A full description of property T of unital C⁎-crossed products

Journal Pre-proof A full description of property T of unital C ∗ -crossed products

Qing Meng, Chi-Keung Ng

PII:

S0022-247X(19)30905-9

DOI:

https://doi.org/10.1016/j.jmaa.2019.123637

Reference:

YJMAA 123637

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

27 May 2019

Please cite this article as: Q. Meng, C.-K. Ng, A full description of property T of unital C ∗ -crossed products, J. Math. Anal. Appl. (2019), 123637, doi: https://doi.org/10.1016/j.jmaa.2019.123637.

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A FULL DESCRIPTION OF PROPERTY T OF UNITAL C ∗ -CROSSED PRODUCTS QING MENG AND CHI-KEUNG NG

Abstract. Let Γ be a discrete group that acts on a unital C ∗ -algebra A through an action β. We study property T as well as its stronger variant of the full and the reduced crossed products. We show that if A has a β-invariant tracial state, then the reduced crossed product A β,r Γ has strong property T if and only if Γ has property T and (A β,r Γ, A) has strong property T . If Γ has the Haagerup property, the strong property T of (A β,r Γ, A) in the above statement can be replaced by that of A. Furthermore, the strong property T of the full crossed product is shown to coincide with that of the reduced one. In the case of property T , we obtain the corresponding result when the group is either ICC or abelian. As an application, a non-compact locally compact group G is amenable if and only if Cb (G) lt,r Gd does not have strong property T , where Gd is the group G equipped with the discrete topology and lt is the left translation action.

1. Introduction ∗

Property T for unital C -algebras was introduced by Bekka in [1]. Since then, this important property has been studied by many authors. In particular, its relations with nuclearity of C ∗ -algebras, Haagerup property of C ∗ -algebras and property T of discrete quantum groups were obtained by Brown, by Suzuki and by Kyed and Solton in [3], [16] and [9] respectively. On the other hand, Leung and Ng gave, in [10], further studies of Bekka’s property T as well as a stronger variant of it. In [12], both this strong property T and the original property T were extended to the case of non-unital C ∗ -algebras. Recently, there has been some interest in the (strong) property T of crossed products of discrete group actions on unital C ∗ -algebras (see e.g. [10], [7], [5] and [11]). In particular, it was shown in [10, Theorem 4.6] that if the group has property T and the C ∗ -algebra has strong property T , then the full crossed product has strong property T . One difficulty of the converse is that the reduced crossed product may have strong property T without the group nor the original C ∗ -algebra having property T . For example, the uniform Roe algebra ∞ (F2 ) r F2 of the free group F2 has strong property T (see e.g. [5, Corollary 2.4]), but neither ∞ (F2 ) nor F2 has property T . In Section 3, we will begin our study by considering actions on C ∗ -algebras that admit invariant tracial states. In particular, we obtain in Proposition 3.4 a relative property T version of the following. Proposition 1. If a discrete group (respectively, a countable discrete group) Γ acts on a unital C ∗ algebra admitting a Γ-invariant tracial state such that the reduced crossed product has strong property T (respectively, has property T ), then Γ has property T . In Section 4, we will first consider relations between property T of the full crossed products and that of the reduced crossed products. In particular, we show in Theorem 4.4 and Proposition 4.5 a more elaborate version of the following results. Theorem 2. (a) The full crossed product of a unital C ∗ -algebra by a discrete group has strong property T if and only if the corresponding reduced crossed product has strong property T . Date: October 30, 2019. 2010 Mathematics Subject Classification. Primary: 46L05, 46L55. Key words and phrases. C ∗ -crossed products; property T ; strong property T . 1

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QING MENG AND CHI-KEUNG NG

(b) If Γ is a discrete group such that elements with finite conjugacy classes are in the center of Γ, then the full crossed product of a unital C ∗ -algebra by Γ has property T if and only if the corresponding reduced crossed product has property T . Combining all the results and techniques in this paper, we obtain Theorem 4.7, which implies the following (see also Remark 3.3). Theorem 3. Let Γ be a discrete group acting on a unital C ∗ -algebra A through an action β such that A admits a β-invariant tracial state. (a) The reduced crossed product A β,r Γ has strong property T if and only if the following hold: a1) Γ has property T ; a2) (A β,r Γ, A) has strong property T . (b) If Γ has the Haagerup property, then A β,r Γ has strong property T if and only if the following hold: b1) Γ is finite; b2) A has strong property T . Notice that if A does not have a β-invariant tracial state, then A β,r Γ has strong property T . In this sense, the above gives a complete description for A β,r Γ to have strong property T . From Theorem 3, we have the following (see Corollary 4.9). Corollary 4. The amenability of a non-compact locally compact group G is equivalent to Cb (G) lt,r Gd not having strong property T . 2. Notations Throughout this article, H is a (complex) Hilbert space, A is a (complex) unital C ∗ -algebra, Γ is a discrete group, Λ is a subgroup of Γ and β is an action of Γ on A by ∗ -automorphisms. We use A β Γ and A β,r Γ to denote, respectively, the full and the reduced C ∗ -crossed products of (A, Γ, β). We will regard A and Γ as subsets of both A β,r Γ and A β Γ through the canonical embeddings. For any s ∈ Γ, we denote by δs ∈ 2 (Γ) the delta function at s. The left regular representation of Γ on 2 (Γ) will be denoted by λ. For any (complex linear) unital -representation Φ : A → B(H), there exists a ∗ -representation Φ × λ of A β,r Γ on the Hilbert space H ⊗ 2 (Γ) satisfying   (2.1) (Φ × λ)(at)(ξ ⊗ δr ) = Φ β(tr)−1 (a) ξ ⊗ δtr (a ∈ A; r, t ∈ Γ; ξ ∈ H). ∗

