Negative definite functions for C*-dynamical systems
aa r X i v : . [ m a t h . OA ] D ec Negative definite functions for C ∗ -dynamical systems Erik B´edos, Roberto ContiOctober 4, 2018
Abstract
Given an action α of a discrete group on a unital C ∗ -algebra A , we introduce anatural concept of α -negative definiteness for functions from G to A , and examinesome of the first consequences of such a notion. In particular, we prove analogs oftheorems due to Delorme-Guichardet and Schoenberg in the classical case where A is trivial. We also give a characterization of the Haagerup property for the action α when G is countable. MSC 2010 : 46L55, 43A50, 43A55.
Keywords : negative definite function, C ∗ -dynamical system, C ∗ -crossed product,equivariant action, one-cocycle, Schoenberg type theorem, semigroup of completelypositive maps, Haagerup property for actions. Given a C ∗ -dynamical system ( A, G, α ), Anantharaman-Delaroche introduced in [1] theconcept of positive definiteness for A -valued continuous functions on G relative to theaction α . She also explained how this notion could be used to characterize the amenabilityof actions of discrete groups on von Neumann algebras and on commutative C ∗ -algebras.More recently, it has been shown [11, 3] that any α -positive definite function on G takingvalues in the center of A naturally induces a completely positive map both on the reducedand the full C ∗ -crossed products associated to a discrete unital system ( A, G, α ).Parallel to the classical notion of positive definiteness for a complex function on a group,it has also been very fruitful to consider negative definite functions. (By negative definitewe always mean the same as what is called conditionally negative definite, or conditionallyof negative type, by some authors). Such functions play an important role in characterizingseveral properties of groups, such as the Haagerup property [8] and property (T) [14, 4].Somewhat surprisingly, a study of negative definite functions for C ∗ -dynamical systemsso far has been missing in the literature. Our main goal in writing this paper is to startfilling this gap by introducing and investigating the first basic concepts. In order to makethis paper easily accessible, we stick to the case of a unital discrete C ∗ -dynamical system( A, G, α ), but we do not see see any serious obstruction in extending most of our resultsto the general case almost mutatis mutandis .We note that (conditionally) negative definiteness for real functions on locally compactgroupoids were introduced by Tu in [20] (see also [18]). As for groups, his definition has1 natural generalization to complex functions. In the case of the transformation groupoidassociated to an action of a discrete group G on a compact Hausdorff space Ω, it is notdifficult to deduce that our concept of negative definiteness for a function from G to C (Ω)(relative to the induced action) is the same as the one obtained after transposing Tu’sdefinition. We also mention a very recent paper [17] of Moslehian where he considersconditionally positive kernels on sets with values in C ∗ -algebras. It should be noted thatour definition of α -negative definiteness may be formulated by using his terminology (seeRemark 3.7), but that there is otherwise little overlap between our paper and his.Among our main results, we mention a Delorme-Guichardet type theorem (cf. Theorem3.17), saying that a function ψ on G taking values in the positive cone of A and vanishingon the identity of G is α -negative definite if and only if it can be represented in the form ψ ( g ) = h c ( g ) , c ( g ) i for a symmetric one-cocycle c relative to an α -equivariant action of G on a Hilbert A -module. We also obtain a natural generalization of the classical Schoenbergtheorem, which provides a bridge between α -positive and α -negative definiteness for center-valued functions on G (cf. Theorem 3.21). As an application, we obtain a characterizationof the Haagerup property for α when G is countable (cf.’Theorem 3.24). This notion wasrecently introduced by Dong and Ruan in [11].We hope that the present work will provide useful tools in noncommutative harmonicanalysis and potential theory, e.g., in the study of C ∗ -dynamical systems, of semigroupsof completely positive maps, and of noncommutative Dirichlet forms. We discuss briefly acouple of examples of this sort, but we expect that other similar applications will appearsoon. In a different direction, it might be interesting to enlarge our set up and studynegative definiteness for functions from G × A into A that are linear in the second variable,as we did for positive definiteness in [3]. We plan to return to this in a subsequent work. Let A be a C ∗ -algebra. We will denote the center of A by Z ( A ), the self-adjoint part of A by A sa , the cone of positive elements in A by A + and the n × n matrices over A forsome natural number n by M n ( A ). By a Hilbert A -module we will mean a right Hilbert C ∗ -module over A , as defined for instance in [15].We record here some lemmas that we will need in the sequel. The first one is provenin [16] (see Lemma 3.1 therein). Lemma 2.1.
The Schur product of a positive matrix in M n ( A ) and a positive element in M n ( Z ( A )) is still positive in M n ( A ) . Lemma 2.2.
Assume B is a commutative C ∗ -algebra, n ∈ N and let [ b ij ] ∈ M n ( B ) + .Then [ b ∗ ij ] ∈ M n ( B ) + .Proof. We may write [ b ij ] = C ∗ C for some C = [ c ij ] ∈ M n ( B ). Consider i, j ∈ { , . . . , n } .Then we have b ij = P nk =1 c ∗ ki c kj . Since B is commutative, we get b ∗ ij = n X k =1 c ∗ kj c ki = n X k =1 ( c ∗ ki ) ∗ c ∗ kj . Thus, setting D = [ c ∗ ij ] ∈ M n ( B ), we get [ b ∗ ij ] = D ∗ D ∈ M n ( B ) + . (cid:3) emma 2.3. Let X be a Hilbert A -module and assume x , . . . , x n ∈ X are such that h x i , x j i ∈ Z ( A ) , for all i, j = 1 , . . . , n . Then the transposed matrix (cid:2) h x j , x i i (cid:3) is positive in M n ( Z ( A )) .Proof. It is well known (cf. [15, Lemma 4.2]) that the matrix [ h x i , x j i ] is positive in M n ( A ).Since this matrix lies in M n ( Z ( A )) by assumption, it follows that (cid:2) h x i , x j i (cid:3) ∈ M n ( Z ( A )) + .Thus, using Lemma 2.2, we get (cid:2) h x j , x i i (cid:3) = (cid:2) h x i , x j i ∗ ] ∈ M n ( Z ( A )) + . (cid:3) Lemma 2.4.
Assume B is a commutative C ∗ -algebra. Let Γ = [ γ ij ] ∈ M n ( B ) and let e ◦ Γ := [ e γ ij ] ∈ M n ( B ) denote its Schur exponential. If Γ is positive, then e ◦ Γ is positivetoo.Proof. It is well known that the assertion is true when B = C . Realizing B as thecontinuous functions on its Gelfand spectrum Ω and identifying M n ( B ) with C (Ω , M n ( C ))in the natural way, we have e ◦ Γ ( ω ) = [ e γ ij ( ω )] = [ e γ ij ( ω ) ] = e ◦ [ γ ij ( ω )] = e ◦ Γ( ω ) for all ω ∈ Ω. Assume now that Γ ∈ M n ( B ) + and let ω ∈ Ω. Then we have Γ( ω ) ∈ M n ( C ) + , so we get e ◦ Γ ( ω ) = e ◦ Γ( ω ) ∈ M n ( C ) + . This shows that e ◦ Γ ∈ M n ( B ) + . (cid:3) Let α : G → Aut( A ) denote an action of a (discrete) group G on A . Following [1, 2] wewill say that a function ϕ : G → A is α -positive definite if for any n ∈ N and g , . . . , g n ∈ G ,we have h α g i (cid:16) ϕ ( g − i g j ) (cid:17)i ≥ M n ( A ). In other words, for any n ∈ N , g , . . . , g n ∈ G and a , . . . , a n ∈ A , we have n X i,j =1 a ∗ i α g i (cid:0) ϕ ( g − i g j ) (cid:1) a j ≥ A . In the scalar case (i.e., A = C ), one recovers the classical notion of positive definite-ness.We recall from [1, Proposition 2.4] that if ϕ : G → A is α -positive definite, then forevery g ∈ G we have α g ( ϕ ( g − )) = ϕ ( g ) ∗ . (1)Moreover, if e denotes the identity of G , we have ϕ ( e ) ∈ A + . (2)We will also need the following two results. Lemma 2.5.
