J2 1 2 11 22 respective∈ly. Thus, for any∈ǫ > 0, one can find i I such that 0 ∈ ξ11 ǫ ξ22 ǫ ξ11 PB1 ξ11 < i0 , ξ22 PB2 ξ22 < i0 (cid:13)(cid:13) i0 − H11(cid:0) i0(cid:1)(cid:13)(cid:13) (cid:13)(cid:13) 2(cid:13)(cid:13) (cid:13)(cid:13) i0 − H22(cid:0) i0(cid:1)(cid:13)(cid:13) (cid:13)(cid:13) 2(cid:13)(cid:13) and ξ12 2+ (cid:13)ξ21 2 < ǫ2. Co(cid:13)nsequently, (cid:13) (cid:13) i0 i0 2 (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13) kξi0 −PHB1⊕B2(ξi0)k ≤qkξi101 −PHB111(ξi101)k2 +kξi202−PHB111(ξi101)k2 +kξi102k2+kξi201k2 < ǫ. In any case, (A A ,B B ) has strong property (T) because of Proposition 2.4(b)(iii). (cid:3) 1 2 1 2 ⊕ ⊕ Our next task is to consider tensor products and crossed products. Let us first recall the followingusefulterminologyofco-rigidityfrom[3,Remark19]. Wewillalsointroduceastronger version of co-rigidity corresponding to strong property (T). Definition 4.3 The pair (A,B) is said to be (a) co-rigid if there exists(Q,β) F(A) R such that for any H Bimod (A) with V (Q,β) + ∗ H ∈ × ∈ ∩ HB = , we have HA = 0 . 6 ∅ 6 { } (b) strongly co-rigid if for any γ > 0, there exists (Q,δ) F(A) R such that for any + ∈ × H Bimod (A) and any ξ V (Q,δ) HB, we have ξ PA(ξ) < γ. ∈ ∗ ∈ H ∩ − H (cid:13) (cid:13) (cid:13) (cid:13) 10