In the case when Φ is injective, the representation Φ × λ : A β,r Γ → B(H ⊗ 2 (Γ)) is also injective (see [14, Theorem 7.7.5]), and is known as the “left regular covariant ∗ -representation” of (A, Γ, β). Remark 2.1. Let λ0 be the left regular representation of a subgroup Λ ⊆ Γ. Suppose that {Λri : i ∈ I} is the set of all right cosets of Λ in Γ, and Φ is a faithful ∗ -representation of A on H. Fix an element i ∈ I. The ∗ -homomorphism Φi : A → B(H) given by   Φi (a)ξ := Φ βr−1 (a) ξ (a ∈ A; ξ ∈ H) i

is clearly injective. We define a unitary Wi : H ⊗ 2 (Λri ) → H ⊗ 2 (Λ) by Wi (ξ ⊗ δsri ) = ξ ⊗ δs

(ξ ∈ H; s ∈ Λ).

It is easy to see that Wi (Φ × λ)(at)|H⊗2 (Λri ) Wi−1 η = (Φi × λ0 )(at)η

(a ∈ A; t ∈ Λ; η ∈ H ⊗ 2 (Λ)).

Consequently, if B ⊆ A is a β|Λ -invariant unital C ∗ -subalgebra, we may identify B β|Λ ,r Λ with the unital C ∗ -subalgebra B β,r Λ of A β,r Γ generated by {(Φ × λ)(bt) : b ∈ B; t ∈ Λ}.

A FULL DESCRIPTION OF PROPERTY T OF UNITAL C ∗ -CROSSED PRODUCTS

3

If u is a unitary representation of Γ on H, then a net {ξi }i∈I of unit vectors in H is called an almost u-invariant unit vector when ut ξi − ξi  → 0 (t ∈ Γ). Recall that the inclusion pair (Γ, Λ) is said to have property T if for any unitary representation u of Γ on H that admits an almost u-invariant unit vector, one has Hu|Λ = {0}, where Hu|Λ is the set of u|Λ -invariant vectors, i.e., Hu|Λ := {ξ ∈ H : ut ξ = ξ, for any t ∈ Λ}. Moreover, Γ is said to have property T if (Γ, Γ) has property T . We call H a Hilbert ∗ -bimodule over A, if there exist a (complex linear) unital ∗ -representation of A on H (considered as a left A-multiplication) as well as a (complex linear) unital ∗ -anti-representation of A on H (considered as a right A-multiplication) with commuting ranges (i.e., the two multiplications are compatible, in the sense that a · (η · b) = (a · η) · b). Remark 2.2. Note that if π : A → B(H) is a unital representation, then π is a ∗ -representation if and only if it is contractive. Therefore, we may also view a Hilbert ∗ -bimodule over A as a Hilbert space equipped with a contractive A-bimodule structure. However, since we need explicitly the fact that the representation and the anti-representation respect the ∗ -operation, we use the term “∗ -bimodules” to avoid confusion. On the other hand, we would like to note the differents between “Hilbert ∗ -bimodules” over A in this paper and “Hilbert C ∗ -modules over A” as well as “Hilbert A-bimodules” in the literature. A net {ηi }i∈I of unit vectors in H is called an almost A-central unit vector if a · ηi − ηi · a → 0

(a ∈ A).

Suppose that B ⊆ A is a unital C ∗ -subalgebra. We recall from [1] that the inclusion pair (A, B) is said to have property T if for any Hilbert ∗ -bimodule H over A, the existence of an almost A-central unit vector in H implies the existence of a unit vector ξ in the subspace of B-central vectors, i.e., ξ ∈ HB := {η ∈ H : b · η = η · b, for any b ∈ B}. Let us also recall from [10] that the inclusion pair (A, B) is said to have strong property T (whose motivation comes from [6, Theorem 1.2(b2)]; see Remark 3.3 below) if for any Hilbert ∗ -bimodule H over A and any almost A-central unit vector {ηi }i∈I in H, one has   ηi − P B (ηi ) → 0, where P B is the projection from H onto HB . When the pair (A, A) has (respectively, strong) property T , we say that A has (respectively, strong) property T . Clearly, if (A, B) has strong property T , then (A, B) has property T . Another obvious fact is that (A, B) has (respectively, strong) property T whenever either A or B has (respectively, strong) property T . We recall in the following two well-known facts from [1, Lemma 10], [10, Lemma 4.1] and [10, Proposition 5.2]. They may be used implicitly in the paper. F1). If Φ is a ∗ -homomorphism from A to another C ∗ -algebra and (A, B) has property T or strong property T , then (Φ(A), Φ(B)) has the same property. F2). If A does not have a tracial state, then A has strong property T .

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QING MENG AND CHI-KEUNG NG

3. Relation with property T of the underlying groups As remarked in the second paragraph of the Introduction, in general, the crossed product having strong property T will not imply that the underlying group have property T . However, we will show in this section (Proposition 3.4) that, with the existence of a β-invariant tracial state, strong property T of the reduced crossed product A β,r Γ will imply Γ having property T . This gives one direction of Theorem 3. Let us begin with the following probably well-known fact concerning “right regular covariant ∗ -antirepresentations”. Since we do not find this statement explicitly stated in the literature, we give a brief account here. Lemma 3.1. For each unital ∗ -anti-representation Ψ : A → B(H), there is a ∗ -anti-representation Ψ × ρ : A β,r Γ → B(H ⊗ 2 (Γ)) satisfying (Ψ × ρ)(at)(ξ ⊗ δr ) = Ψ(βr (a))ξ ⊗ δrt

(a ∈ A; r, t ∈ Γ; ξ ∈ H).