Assume ϕ : G → Z ( A ) is α -positive definite. Define ϕ ∗ : G → Z ( A ) by ϕ ∗ ( g ) = ϕ ( g ) ∗ for each g ∈ G . Then ϕ ∗ is α -positive definite.Proof. Let g , . . . , g n ∈ G . Then (cid:2) α g i (cid:0) ϕ ( g − i g j ) (cid:1)(cid:3) ∈ M n ( Z ( A )) + . Using Lemma 2.2, weget (cid:2) α g i (cid:0) ϕ ∗ ( g − i g j ) (cid:1)(cid:3) = (cid:2) α g i (cid:0) ϕ ( g − i g j ) ∗ (cid:1)(cid:3) = (cid:2) α g i (cid:0) ϕ ( g − i g j ) (cid:1) ∗ (cid:3) ∈ M n ( Z ( A )) + . (cid:3) emma 2.6. Assume ϕ : G → A and ϕ : G → Z ( A ) are both α -positive definite. Thenthe pointwise product ϕ ϕ from G to A is also α -positive definite.Proof. This follows from a straightforward application of Lemma 2.1. (cid:3) C ∗ -dynamical sys-tem Since the concept of (conditionally) negative definiteness for complex functions on groupsis useful in many contexts (see e.g. [5, 8]), it is natural to investigate a notion of negativedefiniteness relative to C ∗ -dynamical systems. Throughout this paper, we let α : G → Aut( A ) denote an action of a (discrete) group G on a unital C ∗ -algebra A and let A α = { a ∈ A | α g ( a ) = a for all g ∈ G } denote the fixed-point algebra of A under α . Theidentity element of G will be denoted by e and the unit of A will be denoted by 1 A .The following definition is the natural generalization of the classical notion. Definition 3.1.
We will say that a function ψ : G → A is α -negative definite if α g ( ψ ( g − )) = ψ ( g ) ∗ for all g ∈ G and, for any n ∈ N , g , . . . , g n ∈ G and b , . . . , b n ∈ A with P ni =1 b i = 0, wehave n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j ≤ A . We will say that an α -negative definite function ψ is normalized when ψ ( e ) = 0.Moreover, we will let ND( α ) denote the set of all α -negative definite functions and setND ( α ) = { ψ ∈ ND( α ) | ψ ( e ) = 0 } .Clearly, ND( α ) contains every constant function from G to A of the form g → t A for some t ∈ R . Also, it follows immediately that ND( α ) is a cone (that is, the sumof α -negative definite functions as well as any positive multiple of an α -negative definitefunction are again α -negative definite) and that ND ( α ) is a subcone of ND( α ). Moreover,we have: Lemma 3.2.
The cones
ND( α ) and ND ( α ) are closed w.r.t. the pointwise norm-topology.Proof. As ND ( α ) is closed in ND( α ) with respect to the pointwise norm-topology, itsuffices to prove the assertion for ND( α ). Assume that { ψ β } is a net in ND( α ) convergingto some ψ : G → A w.r.t. the pointwise norm topology. Then for every g ∈ G we have α g ( ψ ( g − )) = lim β α g ( ψ β ( g − )) = lim β ψ β ( g ) ∗ = ψ ( g ) ∗ . Moreover, let g , . . . , g n ∈ G and let b , . . . , b n ∈ A satisfy P ni =1 b i = 0. Then for every β we have n X i,j =1 b ∗ i α g i (cid:0) ψ β ( g − i g j ) (cid:1) b j ≤ ,
4o we get n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j = lim β (cid:16) n X i,j =1 b ∗ i α g i (cid:0) ψ β ( g − i g j ) (cid:1) b j (cid:17) ≤ A + is norm-closed in A . Thus ψ ∈ ND( α ). (cid:3) Remark 3.3.
Let ψ ∈ ND( α ). Then we have ψ ( e ) ∗ = α e (cid:0) ψ ( e − ) (cid:1) = ψ ( e ) , so ψ ( e ) ∈ A sa .Moreover, taking n = 2, g = e , g = g , b = 1 A = − b in Definition 3.1, we get ψ ( e ) − ψ ( g ) − α g ( ψ ( g − )) + α g ( ψ ( e )) = ψ ( e ) + α g ( ψ ( e )) − (cid:0) ψ ( g ) (cid:1) ≤ , hence Re (cid:0) ψ ( g ) (cid:1) ≥ (cid:0) ψ ( e ) + α g ( ψ ( e )) (cid:1) for all g ∈ G . In particular, if ψ ( e ) ≥
0, then Re (cid:0) ψ ( g ) (cid:1) ∈ A + for all g ∈ G . Remark 3.4.
Let ψ : G → A and define ψ : G → A by ψ ( g ) = ψ ( g ) − ψ ( e )for every g ∈ G , so ψ ( e ) = 0. Assume that ψ ( e ) ∈ A sa ∩ A α . We leave to the reader toverify that ψ ∈ ND( α ) if and and only if ψ ∈ ND ( α ). Remark 3.5.
Let ϕ : G → A be α -positive definite. In particular, ϕ ( e ) ∈ A sa , cf. (2).Assume that we also have ϕ ( e ) ∈ A α . Then the function ψ : G → A defined by ψ ( g ) = ϕ ( e ) − ϕ ( g )belongs to ND ( α ).Indeed, consider g ∈ G . Then, using (1), we have α g ( ψ ( g − )) = α g (cid:0) ϕ ( e ) − ϕ ( g − ) (cid:1) = ϕ ( e ) − α g ( ϕ ( g − )) = ϕ ( e ) ∗ − ϕ ( g ) ∗ = ψ ( g ) ∗ . Moreover, for any g , . . . , g n ∈ G and b , . . . , b n ∈ A with P ni =1 b i = 0, we have n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j = n X i,j =1 b ∗ i α g i (cid:0) ϕ ( e ) − ϕ ( g − i g j ) (cid:1) b j = − n X i,j =1 b ∗ i α g i (cid:0) ϕ ( g − i g j ) (cid:1) b j ≤ , as desired. Proposition 3.6.
