Bekka-type amenabilities for unitary corepresentations of locally compact quantum groups
aa r X i v : . [ m a t h . OA ] M a y BEKKA-TYPE AMENABILITIES FOR UNITARYCOREPRESENTATIONS OF LOCALLY COMPACT QUANTUMGROUPS
XIAO CHEN
Abstract.
In this short note, further to Ng’s study, we extend Bekka amenabil-ity and weak Bekka amenability to general locally compact quantum groups.We generalize some Ng’s results to the general case. In particular, we showthat, a locally compact quantum group G is co-amenable if and only if thecontra-corepresentation of its fundamental multiplicative unitary W G is Bekkaamenable, and G is amenable if and only if its dual quantum group’s funda-mental multiplicative unitary W b G is weakly Bekka amenable. Introduction
The notion of amenability essentially begins with Lebesgue (1904). In 1929,von Neumann introduced and studied the class of amenable groups and usedit to explain why the Banach-Tarski Paradox occurs only for dimension greaterthan or equal to three. In 1950, Dixmier extended the concept of amenability totopological groups (see [11] and [15]). In 1970s, amenability and co-amenabilityfor Kac algebras were introduced by D. Voiculescu, studied further by M. Enockand J.-M. Schwartz and later by Z.-J. Ruan (see [14] and [12]). In [9], follow-ing Bekka’s paper [4], C.-K. Ng introduced Bekka amenability and weak Bekkaamenability for unitary co-representations of Kac algebras, and used them tocharacterize amenability and co-amenability for Kac algebras. Later, amenabil-ity and co-amenability for Hopf C ∗ -algebras was investigated by C.-K. Ng (see[8] and [10]). In 2003, E. B´ e dos, R. Conti and L. Tuset extended amenabilityand co-amenability to algebraic quantum groups and locally compact quantumgroups (see [3] and [2]).In this short note, we give some remarks on Ng’s paper [9]. We extend Bekkaamenability and weak Bekka amenability to general locally compact quantumgroups. Furthermore, we prove that a locally compact quantum group G isco-amenable if and only if the contra-corepresentation of its fundamental mul-tiplicative unitary W G is Bekka amenable, and G is amenable if and only if itsdual group’s fundamental multiplicative unitary W b G is weakly Bekka amenable.These results generalize the corresponding propositions for Kac algebras in Ng’spaper [9]. Copyright 2016 by the Tusi Mathematical Research Group.
Date : Received: 24 January 2017; Revised: 10 April 2017; Accepted: 4 May 2017.2010
Mathematics Subject Classification.
Primary 20G42; Secondary 46L89, 22D25.
Key words and phrases. locally compact quantum group, Bekka amenability, weak Bekkaamenability, co-amenability, amenability.
The notions of Bekka-type amenabilities, studied in this note, originate fromBekka’s paper [4]. In the case of locally compact groups, all of Bekka-typeamenabilities for unitary corepresentations are equal to amenability (introducedby Bekka in [4]) for unitary representations. Remarkably, Bekka showed, in [4],that amenability for a locally compact group is equivalent to the fact that everyunitary representation is amenable. These justify the use of the term “Bekka-typeamenabilities”.This note is organized as follows. After some preliminaries in Section 2, wediscuss in Section 3 Bekka amenability and weak Bekka amenability for locallycompact quantum groups.2.
Notations and definitions
Some notations.
In this note, we use the convention that the inner product h· , ·i of a complex Hilbert space H is conjugate-linear in the second variable. Wedenote by L ( H ) and K ( H ) the set of bounded linear operators and that of compactoperators on H , respectively. For any x, y ∈ H and T ∈ L ( H ), we denote by ω x,y the normal functional given by ω x,y ( T ) := h T x, y i . The symbol ⊗ denotes either a minimal C ∗ -algebraic tensor product or a ten-sor product of Hilbert spaces, and ¯ ⊗ denotes a von Neumann algebraic tensorproduct. Moreover, we denote by id the identity map. Finally, if X and Y are C ∗ -algebras or Hilbert spaces, we use the symbol Σ to denote the canonical flipmap from X ⊗ Y to Y ⊗ X sending x ⊗ y onto y ⊗ x , for all x ∈ X and y ∈ Y .Note that Σ = id.For a C ∗ -algebra A , we use Rep( A ) to denote the collection of unitary equiv-alence classes of non-degenerate ∗ -representations of A . Let us also recall somenotations concerning Rep( A ).Suppose that ( µ, H ) , ( ν, K ) ∈ Rep( A ). We write ν ≺ µ if ker µ ⊂ ker ν .2.2. Locally compact quantum group.