(3.1)

¯ the conjugate Hilbert space of H and by Ψ ¯ the ∗ -representation given ¯ : A → B(H) Proof. Denote by H ∗ ∗ ¯ ¯ ¯ ×λ ¯ by Ψ(a)ξ := Ψ(a )ξ (a ∈ A; ξ ∈ H). If Ψ × λ is the -representation of A β,r Γ as in (2.1), then Ψ ∗ 2 2 2 is a -anti-representation of A β,r Γ on H ⊗  (Γ). Consider V :  (Γ) →  (Γ) to be the complex linear ¯ × λ is the required map Ψ × ρ. isometry satisfying V (δ¯r ) := δr−1 (r ∈ Γ). Then Ad(idH ⊗ V ) ◦ Ψ  We warn the readers that, as demonstrated in the following example, for a Hilbert ∗ -bimodule H over A, the right A β,r Γ-multiplication on H ⊗ 2 (Γ) given by the above does not, in general, commute with the left A β,r Γ-multiplication as given by (2.1). This is one of the reasons why property T of A β,r Γ does not give certain T -like property of A.   1 0 ∈ A. Suppose that β is the action of Z/2Z on A Example 3.2. Let A := M2 (C) and U := 0 −1 2 ∗ induced by Ad U . The space over A in the  H of Hilbert-Schmidt operators on C is a Hilbert  -bimodule  0 1 1 0 ¯ ¯ canonical way. Set a := ∈ A, s := 0 and t := 1 ∈ Z/2Z. If ξ := ∈ H, then, under 1 0 0 1 2 the left and the right A β,r Z/2Z-multiplications on H ⊗  (Z/2Z) as in (2.1) and (3.1), one has        0 −1 1 0 ⊗ δ¯1 • as = ⊗ δ¯1 , at • (ξ ⊗ δ¯0 ) • as = −1 0 0 1        −1 0 0 1 ⊗ δ¯1 . ⊗ δ¯0 = at • (ξ ⊗ δ¯0 ) • as = at • 0 −1 1 0 We will see in Lemma 4.3(a) below that if H satisfies certain covariant condition and if the right multiplication is shifted by a “Fell unitary” as in (3.3) below, then H ⊗ 2 (Γ) becomes a Hilbert ∗ bimodule over A β,r Γ. Let us say that (Γ, Λ) has property Ta if for any unitary representation (u, H) of Γ and any almost u-invariant unit vector {ξi }i∈I , one has ξi − P u|Λ (ξi ) → 0, u|Λ

(3.2)

u|Λ

is the projection from H onto H . We do not use the term “strong property T ” since a where P different property with that name was already introduced in [15, Definition 5.2]. Obviously, property Ta of the pair (Γ, Λ) is stronger than property T . Remark 3.3. It was shown in [6, Theorem 1.2(b2)] that if Γ is countable and (Γ, Λ) has property T , then (Γ, Λ) has property Ta . Since all property T discrete groups are finitely generated, we know that a discrete group Γ has property T if and only if (Γ, Γ) has property Ta . The following is the main result of this section. It generalizes (iii) ⇒ (i) of [1, Theorem 7] and is also a partial generalisation of [10, Theorem 4.5(c)] (in the case when B = Cr∗ (Γ)).

A FULL DESCRIPTION OF PROPERTY T OF UNITAL C ∗ -CROSSED PRODUCTS

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Proposition 3.4. Let Γ be a discrete group acting on a unital C ∗ -algebra A through an action β such that A admits a β-invariant tracial state. Suppose that Λ is a subgroup of Γ and B is a β|Λ -invariant unital C ∗ -subalgebra of A. (a) If (A β,r Γ, B β,r Λ) has strong property T , then (Γ, Λ) has property Ta . (b) If Γ is countable and (A β,r Γ, B β,r Λ) has property T , then (Γ, Λ) has property T . Proof. Suppose that v : Γ → B(K) is a unitary representation of Γ on a Hilbert space K that admits an almost v-invariant unit vector {ζi }i∈I . Until the end of this proof, a, b, x are arbitrary elements in A, r, s, t are arbitrary elements in Γ, and ζ is an arbitrary element in K. Let τ ∈ A∗+ be a β-invariant tracial state, H be the Hilbert space in the GNS construction of τ , and Θ : A → H be the canonical map. Then H is a Hilbert ∗ -bimodule over A under the following ∗ -representation and ∗ -anti-representation: Φ(a)Θ(x) := Θ(ax)

and Ψ(a)Θ(x) := Θ(xa).