Let ψ : G → A and define γ : G × G → A by γ ( g, h ) = ψ ( g ) ∗ + ψ ( h ) − ψ ( e ) − α g ( ψ ( g − h )) (3) for all g, h ∈ G . Then the following two assertions are equivalent: ( i ) ψ ∈ ND( α );( ii ) For every n ∈ N and g , . . . , g n ∈ G , the matrix [ γ ( g i , g j )] is positive in M n ( A ) . roof. Suppose that ψ ∈ N D ( α ) and let g , . . . , g n ∈ G , b , . . . , b n ∈ A . Set b = − P ni =1 b i and g = e . Then P ni =0 b i = 0, so we get n X i,j =0 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j ≤ . This gives that b ∗ ψ ( e ) b − (cid:16) n X i =1 b ∗ i (cid:17) n X j =1 ψ ( g j ) b j − n X i =1 b ∗ i α g i (cid:0) ψ ( g − i ) (cid:1)(cid:16) n X j =1 b j (cid:17) + n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j ≤ . As b ∗ ψ ( e ) b = P ni,j =1 b ∗ i ψ ( e ) b j and α g i (cid:0) ψ ( g − i ) (cid:1) = ψ ( g i ) ∗ for every i , this gives that n X i,j =1 b ∗ i γ ( g i , g j ) b j = − n X i,j =1 b ∗ i (cid:16) ψ ( e ) − ψ ( g j ) − ψ ( g i ) ∗ + α g i (cid:0) ψ ( g − i g j ) (cid:1)(cid:17) b j ≥ . Thus we have shown that ( ii ) holds.Conversely, suppose that ( ii ) is true. Note first that γ ( e, e ) = ψ ( e ) ∗ − ψ ( e ). Since γ ( e, e ) ∈ A + , we get that γ ( e, e ) = γ ( e, e ) ∗ = ψ ( e ) − ψ ( e ) ∗ = − γ ( e, e ), so γ ( e, e ) = 0,hence ψ ( e ) ∗ = ψ ( e ). Let now g ∈ G . Note that γ ( g, e ) = ψ ( g ) ∗ − α g ( ψ ( g − )), while γ ( e, g ) = ψ ( e ) ∗ + ψ ( g ) − ψ ( e ) − ψ ( g ) = ψ ( e ) ∗ − ψ ( e ) = 0. Since (cid:20) γ ( e, e ) γ ( e, g ) γ ( g, e ) γ ( g, g ) (cid:21) ∈ M ( A ) + we have γ ( e, g ) ∗ = γ ( g, e ). Thus, we get ψ ( g ) ∗ − α g ( ψ ( g − )) = γ ( g, e ) = γ ( e, g ) ∗ = 0, thatis, α g ( ψ ( g − )) = ψ ( g ) ∗ .Next, consider g , . . . , g n ∈ G and b , . . . , b n ∈ A with P ni =1 b i = 0. Then n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j = n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j − (cid:16) n X i =1 b i (cid:17) ∗ n X j =1 ψ ( g j ) b j − n X i =1 b ∗ i α g i ( ψ ( g − i )) (cid:16) n X j =1 b j (cid:17) = − (cid:16) n X i,j =1 b ∗ i γ ( g i , g j ) b j (cid:17) ≤ , since (cid:2) γ ( g i , g j ) (cid:3) ∈ M n ( A ) + . Thus ψ ∈ N D ( α ), as desired. (cid:3) Remark 3.7.
In a very recent work [17], Moslehian studies positive and conditionallypositive kernels on sets with values in C ∗ -algebras. One easily sees that a function ψ : G → A is α -negative definite in our sense if and only if the kernel from G × G into A given by K ( g, h ) = − α g ( ψ ( g − h )) is Hermitian and conditionally positive as defined in[17]. Proposition 3.6, which says that ψ is α -negative definite if and only if the kernel γ ispositive, may then be deduced from [17, Theorem 2.4]. We have included a self-containedproof of Proposition 3.6 for the ease of the reader. There is otherwise very little overlapbetween our paper and [17]. 6 emark 3.8. Let f : G → C and consider e f : G → A defined by e f ( g ) = f ( g ) 1 A . Ifthe function e f is α -negative definite, then it is immediate that f is negative definite.Conversely, if f is negative definite, then the kernel F ( g, h ) := f ( g ) + f ( h ) − f ( e ) + f ( g − h ) ∈ C on G × G is positive (cf. Proposition 3.6). But this implies that the kernel e F ( g, h ) := F ( g, h ) 1 A ∈ A is positive, and Proposition 3.6 gives now that e f is α -negative definite.We will let α ′ denote the action of G on Z ( A ) obtained by restricting each α g to Z ( A ). If ψ ∈ ND( α ) is Z ( A )-valued, then obviously ψ ∈ ND( α ′ ). Conversely, assume that ψ ∈ ND( α ′ ). Then ψ is Z ( A )-valued, and α g ( ψ ( g − )) = α ′ g ( ψ ( g − )) = ψ ( g ) ∗ for every g ∈ G . Moreover, let γ be defined as in (3). Of course, we also have γ ( g, h ) = ψ ( g ) ∗ + ψ ( h ) − ψ ( e ) − α ′ g ( ψ ( g − h ))for all g, h ∈ G . Now, consider g , . . . , g n ∈ G . Using Proposition 3.6 (with α ′ insteadof α ), we get that Γ := (cid:2) γ ( g i , g j ) (cid:3) ∈ M n ( Z ( A )) + , and this implies that Γ ∈ M n ( A ) + .Proposition 3.6 (now with α ) gives that ψ ∈ ND( α ). This means that we have: Proposition 3.9.
One has
ND( α ′ ) = (cid:8) ψ ∈ ND( α ) | ψ is Z ( A ) -valued (cid:9) . Assume that there exists a conditional expectation E : A → Z ( A ) satisfying α ′ ◦ E = E ◦ α . Then the map ψ → E ◦ ψ gives a surjection from ND( α ) onto ND( α ′ ). Thisfollows readily from the fact that a conditional expectation is completely positive (cf. [7]).Similarly, if there exists a state ω on A which is α -invariant, then the map ψ → ω ◦ ψ givesa surjection from ND( α ) onto ND( G ), the complex negative definite functions on G .We also record the following: Proposition 3.10. If ψ ∈ ND( α ′ ) , then Re ψ ∈ ND( α ′ ) . If, in addition, ψ is normalized,then Re ψ is Z ( A ) + -valued.Proof. We may assume that A is commutative and α ′ = α . So consider ψ ∈ ND( α ). Wehave to show that Re ψ ∈ ND( α ).One readily verifies that for each g ∈ G we have α g (cid:0) (Re ψ )( g − ) (cid:1) = (Re ψ )( g ) ∗ . Next,consider g , . . . , g n ∈ G and b , . . . , b n ∈ A with P ni =1 b i = 0. Then n X i,j =1 b ∗ i α g i (cid:0) (Re ψ )( g − i g j ) (cid:1) b j = 12 n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j + 12 n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) ∗ (cid:1) b j Since the first term on the right hand-side of this equality is negative, it suffices to showthat the second term is also negative. Now, using that A is commutative, we have n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) ∗ (cid:1) b j = n X i,j =1 (cid:0) b ∗ j α g i (cid:0) ψ ( g − i g j ) (cid:1) b i (cid:1) ∗ = (cid:16) n X i,j =1 ( b ∗ i ) ∗ α g i (cid:0) ψ ( g − i g j ) (cid:1) b ∗ j (cid:17) ∗ which is negative since P ni =1 b ∗ i = 0 and ψ ∈ ND( α ). Thus, Re ψ ∈ ND( α ).Finally, if ψ ∈ ND ( α ), then Remark 3.3 gives that Re ψ is A + -valued. (cid:3)
7n general, we do not know whether Re ψ belongs to ND( α ) whenever ψ ∈ ND( α ). Remark 3.11.