∗

Since Θ(x) = τ (x x) (x ∈ A), we know that for each t ∈ Γ, the operator ut : Θ(A) → Θ(A) given by ut (Θ(x)) := Θ(βt (x)) (x ∈ A; t ∈ Γ) is a well-defined bijective isometry, and hence it extends to a unitary ut ∈ B(H). In this case, t → ut is a unitary representation of Γ. Define U : K ⊗ H ⊗ 2 (Γ) → K ⊗ H ⊗ 2 (Γ) by 2

U (ζ ⊗ ξ ⊗ δr ) := vr ζ ⊗ ur ξ ⊗ δr (ξ ∈ H). ˇ If Ψ × ρ is the -anti-representation as in Lemma 3.1 and Ψ := ad(U ∗ ) ◦ (idK ⊗ (Ψ × ρ)), then ˇ Ψ(bs)(ζ ⊗ Θ(x) ⊗ δr ) = U ∗ (vr ζ ⊗ Ψ(βr (b))Θ(βr (x)) ⊗ δrs )

(3.3)

∗

= vs−1 ζ ⊗ Θ (βs−1 (xb)) ⊗ δrs . ˇ := idK ⊗ (Φ × λ) (see (2.1)), then On the other hand, if we set Φ     ˇ ˇ ˇ ζ ⊗ Θ(β(tr)−1 (a)x) ⊗ δtr Ψ(bs) Φ(at)(ζ ⊗ Θ(x) ⊗ δr ) = Ψ(bs)   = vs−1 ζ ⊗ Θ β(trs)−1 (a)βs−1 (xb) ⊗ δtrs ˇ = Φ(at) (vs−1 ζ ⊗ Θ(βs−1 (xb)) ⊗ δrs )   ˇ ˇ = Φ(at) Ψ(bs)(ζ ⊗ Θ(x) ⊗ δr ) .

(3.4)

(3.5)

Hence, K ⊗ H ⊗ 2 (Γ) becomes a Hilbert ∗ -bimodule over A β,r Γ. Moreover, using an approximation argument, it is not hard to check that the net {ζi ⊗ Θ(1) ⊗ δe }i∈I is an almost A β,r Γ-central unit vectors in K ⊗ H ⊗ 2 (Γ). Observe that if η ∈ (K ⊗ H ⊗ 2 (Γ))Bβ,r Λ and r0 ∈ Γ, then for any t ∈ Λ, the relation t • η = η • t (t ∈ Λ) implies (3.6) (vt−1 ⊗ ut−1 )(η(r0 )) = η(t−1 r0 t). (a) Since (A β,r Γ, B β,r Λ) has strong property T , if we set   ηi := P Bβ,r Λ ζi ⊗ Θ(1) ⊗ δe , then ζi ⊗ Θ(1) ⊗ δe − ηi  → 0, which implies ζi ⊗ Θ(1) − ηi (e)2 → 0.

(3.7)

Relation (3.6) tells us that ηi (e) is v|Λ ⊗ u|Λ -invariant. Let P be the orthogonal projection from H onto C · Θ(1). Then Relation (3.7) implies ζi ⊗ Θ(1) − (idK ⊗ P )(ηi (e))2 → 0. For each i ∈ I, if

ζiv

∈ K is the element satisfying ζiv ⊗ Θ(1) = (idK ⊗ P )(ηi (e)),

(3.8)

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QING MENG AND CHI-KEUNG NG

then ζiv is v|Λ -invariant, since C · Θ(1) is u-invariant. Consequently, we may conclude from (3.8) that ζi − P v|Λ (ζi ) → 0 as required. (b) By the property T of (A β,r Γ, B β,r Λ), one can find a non-zero vector η ∈ (K ⊗ H ⊗ 2 (Γ))Bβ,r Λ . Fix r0 ∈ Γ with η(r0 ) = 0 and let CΛ (r0 ) := {tr0 t−1 : t ∈ Λ}. Then Equality (3.6) and the relation  η(s)2 < ∞ s∈CΛ (r0 )

imply that the set CΛ (r0 ) is finite. Consequently, the subspace generated by {(vt ⊗ ut )(η(r0 )) : t ∈ Λ} is finite dimensional. Let us denote v 0 := v|Λ and u0 := u|Λ . By the above, as well as [2, Proposition A.1.12], the unitary representation v 0 ⊗ u0 ⊗ v 0 ⊗ u0 contains the trivial one dimensional representation of Λ. Using [2, Proposition A.1.12] again, we know that v 0 contains a finite dimensional subrepresentation of Λ. Now, [1, Theorem 9] tells us that (Γ, Λ) has property T .  The above gives the converse of the statement in the paragraph after [1, Definition 6]. More precisely, one has the following result. Notice that in the case of possibly uncountable group, we also have the corresponding statement for strong property T of (A β,r Γ, Cr∗ (Λ)) and property Ta of (Γ, Λ). Corollary 3.5. Let Γ be a countable discrete group acting on a unital C ∗ -algebra A through an action β such that there is a β-invariant tracial state on A. If Λ is a subgroup of Γ, then (A β,r Γ, Cr∗ (Λ)) has property T if and only if (Γ, Λ) has property T . The above corollary can be regarded as a generalization of the following easy fact (which can be verified by using the corresponding statement as (F1) for groups): if Γ acts on another group Ξ by group automorphisms, then for a subgroup Λ of Γ, the pair (Ξ  Γ, Λ) has property T if and only if (Γ, Λ) has property T . However, unlike the group case, Cr∗ (Γ) is not a quotient C ∗ -algebra of A β,r Γ in general, and the above corollary is not at all obvious. To see an example of this, we consider Γ to be a finite group and n := |Γ|. Suppose that A is the unital C ∗ -algebra c0 (Γ), equipped with the left translation action β by Γ. The averaging of coefficients of functions in A produces a β-invariant mean on A. However, since A β,r Γ = Mn (C) is a simple C ∗ -algebra, the reduced group C ∗ -algebra Cr∗ (Γ) cannot be a quotient C ∗ -algebra of A β,r Γ. If Γ is amenable and A has a tracial state, then, by [2, Theorem G.1.7], there always exists a βinvariant tracial state on A. This, together with Proposition 3.4 and [2, Theorem 1.1.6], gives the following partial generalization of the main result in [5]. Observe that even in the case when Γ is the trivial group, one cannot get the conclusion that A is finite dimensional (as in [5, Theorem 2.3]) without the assumption of A being nuclear and “having enough tracial states”. Corollary 3.6. Let Γ be an amenable discrete group and A be a unital C ∗ -algebra having a tracial state. If there exists an action β of Γ on A such that A β,r Γ has strong property T , then Γ is finite. If, in addition, Γ is countable, we can replace strong property T by property T . 4. (Strong) property T for full and reduced crossed products The aims of this section are to obtain relations between property T of full crossed products and reduced crossed products, as well as to give an equivalent description for strong property T of C ∗ crossed products. As in the above, Λ ⊆ Γ is a subgroup and B ⊆ A is a β|Λ -invariant unital C ∗ -subalgebra. We denote ι : B β|Λ Λ → A β Γ to be the canonical map. It is known that ι is not injective in general, but it is true in some special cases (see [13], [8, Lemma 2.3(i)], [4, Proposition 3.1]).

A FULL DESCRIPTION OF PROPERTY T OF UNITAL C ∗ -CROSSED PRODUCTS

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Definition 4.1. (a) Let H be a Hilbert ∗ -bimodule over A and w : Γ → B(H) be a unitary representation. Then (H, w) is called a β-covariant ∗ -bimodule over A if wt (a · ξ) = βt (a) · wt (ξ) and wt (ξ · a) = wt (ξ) · βt (a)

(a ∈ A; t ∈ Γ; ξ ∈ H).

(4.1)

(b) (A, Γ, β; Λ, B) is said to have strong property T if for any β-covariant ∗ -bimodule (H, w) of A and any almost A-central almost w-invariant unit vector {ξi }i∈I in H, one has ξi − P B,w|Λ (ξi ) → 0, where P B,w|Λ is the projection from H onto the subspace, HB,w|Λ , of B-central w|Λ -invariant vectors. (c) The action β is said to have strong property T if (A, Γ, β; Γ, A) has strong property T . As will be seen in Example 4.2 below, β-covariant ∗ -bimodules over A are different from Hilbert -bimodules over A β Γ. The problem is that although the two equalities in (4.1) produce both a left and a right A β Γ-multiplications on H, these two multiplications do not commute in general. In fact, a contractive A β Γ-bimodule structure on a Hilbert space H requires a ∗ -bimodule structure over A together with two (instead of one) unitary representations of Γ on H satisfying certain compatibility conditions. ∗

In the view of this, the definition of strong property T of an action β is not a verbal translation of the strong property T of Aβ Γ (although we will see in Theorem 4.4 below that they are indeed equivalent). Example 4.2. Suppose that A := C ⊕ C and β is the action of Z/2Z on A such that β¯1 (x, y) := (y, x) (x, y ∈ C). If τ is the state on A defined by τ (x, y) := (x+y)/2 (x, y ∈ C), then τ is β-invariant. Suppose that H and u are as in in the proof of Proposition 3.4 for the state τ . Then H is two dimensional and is canonically a Hilbert ∗ -bimodule over A. It is not hard to see that (H, u) is a β-covariant ∗ -bimodule over A. Let us set a := (1, 0) ∈ A, s := ¯0, t := ¯1 ∈ Z/2Z and ξ := (0, 1) ∈ H. Then one has (at · ξ) · as = (1, 0) but at · (ξ · as) = (0, 0). Consequently, H is not a Hilbert ∗ -bimodule over A β Γ. Nevertheless, if (H, w) is a β-covariant ∗ -bimodule, one can obtain a contractive A β,r Γ-bimodule structure on H ⊗ 2 (Γ), instead of on H. More precisely, let Φ and Ψ be the ∗ -representation and ∗ -antirepresentation of A corresponding to the left and the right A-multiplications on H. Then by considering the unitary U ∈ B(H ⊗ 2 (Γ)) as in (3.3) when v ⊗ u is replaced by w, together with a similar argument as (3.4), we know that the assignment ¯ Ψ(bs) : ξ ⊗ δr → ws−1 (ξ · b) ⊗ δrs (ξ ∈ H; b ∈ A; r, s ∈ Γ) ¯ : A β,r Γ → B(H ⊗ 2 (Γ)). Furthermore, using a similar lines of induces a ∗ -anti-representation Ψ ¯ argument as in (3.5), we know that Ψ(bs) commutes with (Φ × λ)(at). This gives part (a) of the following lemma. Parts (b) and (c) are easy to verify. Lemma 4.3. Let (H, w) be a β-covariant ∗ -bimodule over A, and η ∈ H ⊗ 2 (Γ). (a) H ⊗ 2 (Γ) is a Hilbert ∗ -bimodule over A β,r Γ under the following multiplications: at • (ξ ⊗ δr ) = β(tr)−1 (a) · ξ ⊗ δtr and (ξ ⊗ δr ) • bs = ws−1 (ξ · b) ⊗ δrs . (b) η is B-central if and only if βs−1 (b) · η(s) = η(s) · b

(b ∈ B; s ∈ Γ).

(4.2)

(c) η is B β,r Λ-central if and only if it satisfies (4.2) as well as wt (η(s)) = η(tst−1 )

(s ∈ Γ; t ∈ Λ).

(4.3)

Theorem 4.4. Let A, Γ, β, Λ, B and ι be as in the beginning of this section. Then the following statements are equivalent. S1). (A,  property T .  Γ, β; Λ,B) has strong S2). A β Γ, ι B β|Λ Λ has strong property T . S3). (A β,r Γ, B β,r Λ) has strong property T .