Let ψ ∈ ND( α ) and assume ψ takes its values in A + (or in Z ( A ) + ). When A = C , it is known that ψ / (or, more generally, ψ β with 0 < β <
1) is still α -negativedefinite, see for example [5, Corollary 2.10]. One might wonder whether this holds ingeneral. The first condition for α -negative definitess of ψ / is satisfied since for every g ∈ G we have α g ( ψ / ( g − )) = α g (( ψ ( g − )) / ) = (cid:0) α g ( ψ ( g − )) (cid:1) / = ψ ( g ) / = ψ / ( g ).However, it is not obvious how to proceed to handle the second condition.It is now time to introduce a natural class of normalized α -negative definite functionsrelated to α -equivariant actions of G on Hilbert A -modules and one-cocycles for suchactions, much in the same way as normalized complex negative definite functions on G are related to unitary representations of G on Hilbert spaces and their associated one-cocycles. We recall from [1] (see also [9]) that an α -equivariant action u of G on a Hilbert A -module X is a homomorphism u : g u g from G into the group I ( X ) of bijective C -linear isometries from X into itself, satisfying:(i) α g (cid:0) h x, y i (cid:1) = h u g x, u g y i , and(ii) u g ( x · a ) = ( u g x ) · α g ( a ),for all g ∈ G, x, y ∈ X , and a ∈ A . Definition 3.12.
We will say that x ∈ X is u - symmetric if h x, u g x i ∈ A sa for all g ∈ G ,and that it is u - central if h x, u g x i ∈ Z ( A ) for all g ∈ G .It follows easily by using property (i) of u that x ∈ X is u -symmetric (resp. u -central)if and only if h u g x, u h x i belongs to A sa (resp. Z ( A )) for all g, h ∈ G . Example 3.13.
Let x ∈ X be u -symmetric (resp. u -central) and let ψ : G → A + (resp. Z ( A ) + ) be defined by ψ ( g ) = h u g x − x, u g x − x i . Then ψ ∈ ND ( α ). Indeed, it is clear that ψ ( e ) = 0, and for every g ∈ G we have α g (cid:0) ψ ( g − (cid:1) = α g (cid:0) h u g − x − x, u g − x − x i (cid:1) = h x − u g x, x − u g x i = ψ ( g ) = ψ ( g ) ∗ . Moreover, for any g , . . . , g n ∈ G and b , . . . , b n ∈ A with P ni =1 b i = 0, we have n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j = n X i,j =1 b ∗ i α g i (cid:0) h u g − i g j x − x, u g − i g j x − x i (cid:1) b j = n X i,j =1 b ∗ i h u g j x − u g i x, u g j x − u g i x i b j = (cid:16) n X i =1 b ∗ i (cid:17)(cid:16) n X j =1 h u g j x, u g j x i b j (cid:17) − (cid:16) n X i,j =1 b ∗ i h u g j x, u g i x i b j (cid:17) − (cid:16) n X i,j =1 b ∗ i h u g i x, u g j x i b j (cid:17) + (cid:16) n X i =1 b ∗ i h u g i x, u g i x i (cid:17)(cid:16) n X j =1 b j (cid:17) = − n X i,j =1 b ∗ i h u g j x, u g i x i b j − n X i,j =1 b ∗ i h u g i x, u g j x i b j . u -symmetric (resp. u -central) vector x gives rise to a symmetric (resp.central) one-cocycle c : G → X (w.r.t. u ), as defined below, by setting c ( g ) = u g x − x foreach g ∈ G . Definition 3.14.
A map c : G → X will be called a one-cocycle (w.r.t. u ) if it satisfiesthat c ( gh ) = c ( g ) + u g ( c ( h )) , for all g, h ∈ G .
Moreover, such a one-cocycle c will be called(i) symmetric if h c ( g ) , c ( h ) i ∈ A sa for all g, h ∈ G , or, equivalently, if h c ( g ) , c ( h ) i = h c ( h ) , c ( g ) i for all g, h ∈ G ;(ii) central if h c ( g ) , c ( h ) i ∈ Z ( A ) for all g, h ∈ G .One-cocycles (wr.r.t. u ) of the form c ( g ) = u g x − x should be thought of as cobound-aries. Remark 3.15.
Assume that c : G → X is a one-cocycle (w.r.t. u ). For each g ∈ G onemay define a bijective affine map a g : X → X by a g x = u g x + c ( g ) . Then each a g is isometric in the sense that k a g x − a g y k = k x − y k for all x, y ∈ X , and oneeasily checks that a gh = a g a h for all g, h ∈ G . Hence g a g is a homomorphism from G into the group of affine isometric bijections from X into itself. Proposition 3.16.
Let c : G → X be a one-cocycle ( w.r.t. u ) and suppose that c issymmetric ( resp. central ) . Let ψ : G → A + ( resp. Z ( A ) + ) be defined for each g ∈ G by ψ ( g ) = h c ( g ) , c ( g ) i . Then ψ ∈ ND ( α ) .Proof. One has c ( g ) = c ( ge ) = c ( g ) + u g ( c ( e )), thus c ( e ) = 0 and it follows at once that ψ ( e ) = 0. Now, 0 = c ( e ) = c ( g − g ) = c ( g − ) + u g − ( c ( g )), that is c ( g − ) = − u g − ( c ( g ))and therefore α g ( ψ ( g − )) = α g (cid:0) h c ( g − ) , c ( g − ) i (cid:1) = h u g ( c ( g − )) , u g ( c ( g − )) i = h c ( g ) , c ( g ) i = ψ ( g ) = ψ ( g ) ∗ for every g ∈ G . Now, for g, h ∈ G , we have h c ( g − h ) , c ( g − h ) i = (cid:10) c ( g − ) + u g − ( c ( h )) , c ( g − ) + u g − ( c ( h )) (cid:11) = (cid:10) u g − ( c ( h ) − c ( g )) , u g − ( c ( h ) − c ( g )) (cid:11) , so we get α g ( ψ ( g − h )) = α g (cid:0) h c ( g − h ) , c ( g − h ) i (cid:1) = h c ( h ) − c ( g ) , c ( h ) − c ( g ) i . g , . . . , g n ∈ G and b , . . . , b n ∈ A with P ni =1 b i = 0, we have n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j = n X i,j =1 b ∗ i (cid:10) c ( g j ) − c ( g i ) , c ( g j ) − c ( g i ) (cid:11) b j . The last sum above is negative if c is symmetric or central. Indeed, if c is symmetric, then n X i,j =1 b ∗ i (cid:10) c ( g j ) − c ( g i ) , c ( g j ) − c ( g i ) (cid:11) b j = − n X i,j =1 b ∗ i h c ( g i ) , c ( g j ) i b j , which is negative since the matrix [ h c ( g i ) , c ( g j ) i ] is positive (cf. [15]). If c is central, then n X i,j =1 b ∗ i (cid:10) c ( g j ) − c ( g i ) , c ( g j ) − c ( g i ) (cid:11) b j = − n X i,j =1 b ∗ i h c ( g i ) , c ( g j ) i b j − n X i,j =1 b ∗ i b j h c ( g j ) , c ( g i ) i which is seen to be negative by using Lemma 2.1 and Lemma 2.3. (cid:3) A well known result of Delorme and Guichardet [10, 13] says that any normalizednegative definite function f : G → R + can be written in the form f ( s ) = k c ( s ) k for asuitable unitary representation π of G on a Hilbert space H and a one-cocycle c for π , i.e.,a map c : G → H satisfying c ( gh ) = c ( g ) + π g (cid:0) c ( h ) (cid:1) for all g, h ∈ G . In our context, as aconverse to Proposition 3.16, we have the following analogous result: Theorem 3.17.