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QING MENG AND CHI-KEUNG NG

In particular, A β Γ (equivalently, A β,r Γ) has strong property T if and only if the action β has strong property T . Proof. (S1) ⇒ (S2). Let H be a Hilbert ∗ -bimodule over A β Γ and {ξi }i∈I be an almost A β Γ-central unit vector in H. Obviously, H is a Hilbert ∗ -bimodule over A under the induced module structures. If we define w : Γ → B(H) by wt ξ := t · ξ · t−1 (ξ ∈ H; t ∈ Γ), (4.4) then it is easy to check that (H, w) is a β-covariant ∗ -bimodule, and {ξi }i∈I is an almost A-central almost w-invariant unit vector. The required implication now follows from the easy fact that Hι(Bβ|Λ Λ) = HB,w|Λ . (S2) ⇒ (S3). This follows from Fact (F1). (S3) ⇒ (S1). Let {ξi }i∈I be an almost A-central almost w-invariant unit vector in H. By Lemma 4.3(a), the Hilbert space H ⊗ 2 (Γ) becomes a Hilbert ∗ -bimodule over A β,r Γ. Furthermore, we have at • (ξi ⊗ δe ) − (ξi ⊗ δe ) • at = βt−1 (a) · ξi ⊗ δt − wt−1 (ξi ) · βt−1 (a) ⊗ δt  → 0, which, together with an approximation argument, verifies that {ξi ⊗ δe }i∈I is an almost A β,r Γ-central unit vector. Since (Aβ,r Γ, Bβ,r Λ) has strong property T , if we set ηi := P Bβ,r Λ (ξi ⊗δe ), then ξi ⊗δe −ηi  → 0. This means that ξi − ηi (e)2 + ηi − ηi (e) ⊗ δe 2 → 0. (4.5) Fix any i ∈ I. We learn from Lemma 4.3(c) that b · ηi (e) = ηi (e) · b (b ∈ B) Consequently, ηi (e) ∈ H ξi − ηi (e) → 0. ∗

B,w|Λ

and

wt (ηi (e)) = ηi (e) (t ∈ Λ).

. This, together with Relation (4.5), tells us that ξi − P B,w|Λ (ξi ) ≤ 

As said in the paragraph following Definition 4.1, the above theorem is not obvious since β-covariant -bimodules over A are different from Hilbert ∗ -bimodules over A β Γ.

For the same reason, we do not know if Statement (T2) below implies Statement (T1), without the assumption of Γfc = {e} (even in the case when Λ = Γ and B = A). Consequently, we do not make Statement (T1) as the definition of the action β having property T . Proposition 4.5. Let A, Γ, β, Λ, B and ι be as in the beginning of this section. Set CΛ (s) := {tst−1 : t ∈ Λ}

and

ΓΛ fc := {s ∈ Γ : CΛ (s) is a finite set}.

Consider the following statements. T1). If (H, w) is a β-covariant ∗ -bimodule over A containing an almost A-central almost w-invariant B,w|Λ unit vector, then there .  is a non-zero vector in H  T2). A β Γ, ι B β|Λ Λ has property T . T3). (A β,r Γ, B β,r Λ) has property T . (a) One has (T1) ⇒ (T2) ⇒ (T3). (b) If ΓΛ fc = {e}, then (T3) ⇒ (T1). Λ (c) If Γfc = {s ∈ Γ : st = ts, for any t ∈ Λ}, then (T3) ⇒ (T2). Proof. (a) The implication (T1) ⇒ (T2) follows from the same argument as that for (S1) ⇒ (S2) in Theorem 4.4, while the implication (T2) ⇒ (T3) follows from Fact (F1). (b) As in the beginning of the proof for (S3) ⇒ (S1) in Theorem 4.4, if {ξi }i∈I is an almost A-central almost w-invariant unit vector in a β-covariant ∗ -bimodule H, then {ξi ⊗ δe }i∈I is an almost A β,r Γcentral unit vector in H ⊗ 2 (Γ).

A FULL DESCRIPTION OF PROPERTY T OF UNITAL C ∗ -CROSSED PRODUCTS

9

 Bβ,r Λ Hence, property T of (A β,r Γ, B β,r Λ) produces a non-zero element η ∈ H ⊗ 2 (Γ) . The being trivial, Relation (4.3) and the condition hypothesis of ΓΛ fc  η(r)2 < ∞ (s ∈ Γ) r∈CΛ (s)

will force η(s) = 0 whenever s = e. In other words, η = η(e) ⊗ δe . It now follows from Lemma 4.3(c) (see (4.2) and (4.3)) that η(e) ∈ HB,w|Λ \ {0}. (c) Let H be a Hilbert ∗ -bimodule over A β Γ and {ξi }i∈I be an almost A β Γ-central unit vector in H. As in the proof of part (a) above, one knows that (H, w) is a β-covariant ∗ -bimodule if w is as defined in (4.4), and that {ξi }i∈I is almost A-central and almost w-invariant. The argument of part (b) will then  Bβ,r Λ produce an element η ∈ H ⊗ 2 (Γ) \ {0}. Consider r0 ∈ Γ with η(r0 ) = 0, and set ξ0 := r0 · η(r0 ). For any b ∈ B, as r0−1 br0 · η(r0 ) = η(r0 ) · b (because of (4.2)), one has b · ξ0 = ξ0 · b (b ∈ B).  On the other hand, (4.3) and the requirement s∈CΛ (r0 ) η(s)2 < ∞ tells us that r0 ∈ ΓΛ fc . Hence, from the hypothesis and (4.3), one concludes that t · ξ0 = ξ0 · t

(t ∈ Λ).