Let ψ : G → A + be a normalized α -negative definite function. Thenthere exists a Hilbert A -module X , an α -equivariant action u of G on X and a symmetricone-cocycle c : G → X ( w.r.t. u ) such that ψ ( g ) = h c ( g ) , c ( g ) i for all g ∈ G . Moreover, the A -submodule of X generated by the c ( g ) ’s is dense in X .Finally, if ψ takes values in Z ( A ) + , then c is also central.Proof. For every ( g, h ) ∈ G × G , we set γ ( g, h ) = 12 (cid:16) ψ ( g ) + ψ ( h ) − α g ( ψ ( g − h )) (cid:17) ∈ A sa . Note that since ψ ( e ) = 0, this agrees with the expression for γ ( g, h ) given in (3), except forthe normalization factor 1 /
2. Since ψ is α -negative definite, we have that α g ( ψ ( g − h )) = α h (cid:0) α h − g ( ψ (( h − g ) − )) (cid:1) = α h ( ψ ( h − g )) and it readily follows from this equality that γ ( g, h ) = γ ( h, g ) for all g, h ∈ G . Moreover, according to Proposition 3.6, we have n X i,j =1 b ∗ i γ ( g i , g j ) b j ≥ g , . . . , g n ∈ G and b , . . . , b n ∈ A .Let now X := C c ( G, A ) denote the space of all A -valued, finitely supported functionson G . We can then define a right action of A on X by ( f · a )( g ) = f ( g ) a for every f ∈ X and every a ∈ A , and an A -valued semi-inner product on X by h f , f i := X g,h ∈ G f ( g ) ∗ γ ( g, h ) f ( h ) .
10s usual, setting N = { f ∈ X | h f, f i = 0 } and defining h f + N, f + N i := h f , f i ,X /N becomes an inner product A -module. We let X be its Hilbert A -module completionand identify X /N with its canonical image in X .Next, we define c : G → X by c ( g ) := ( δ g ⊙ A ) + N for each g ∈ G , where δ g ⊙ A denotes the function in X which takes the value 1 A at g and is zerootherwise. Then we clearly have that X /N = Span (cid:8) c ( g ) · a | g ∈ G, a ∈ A (cid:9) , so the A -submodule of X generated by the c ( g )’s is dense in X . We also note that h c ( g ) , c ( h ) i = h δ g ⊙ A , δ h ⊙ A i = γ ( g, h ) (4)for all g, h ∈ G , which immediately yields that c is symmetric. This also gives that h c ( e ) , c ( e ) i = γ ( e, e ) = ψ ( e ) / c ( e ) = 0. Moreover, using (4), we get that forall g, h, h ′ ∈ G , (cid:10) c ( gh ) − c ( g ) , c ( gh ′ ) − c ( g ) (cid:11) = γ ( gh, gh ′ ) − γ ( gh, g ) − γ ( g, gh ′ ) + γ ( g, g )= 12 h ψ ( gh ) + ψ ( gh ′ ) − α gh ( ψ (( gh ) − gh ′ )) − ψ ( gh ) − ψ ( g ) + α gh ( ψ (( gh ) − g )) − ψ ( g ) − ψ ( gh ′ ) + α g ( ψ ( g − gh ′ )) + 2 ψ ( g ) i = 12 α g (cid:16) − α h ( ψ ( h − h ′ )) + α h ( ψ ( h − )) + ψ ( h ′ ) (cid:17) = α g (cid:0) γ ( h, h ′ ) (cid:1) = α g (cid:16)(cid:10) c ( h ) , c ( h ′ ) (cid:11)(cid:17) . Consider now g, g , . . . , g n , g ′ , . . . , g ′ m ∈ G , a , . . . , a n , a ′ , . . . , a ′ m ∈ A , and F = n X i =1 c ( g i ) · a i , F ′ = m X j =1 c ( g ′ j ) · a ′ j ,U = n X i =1 (cid:0) c ( gg i ) − c ( g ) (cid:1) · α g ( a i ) , U ′ = m X j =1 (cid:0) c ( gg ′ j ) − c ( g ) (cid:1) · α g ( a ′ j )in X /N . Then, using our previous observation, we get h U, U ′ i = X i,j (cid:10) ( c ( gg i ) − c ( g )) · α g ( a i ) , ( c ( gg ′ j ) − c ( g )) · α g ( a ′ j ) (cid:11) = X i,j α g ( a i ) ∗ (cid:10) c ( gg i ) − c ( g ) , c ( gg ′ j ) − c ( g ) (cid:11) α g ( a ′ j )= X i,j α g ( a i ) ∗ α g (cid:0) h c ( g i ) , c ( g ′ j ) i (cid:1) α g ( a ′ j )= α g (cid:16) X i,j h c ( g i ) · a i , c ( g ′ j ) · a ′ j i (cid:17) = α g ( h F, F ′ i ) . k U k = kh U, U ik / A = k α g ( h F, F i ) k / A = kh F, F ik / A = k F k .As X /N = Span (cid:8) c ( g ) · a | g ∈ G, a ∈ A (cid:9) , we see that, for each g ∈ G , the map u g : X /N → X /N given by u g (cid:16) n X i =1 c ( g i ) · a i (cid:17) = n X i =1 (cid:0) c ( gg i ) − c ( g ) (cid:1) · α g ( a i )is well-defined, isometric and satisfies h u g F, u g F ′ i = α g (cid:0) h F, F ′ i (cid:1) for all F, F ′ ∈ X /N . Ittherefore extends to an isometry on X , that we also denote by u g , satisfying h u g x, u g y i = α g ( h x, y i )for all x, y ∈ X (by continuity). For F as above and a ∈ A , we have u g ( F · a ) = u g (cid:16) n X i =1 c ( g i ) · ( a i a ) (cid:17) = n X i =1 (cid:0) c ( gg i ) − c ( g ) (cid:1) · α g ( a i a ) = ( u g F ) · α g ( a ) . Therefore, u g ( x · a ) = u g ( x ) · α g ( a ) for all x ∈ X and a ∈ A (by continuity).Consider now g, h ∈ G . For every k ∈ G and a ∈ A , we have( u g u h )( c ( k ) · a ) = u g (cid:0) ( c ( hk ) − c ( h )) · α h ( a ) (cid:1) = (cid:0) c ( ghk ) − c ( g ) (cid:1) · α g ( α h ( a )) − (cid:0) c ( gh ) − c ( g ) (cid:1) α g ( α h ( a ))= ( c ( ghk ) − c ( gh )) · α gh ( a ) = u gh ( c ( k ) · a ) . Thus, by linearity, density and continuity, we get that u g u h = u gh . In particular, u g u g − = u g − u g = u e = id X (since u e ( c ( k ) · a ) = ( c ( k ) − c ( e )) · a = c ( k ) · a , as c ( e ) = 0). Hence each u g is invertible.Altogether, we have shown that u : g u g is an α -equivariant action of G on X .Finally, by the definition of u , for all g, h ∈ G , we have u g ( c ( h )) = (cid:0) c ( gh ) − c ( g ) (cid:1) · α g (1 A ) = c ( gh ) − c ( g ) . So c is a symmetric one-cocycle (w.r.t. u ). Since h c ( g ) , c ( g ) i = γ ( g, g ) = ψ ( g ) for all g ∈ G ,we are done with the first two assertions of the theorem.If ψ is assumed to be Z ( A ) + -valued, then we see from (4) that h c ( g ) , c ( h ) i belongs to Z ( A ) for all g, h ∈ G , i.e., c is central. (cid:3) Theorem 3.17 may probably be generalized to give a representation of any α -negativedefinite function (see [10, 13] for the classical case). However, for the time being we leavethis as an open problem. Remark 3.18.