This completes the proof. An obstruction in using the proof of part (c) to get the general case is that zero, when r0 , . . . , rn are all the distinct elements in the finite set CΛ (r0 ).

n

k=0 rk

 · η(rk ) could be

In the case when Λ = Γ, the assumption in part (b) above means that Γ is ICC, while the assumption of part (c) holds when Γ is either ICC or abelian. The final theorem in this paper gives an equivalent description of strong property T of arbitrary unital C ∗ -crossed products. Part (a) of it can be regarded as a generalization of the well-known fact that if Γ is a group acting on another group Ξ, then Ξ  Γ has property T if and only if Γ has property T and (Ξ  Γ, Ξ) has property T . This theorem needs the following lemma. Lemma 4.6. If Γ is a finite group and (A β,r Γ, B) has strong property T , then (A, B) has strong property T . Proof. Let H be a Hilbert ∗ -bimodule over A. Let Φ and Ψ be the ∗ -representation and the ∗ -antirepresentation of A on H corresponding to the left and the right A-multiplications. Employing the  ¯ × λ : A β,r Γ → B H ⊗ argument as in the proof of Lemma 3.1, one obtains a ∗ -anti-representation Ψ  2 (Γ) that induces the following right A β,r Γ-multiplication on H ⊗ 2 (Γ) ⊗ 2 (Γ): (ξ ⊗ δr ⊗ δ¯r ) • bs = ξ · βr−1 (b) ⊗ δr ⊗ δ¯s−1 r

(ξ ∈ H; b ∈ A; r, r , s ∈ Γ).

It is not hard to check that this multiplication commutes with the left A β,r Γ-multiplication induced by (Φ × λ) ⊗ id2 (Γ) . Hence, H ⊗ 2 (Γ) ⊗ 2 (Γ) becomes a Hilbert ∗ -bimodule over A β,r Γ.  Suppose that¯ {ξi }i∈I is an almost A-central unit vector in H. As |Γ| < ∞, one can define ηi := r∈Γ ξi ⊗ δr ⊗ δr (i ∈ I). For any a ∈ A and t ∈ Γ, one has        at • ηi − ηi • at =  β(ts)−1 (a) · ξi ⊗ δts ⊗ δ¯s − ξi · βr−1 (a) ⊗ δr ⊗ δ¯t−1 r    s∈Γ r∈Γ         βr−1 (a) · ξi − ξi · βr−1 (a) ⊗ δr ⊗ δ¯t−1 r  → 0. =   r∈Γ

i is an almost A β,r Γ-central unit vector. Hence, |Γ|η1/2 i∈I

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QING MENG AND CHI-KEUNG NG

As (A β,r Γ, B) has strong property ηi − P B (ηi ) → 0. For every i ∈ I, if we  T , wer,sknow that B consider the decomposition P (ηi ) = r,s∈Γ ζi ⊗ δr ⊗ δ¯s , then  2       2 r,r r,s ¯  ξi − ζi  +  ζi ⊗ δr ⊗ δ s  (4.6)  → 0.  r=s  r∈Γ  B Since P B (ηi ) ∈ H ⊗ 2 (Γ) ⊗ 2 (Γ) , we have ζir,r

βr−1 (a) · ζir,s = ζir,s · βs−1 (a)

(a ∈ B; r, s ∈ Γ),

B

which gives ∈ H , for all r ∈ Λ (note that B is only assumed to be β|Λ -invariant). This, together 2  with (4.6), produces ξi − P B (ξi )2 ≤ ξi − ζie,e  → 0 as required. Theorem 4.7. Let A, Γ, β, Λ, B and ι be as in the beginning of this section. Suppose that A has a β-invariant tracial state. (a) The pair (A β,r Γ, B β,r Λ) has strong property T if and only if the following two conditions hold: i). (Γ, Λ) has property Ta ; ii). (A β,r Γ, B) has strong property T . (b) If Γ has the Haagerup property and B is a β-invariant C ∗ -subalgebra, then (A β,r Γ, B β,r Γ) has strong property T if and only if the following two conditions hold: i). Γ is a finite group; ii). (A, B) has strong property T . Proof. (a) The “only if” part follows from Proposition 3.4(a). To prove the “if” part, assume that Statements (i) and (ii) hold. Let H be a Hilbert ∗ -bimodule over A β,r Γ and {ξi }i∈I be an almost A β,r Γ-central unit vector in H. The strong property T of (A β,r Γ, B) gives ξi − P B (ξi ) → 0. Consider w to be the representation of Γ defined by wt ξ := t · ξ · t−1

(ξ ∈ H; t ∈ Γ).

Since {ξi }i∈I is an almost w-invariant unit vector in H, if P w|Λ is the projection from H onto Hw|Λ , then property Ta of (Γ, Λ) imply that ξi − P w|Λ (ξi ) → 0 (see (3.2)). As B is β|Λ -invariant and w satisfies (4.1), we know that HB is w|Λ -invariant. For each ζ ∈ H, t ∈ Λ and η ∈ HB , we have wt (ζ) − wt (P B (ζ)) = ζ − P B (ζ) ≤ ζ − wt−1 (η) = wt (ζ) − η, which implies that P B (wt (ζ)) = wt (P B (ζ)). Thus, P B (Hw|Λ ) ⊆ HB,w|Λ . Consequently, P B (P w|Λ (ξi )) ∈ HB,w|Λ = HBβ,r Λ , and       ξi − P Bβ,r Λ (ξi ) ≤  ξi − P B P w|Λ (ξi )         ≤ ξi − P B (ξi ) + P B (ξi ) − P B P w|Λ (ξi )  which goes to zero. (b) Suppose that Γ has the Haagerup property. If (A β,r Γ, B β,r Γ) has strong property T , it follows from part (a) that Γ is finite. Now, Lemma 4.6, together with part (a), ensures that (A, B) has strong property T . The converse follows from [12, Proposition 3.5].  Suppose that Γ is amenable. Then, by [2, Theorem G.1.7], the existence of a tracial state on A will imply the existence of a β-invariant tracial state. Therefore, in this case, we may assume that A has a tracial state in the hypothesis. Notice that if A does not have a β-invariant tracial state, then A β,r Γ cannot have a tracial state, and hence A β,r Γ has strong property T . Therefore, Theorem 4.7 gives a complete description for the pair (A β,r Γ, B β,r Λ) to have strong property T .