The triple (
X, u, c ) associated to ψ in the previous theorem is unique in thefollowing sense. If X ′ is another Hilbert A -module, equipped with an α -equivariant action u ′ of G and a symmetric one-cocycle c ′ : G → X ′ (w.r.t. u ′ ) such that ψ ( g ) = h c ′ ( g ) , c ′ ( g ) i ′ g ∈ G and the A -submodule of X ′ generated by the c ′ ( g )’s is dense in X ′ , thenthere exists a unitary operator V from X to X ′ satisfying V u g V ∗ = u ′ g and V c ( g ) = c ′ ( g )for all g ∈ G .To see this, the main observation is that we have h c ( g ) , c ( h ) i = h c ′ ( g ) , c ′ ( h ) i ′ for all g, h ∈ G ; indeed,2 h c ( g ) , c ( h ) i = ψ ( g ) + ψ ( h ) − α g ( ψ ( g − h ))= h c ′ ( g ) , c ′ ( g ) i ′ + h c ′ ( h ) , c ′ ( h ) i ′ − h u ′ g ( c ′ ( g − h )) , u ′ g ( c ′ ( g − h )) i ′ = h c ′ ( g ) , c ′ ( g ) i ′ + h c ′ ( h ) , c ′ ( h ) i ′ − h c ′ ( h ) − c ′ ( g ) , c ′ ( h ) − c ′ ( g ) i ′ = h c ′ ( h ) , c ′ ( g ) i ′ + h c ′ ( g ) , c ′ ( h ) i ′ = 2 h c ′ ( g ) , c ′ ( h ) i . It is then easy to check that the map V : X → X ′ determined by V (cid:16) X i c ( g i ) · a i (cid:17) = X i c ′ ( g i ) · a i , g i ∈ G, a i ∈ A will do the job. Remark 3.19.
It follows readily from Proposition 3.16 and Theorem 3.17 that the coneof A + -valued normalized α -negative definite coincides with the set of functions of theform g
7→ h c ( g ) , c ( g ) i where c ranges over all symmetric one-cocycles (with respect to α -equivariant actions of G ). Similarly, the subcone of Z ( A ) + -valued normalized α -negativedefinite coincides with the set of functions of the form g
7→ h c ( g ) , c ( g ) i where c rangeseither over all symmetric and central one-cocycles, or over all central one-cocycles (withrespect to α -equivariant actions of G ). Remark 3.20.
Consider a function ψ : G → A + given by ψ ( g ) = h c ( g ) , c ( g ) i for some α -equivariant action u of G on Hilbert A -module X and a one-cocycle c : G → X (w.r.t. u ).Such a function will satisfy the first requirement, but not necessarily the second, in thedefinition of α -negative definiteness. Instead of the second requirement, it will satisfy n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j ≤ g , . . . , g n ∈ G and b , . . . , b n ∈ Z ( A ) with P ni =1 b i = 0. It might be worth to havea closer look at this class of functions in the future.A well known consequence of Schoenberg’s theorem (see e.g. [5, 4]) is that a function ψ : G → C is negative definite if and only if the function ϕ t := e − tψ is positive definite forall t >
0. We now proceed to show that a version of this result continues to hold in ourgeneralized setting, at least for central-valued functions.
Theorem 3.21.
Let ψ : G → A and consider the following two claims: ( i ) ψ is α -negative definite; ( ii ) e − tψ is α -positive definite for all t > . hen ( ii ) implies ( i ) .Moreover, suppose that ψ is Z ( A ) -valued Then ( i ) implies ( ii ) . Thus, in this case, the twoclaims above are equivalent.Proof. Assume that ( ii ) holds. Let g ∈ G . From [1, Proposition 2.4] we get that α g (cid:0) e − tψ ( g − ) (cid:1) = (cid:0) e − tψ ( g ) (cid:1) ∗ , hence e − tα g ( ψ ( g − )) = e − tψ ( g ) ∗ for all t >
0. Then [12, Theorem VIII.1.2] gives that α g ( ψ ( g − )) = ψ ( g ) ∗ . (5)Next, suppose g , . . . , g n ∈ G , b , . . . , b n ∈ A with P ni =1 b i = 0 and let ω be a state on A .By assumption, the scalar-valued function R ∋ t ω (cid:16) n X i,j =1 b ∗ i α g i (cid:0) e − tψ ( g − i g j ) (cid:1) b j (cid:17) is non-negative for t > t = 0. Thus its right-derivative at t = 0 must benon-negative, i.e., − ω (cid:16) n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j (cid:17) ≥ . (6)Now, using (5) with g = g − i g j , we get (cid:16) n X i,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j (cid:17) ∗ = n X i,j =1 b ∗ j α g i (cid:0) ψ ( g − i g j ) ∗ (cid:1) b i = n X i,j =1 b ∗ j α g i (cid:0) α g − i g j (cid:0) ψ ( g − j g i ) (cid:1)(cid:1) b i = n X i,j =1 b ∗ j α g j (cid:0) ψ ( g − j g i ) (cid:1) b i , which shows that P ni,j =1 b ∗ i α g i (cid:0) ψ ( g − i g j ) (cid:1) b j is self-adjoint. As (6) holds for every state ω on A we can therefore conclude that n X i,j =1 b ∗ i α g i (cid:0) e − tψ ( g − i g j ) (cid:1) b j ≤ . Thus we have shown that ψ ∈ ND( α ), that is, ( i ) holds.Suppose now that ψ ∈ ND( α ) is Z ( A )-valued. In order to show that ( ii ) holds in thiscase, it is enough to show that e − ψ is α -positive definite, i.e., that the Z ( A )-valued matrix (cid:2) α g i ( e − ψ ( g − i g j ) ) (cid:3) is positive in M n ( Z ( A )) for any given g , . . . , g n ∈ G . To this end, usingthe properties of the exponential function, we may write α g i (cid:0) e − ψ ( g − i g j ) (cid:1) = e − α gi ( ψ ( g − i g j )) = e ψ ( g i ) ∗ + ψ ( g j ) − ψ ( e ) − α gi ( ψ ( g − i g j )) e − ψ ( g i ) ∗ − ψ ( g j )+ ψ ( e ) i, j . Setting b i = e ψ ( e ) − ψ ( g i ) ∈ Z ( A ) for i = 1 , . . . , n , we get that h e − ψ ( g i ) ∗ − ψ ( g j )+ ψ ( e ) i = (cid:2) b ∗ i b j (cid:3) is positive in M n ( Z ( A )). Therefore, using Lemma 2.1, it is enough to show that h e ψ ( g i ) ∗ + ψ ( g j ) − α gi ( ψ ( g − i g j )) i ≥ . (7)Now, Proposition 3.6 gives that the matrix h ψ ( g i ) ∗ + ψ ( g j ) − α g i ( ψ ( g − i g j )) i is positive in M n ( Z ( A )). Since Lemma 2.4 says that the Schur exponential of a Z ( A )-valued positivematrix is still positive, we see that (7) is satisfied. Hence we are done. (cid:3) Corollary 3.22.
Let ψ be a normalized Z ( A ) -valued α -negative definite function. Thenthere exists a one-parameter semigroup ( M t ) t ≥ of unital completely positive maps on thefull crossed product C ∗ ( A, G, α ) satisfying M t ( F ) = e − tψ F for all t ≥ and all F ∈ C c ( G, A ) . Moreover, if Λ denotes the canonical homomorphismfrom C ∗ ( A, G, α ) onto the reduced crossed product C ∗ r ( A, G, α ) , then there also exists a one-parameter semigroup ( M rt ) t ≥ of unital completely positive maps on C ∗ r ( A, G, α ) satisfying M rt (cid:0) Λ( F ) (cid:1) = Λ( e − tψ F ) for all t ≥ and all F ∈ C c ( G, A ) , i.e., M rt ◦ Λ = Λ ◦ M t for all t ≥ .Proof. Both statements follow by combining Theorem 3.21 with [3, Proposition 4.3]. Thesecond statement can also be deduced from Theorem 3.21 and [11, Theorem 3.2]. (cid:3)
Remark 3.23.
Let ( M t ) t ≥ be as described in Corollary 3.22. Arguing as in [18, Proposi-tion 4.5] one obtains that the generator − ∆ of this semigroup has the dense ∗ -subalgebra C c ( G, A ) as its essential domain, and we have ∆ F = ψF for all F ∈ C c ( G, A ). (A similarremark is true for the semigroup ( M rt ) t ≥ .) Following Sauvageot [19] (see also [18]), onemay then associate to ( M t ) t ≥ a Dirichlet form L on C c ( G, A ), which may be describedin terms of a C ∗ -correspondence E over C ∗ ( A, G, α ) and a derivation δ : C c ( G, A ) → E .When A = C (Ω) is commutative, one may identify C ∗ ( A, G, α ) with the full C ∗ -algebra ofthe associated transformation groupoid ( G, Ω). In this case, Renault gives in [18, Theorem4.6] a concrete description of the pair (
E, δ ). We believe it should be possible to obtainan analogous description also when A is noncommutative.We recall that a function f : G → A is said to go to zero at infinity if, for any ǫ >
0, there exists a finite subset F ⊂ G such that k f ( g ) k < ǫ for all g / ∈ F (that is, g
7→ k f ( g ) k ∈ C ( G )). We denote by C ( G, A ) the space of all such functions.Next, assume that G is countable. We recall from [11] that α is said to have the Haagerup property if there exists a sequence ( h n ) of α -positive definite Z ( A )-valued func-tions on G such that h n ( e ) = 1 A , h n ∈ C ( G, A ) for all n ∈ N and k h n ( g ) − A k → n → ∞ for all g ∈ G . (Note that Dong and Ruan’s definition of α -positive definiteness in1511] is slightly different than the one introduced in [1], but this is essentially a matter ofconvention and does not affect the definition of the Haagerup property for α ). It is easyto check that α has the Haagerup property if the same property holds for a net ( h ι ) ι ∈ I instead of a sequence. It is a simple exercise to check that if G has the Haagerup property,then α has the Haagerup property. On the other hand, if α has the Haagerup propertyand there exists an α -invariant state ω on A , then G has the Haagerup property (for if( h n ) is sequence that works for α , then ( ω ◦ h n ) will work for G ).We will say that a function ψ : G → A + is spectrally proper if the function ℓ ψ : g inf sp (cid:0) ψ ( g ) (cid:1) is proper as a function from G to R + . Notice that this is a stronger property than requiringthat the function g
7→ k ψ ( g ) k is proper in the usual sense.The Haagerup property for a countable group G may be characterized by the existenceof a proper normalized negative definite function from G into R + (see [8] and referencestherein). Analogously, we have: Theorem 3.24.
Assume that G is countable. Then α has the Haagerup property if andonly if and there exists there exists a spectrally proper Z ( A ) + -valued normalized α -negativedefinite function on G .Proof. Assume first that α has the Haagerup property and let ( h n ) be a sequence as inthe definition. For each n ∈ N define ϕ n : G → Z ( A ) + by ϕ n ( g ) = h n ( g ) ∗ h n ( g ) for all g ∈ G . Then using Lemmas 2.5 and 2.6 we get that ( ϕ n ) is a sequence in C ( G, A ) of Z ( A ) + -valued α -positive definite functions satisfying ϕ n ( e ) = 1 A , and k ϕ n ( g ) − A k → n → ∞ for all g ∈ G .Let now ( K n ) be an increasing and exhausting sequence ( K n ) of finite subsets of G .Passing to a subsequence of ( ϕ n ) if necessary, we can assume that k A − ϕ n ( g ) k ≤ / n for all n ∈ N and g ∈ K n . Since k ϕ n ( g ) k ≤ k ϕ n ( e ) k = 1 (cf. [1, Proposition 2.4 ii)]), weget that 1 A − ϕ n ( g ) ∈ Z ( A ) + for all n and g . Moreover, (1 − / n )1 A ≤ ϕ n ( g ) ≤ A for all n ∈ N and g ∈ K n . Now, each function 1 − ϕ n is a Z ( A ) + -valued normalized α -negativedefinite function, cf. Remark 3.5. Since P ∞ j =1 k A − ϕ j ( g ) k < + ∞ for all g ∈ G , we candefine ψ : G → Z ( A ) + by ψ ( g ) = P ∞ j =1 (cid:0) A − ϕ j ( g ) (cid:1) . Using Lemma 3.2 we get that ψ isa normalized α -negative definite function. It remains to show that ψ is spectrally proper.For each n ∈ N , using that ϕ n ∈ C ( G, A ), we can find a finite subset F n ⊂ G suchthat k ϕ n ( g ) k < / g / ∈ F n . Since ϕ n ( g ) ≥
0, we have ϕ n ( g ) < A for all g / ∈ F n and K n ⊂ F n for each n .Define G n = S nj =1 F j , so K n ⊂ G n and ( G n ) is an increasing and exhausting sequenceof finite subsets of G . Consider g / ∈ G n . Then ϕ j ( g ) < A for j = 1 , . . . , n , so ψ ( g ) = ∞ X j =1 (1 A − ϕ j ( g )) ≥ n X j =1
12 1 A = n A . Thus ℓ ψ ( g ) ≥ n/
2. It is now clear that ℓ ψ is proper, i.e., ψ is spectrally proper, as desired.Conversely, assume that there exists a spectrally proper Z ( A ) + -valued normalized α -negative definite function ψ on G , and consider the net ( e − tψ ) t> . By Theorem 3.21, each16 − tψ is α -positive definite and takes its values in Z ( A ) + . Clearly, e − tψ ( e ) = 1 A for every t >
0. Moreover, for t > g ∈ G , we have k e − tψ ( g ) k = sup (cid:8) e − tλ | λ ∈ sp( ψ ( g )) (cid:9) = e − t ℓ ψ ( g ) , which goes to 0 as g → ∞ for each t > ℓ ψ is proper. Thus e − tψ ∈ C ( G, A ) for all t >
0. Finally, it is clear that lim t → k e − tψ ( g ) − A k = 0 for all t ∈ R + . Hence we concludethat α has the Haagerup property. (cid:3) Example 3.25.