A FULL DESCRIPTION OF PROPERTY T OF UNITAL C ∗ -CROSSED PRODUCTS

11

In the special case when A = B = C, the algebra A always has a β-invariant tracial state and the Statement (ii) of part (a) above always holds. In this case, part (a) gives the equivalence of Statements (i) and (ii) of [10, Theorem 5.4]. Note that, in general, one cannot replace strong property T of (A β,r Γ, B) in part (a) by strong property T of (A, B), even when B = A and Λ = Γ. For example, if Γ is an infinite property T group, A = ∞ (Γ) ⊕ C, and the action β is the direct sum of the left translation action with the trivial action, then A has a β-invariant tracial state and A β,r Γ = ∞ (Γ) βΓ ,r Γ ⊕ Cr∗ (Γ) has strong property T (by [5, Corollary 2.4]), but A does not have property T . Corollary 4.8. (a) If Λ = Γ and is not finitely generated, then (A β,r Γ, B β,r Γ) has strong property T if and only if there is no β-invariant tracial state on A. (b) Suppose that A β,r Γ does not have strong property T . Then (A β,r Γ, B β,r Λ) has strong property T if and only if (Γ, Λ) has property Ta and (A β,r Γ, B) has strong property T . If, in addition, Λ = Γ and has the Haagerup property, then (A β,r Γ, B β,r Γ) has strong property T if and only if Γ is a finite group and (A, B) has strong property T . Let us end this article with the following characterization of amenability of G in terms of crossed product of Cb (G). Note that one cannot obtain this corollary through results in [7] or [5], because the amenability of G does not imply the amenability of Gd . Corollary 4.9. A non-compact locally compact group G is amenable if and only if Cb (G) βG ,r Gd does not have strong property T , where Gd is the group G equipped with the discrete topology and βG is the left translation action. Proof. If G is non-amenable, then there is no βG -invariant mean on Cb (G) and hence Cb (G) βG ,r Gd has strong property T . Conversely, suppose that G is amenable but Cb (G) βG ,r Gd has strong property T . Then Theorem 4.7(a) implies that Gd has property T (see also Remark 3.3). Hence, G has property T (cf. [2, Theorem 1.3.4]) and we arrive at the contradiction that G is compact.  Acknowledgement The authors are supported by the National Natural Science Foundation of China (11471168) and (11871285), the China Postdoctoral Science Foundation (2018M642633) as well as the Shandong Province Higher Educational Science and Technology Program (J18KA238). References M.B. Bekka, Property (T) for C ∗ -algebras, Bull. London Math. Soc. 38 (2006), 857-867. M.B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T), Cambridge (2008). N.P. Brown, Kazhdan’s property T and C ∗ -algebras, J. Funct. Anal., 240 (2006), 290-296. S. Itoh, Conditional expectations in C ∗ -crossed products, Trans. Amer. Math. Soc. 267 (1981), 661-667. B.J. Jiang and C.K. Ng, Property T of reduced C ∗ -crossed products by discrete groups, Ann. Funct. Anal., 7(3) (2016), 381-385. [6] P. Jolissaint, On Property (T) for Pairs of Topological Groups, Enseign. Math. (2) 51 (2005), 31-45. [7] F. Kamalov, Property T and amenable transformation group C ∗ -algebras, Canad. Math. Bull. 58 (2015), 110-114. [8] E. Kirchberg and S. Wassermann, Exact groups and continuous bundles of C ∗ -algebras, Math. Ann. 315 (1999), 169–203. [9] D. Kyed and P. M. Soltan, Property (T) and exotic quantum group norms, J. Noncommut. Geom. 6 (2012), 773-800. [10] C.W. Leung and C.K. Ng, Property (T) and strong property (T) for unital C ∗ -algebras, J. Funct. Anal. 256 (2009), 3055-3070. [11] Q. Meng and C.K. Ng, Invariant means on measure spaces and property T of C ∗ -algebra crossed products, preprint. [12] C.K. Ng, Property T for general C ∗ -algebras, Math. Proc. Camb. Phil. Soc. 156 (2014), 229-239. [13] N. Ozawa, Amenable actions and exactness for discrete groups, C. R. Acad. Sci. Paris S´ er. I Math. 330 (2000), 691–695. [14] G. K. Pedersen, C ∗ -algebras and their automorphism groups, Academic Press (1979). [15] Y. Shalom, Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s Property (T), Trans. Amer. Math. Soc. 351 (1999), 3387-3412.

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[16] Y. Suzuki, Haagerup property for C ∗ -algebras and rigidity of C ∗ -algebras with property (T), J. Funct. Anal. 265 (2013), 1778-1799. (Qing Meng) School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong province 273165, China. E-mail address: [email protected] (Chi-Keung Ng) Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China. E-mail address: [email protected]; [email protected]