Let us say that α is centrally amenable if there exists a net ( h i ) of finitelysupported α -positive definite Z ( A )-valued functions on G such that h i ( e ) = 1 A for all i and k h i ( g ) − A k → n → ∞ for all g ∈ G . Clearly, this is a stronger property than theHaagerup property for α . We also note that if α is centrally amenable, then α is amenablein the sense of Anantharaman-Delaroche [1] (and also as defined in [3]).Now, assume that α is amenable as defined by Brown and Ozawa in their book [7].Then α is centrally amenable. Indeed, if ( ξ i ) is a net satisfying the requirements of [7,Definition 4.3.1], then it is not difficult to see that the net ( h i ) in C c ( G, A ) defined by h i ( g ) = (cid:10) ξ i , e α g ( ξ i ) (cid:11) , where (cid:10) ξ, η (cid:11) = X s ∈ G ξ ( s ) ∗ η ( s ) and [ e α g ( ξ )]( h ) = α g (cid:0) ξ ( g − h ) (cid:1) for ξ, η ∈ C c ( G, A ) and g, h ∈ G , satisfies all the conditions needed for showing that α iscentrally amenable. The main point is that each h i is α -positive definite, as follows from [1,p. 300-301]. Hence, if G is countable, we can conclude that α has the Haagerup property,and Theorem 3.24 gives that there exists a spectrally proper Z ( A ) + -valued normalized α -negative definite function on G . Remark 3.26.
Recall (see e.g. [8, 4]) that when G is countable, then G has property(T) if and only if every negative definite function from G to C is bounded. One couldtherefore say that an action α has property (T) (resp. has the central property (T)) ifevery α -negative definite function (resp. every center-valued α -negative definite function)is bounded. Clearly α will have the central property (T) whenever it has property (T).Moreover, G will have property (T) whenever α has the central property (T).Indeed, assume α has the central property (T) and let f : G → C be negative definite.Define f : G → C by f ( g ) = f ( g ) − f ( e ). Then f is normalized and negative definite.Now let ψ : G → A be given by ψ ( g ) = f ( g ) 1 A . Then ψ is center-valued and normalized,and it follows from Remark 3.8 that ψ is α -negative definite. Using the assumption, ψ hasto be bounded. So f is bounded, and this clearly implies that f is bounded too. Hence, G has property (T).Note that if A has the strong property (T), as defined by Leung-Ng in [16], and G hasproperty (T), then any C ∗ -crossed product of A by G also has the strong property (T)[16, Theorem 4.6]. If one assumes that α has property (T), or the central property (T), itwould be interesting to know if one can find some (weaker) conditions on A ensuring that C ∗ ( A, G, α ) (or C ∗ r ( A, G, α )) still has the strong property (T).17 cknowledgements.
Most of the present work was done during visits made by E.B. atthe Sapienza University of Rome and by R.C. at the University of Oslo in 2015 and 2016.Both authors would like to thank these institutions for their kind hospitality.
References [1] C. Anantharaman-Delaroche: Syst`emes dynamiques non commutatifs etmoyennabilit´e.
Math. Ann. (1987), 297–315.[2] C. Anantharaman-Delaroche: Amenability and exactness for dynamical systems andtheir C ∗ -algebras, Trans. Amer. Math. Soc. (2002), 4153–4178.[3] E. B´edos, R. Conti: The Fourier-Stieltjes algebra of a C ∗ -dynamical system. Inter-nat. J. Math. (2016), 1650050 [50 pages].[4] B. Bekka, P. de la Harpe, A. Valette: Kazhdan’s property (T). New MathematicalMonographs, 11. Cambridge University Press, Cambridge, 2008.[5] C. Berg, J. P. Reus Christensen, P. Ressel: Harmonic analysis on semigroups. GTM , Springer-Verlag, Berlin-Heidelberg-New York, 1984.[6] B. Blackadar: Operator algebras. Theory of C ∗ -algebras and von Neumann alge-bras. Encyclopaedia of Mathematical Sciences, 122. Operator Algebras and Non-commutative Geometry, III. Springer-Verlag, Berlin, 2006.[7] N.P. Brown, N. Ozawa: C ∗ -algebras and finite-dimensional approximations. GraduateStudies in Mathematics , . American Mathematical Society, Providence, RI, 2008.[8] P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, A. Valette: Groups with the Haagerupproperty. Gromov’s a-T-menability. Progress in Mathematics, . Birkh¨auser Verlag,Basel, 2001.[9] F. Combes: Crossed products and Morita equivalence. Proc. London Math. Soc. (1984), 289–306.[10] P. Delorme: 1-cohomologie des repr´esentations unitaires des groupes deLie semi-simples et r´esolubles. Produits tensoriels continus de repr´esentations. Bull. Soc. Math. France (1977), 281–336.[11] Z. Dong, Z.-J. Ruan: A Hilbert module approach to the Haagerup property.
In-tegr. Equ. Oper. Theory (2012), 431–454.[12] N. Dunford, J.T. Schwartz: Linear Operators. I. General Theory. Pure and AppliedMathematics, Vol. 7 Interscience Publishers, Inc., New York; Interscience Publishers,Ltd., London, 1958.[13] A. Guichardet: Symmetric Hilbert spaces and related topics. Lecture Notes in Math-ematics, Vol. . Springer-Verlag, Berlin-New York, 1972.1814] P. de la Harpe, A. Valette: La propri´et´e (T) de Kazhdan pour les groupes localementcompacts (avec un appendice de Marc Burger). Ast´erisque (1989).[15] C. Lance: Hilbert C ∗ -modules. A toolkit for operator algebraists. London Mathe-matical Society Lecture Note Series, , Cambridge University Press, Cambridge,1995.[16] C.-W. Leung, C.-K. Ng: Property (T) and strong property (T) for unital C ∗ -algebras. J. Funct. Anal. (2009), 3055–3070.[17] M. Moslehian. Conditionally positive definite kernels on Hilbert C ∗ -modules.Preprint, arXiv 1611.08382.[18] J. Renault: Groupoid cocycles and derivation. Ann. of Funct. Anal. (2012), 1–20.[19] J.-L. Sauvageot: Tangent bimodule and locality for dissipative operators on C ∗ -algebras. Quantum probability and applications IV. Lecture Notes in Mathematics (1989), 322–338.[20] J.L. Tu: La conjecture de Baum-Connes pour les feuilltages moyennables.
K-Theory (1999), 215–264.[21] D.P. Williams: Crossed products of C ∗∗