Let ( C ( G ) , ∆ , ϕ, ψ ) be a reducedlocally compact quantum group as introduced in [6, Definition 4.1] (for simplicity,we denote it by G ). The dual locally compact quantum group of G is denoted by( C ( b G ) , b ∆ , b ϕ, b ψ ) (or simply, b G ). We use L ( G ) to denote the Hilbert space givenby the GNS construction of the left invariant Haar weight ϕ and consider both C ( G ) and C ( b G ) as C ∗ -subalgebras of L ( L ( G )). Notice that L ( G ) = L ( b G ).Let be the identity of M ( C ( G )). There is a unitary W G ∈ M ( C ( G ) ⊗ C ( b G )) ⊆ L ( L ( G ) ⊗ L ( G )) , called the fundamental multiplicative unitary , that implements the comultiplica-tion: ∆( x ) = W ∗ G ( ⊗ x ) W G ( x ∈ C ( G )) . We denote by W b G the fundamental multiplicative unitary for the dual quantumgroup b G given by Σ W ∗ G Σ, where Σ is the flip map as defined above. For moredetails, the readers may refer to [6] and [13].
EKKA-TYPE AMENABILITIES FOR UNITARY COREPRESENTATIONS 3
The von Neumann subalgebra L ∞ ( G ) generated by C ( G ) in L ( L ( G )) is aHopf von Neumann algebra under a comultiplication e ∆ defined by W G as in theabove (see [7] or [13, Section 8.3.4]). We usually call L ∞ ( G ) the von Neumannalgebraic quantum group of G . Then L ( G ) denotes the predual of L ∞ ( G ), and L ∗ ( G ) := { ω ∈ L ( G ) | ∃ η ∈ L ( G ) s.t. ( ω ⊗ id)( W G ) ∗ = ( η ⊗ id)( W G ) } is a dense ∗ -subalgebra of L ( G ) as introduced in [5, Page 294-295].2.3. Corepresentation.
For any Hilbert space H U , a unitary U ∈ M ( K ( H U ) ⊗ C ( G )) is called a unitary corepresentation of G on H U if(id ⊗ ∆)( U ) = U U , (2.1)where U ij is the usual “leg notation” (see [6, Page 13] and [13, Section 7.1.2]).Let Corep( G ) denote the collection of unitary corepresentations of G .For U, V ∈ Corep( G ), T is called an intertwiner between U and V , and wewrite T ∈ Intw(
U, V ), if T ∈ L ( H U , H V ) such that T (id ⊗ ω )( U ) = (id ⊗ ω )( V ) T, for any ω ∈ L ∗ ( G ) . We say that U is unitarily equivalent to V and write U ∼ = V , if there exists T ∈ Intw(
U, V ) such that T is a unitary.2.4. Universal quantum group.
The universal quantum group C ∗ -algebra of b G is denoted by ( C u0 ( b G ) , b ∆ u ) (see [5, Section 4 and 5]). As shown in [5, Proposition5.2], there exists a unitary V u G ∈ M ( C u0 ( b G ) ⊗ C ( G ))that implements a bijection between unitary corepresentations U of G on H andnon-degenerate ∗ -representations π U of C u0 ( b G ) on H through the correspondence U = ( π U ⊗ id)( V u G ) . The identity G = 1 ⊗ of L ( C ) ⊗ M ( C ( G )) ∼ = M ( K ( C ) ⊗ C ( G )) ∼ = M ( C ( G ))is a trivial unitary corepresentation of G on C and π G is a character of C u0 ( b G ).As in the literature, we write U ≺ W when π U ≺ π W (see, e.g., [2, Section 5]and Section 2.1).2.5. Contra-corepresentation.
Let U be a unitary coreprsentation of G on aHilbert space H U . As in [2, Page 871], we define the contra-corepresentation U of U by U := ( τ ⊗ R )( U ) , where τ is the canonical anti-isomorphism from L ( H U ) to L ( H U ) (with H U beingthe conjugate Hilbert space of H U ) and R is the unitary antipode on C ( G ).Then U is a unitary corepresentation of G on H U . Notice that, it is unique up toequivalence ∼ =, and that U ∼ = U. If W is another unitary corepresentation of G on a Hilbert space K , we denoteby U ⊤ (cid:13) W the unitary corepresentation U W on H ⊗ K and call it the tensorproduct of U and W . In this case, π U ⊤ (cid:13) W = ( π U ⊗ π W ) ◦ b ∆ u . (2.2) X. CHEN Amenability, co-amenability and Bekka-type amenabilities
Let us first recall the following definitions of amenability and co-amenabilityof a locally compact quantum group.
Definition 3.1. ([2, Definition 3.1 and 3.2])
Let G be a locally compact quantumgroup.(a) We say that G is co-amenable if there exists a state ǫ of C ( G ) ) such that (id ⊗ ǫ )∆ = id .(b) A left invariant mean for a locally compact quantum group G is a state m on L ∞ ( G ) such that m ( ω ⊗ id)∆ = ω ( ) m , for all ω ∈ L ( G ) . We say that a locallycompact quantum group G is amenable if it has a left invariant mean. Remark 3.2.
Similarly, we also can define: a right invariant mean for G is astate m on L ∞ ( G ) such that m (id ⊗ ω )∆ = ω ( ) m , for all ω ∈ L ( G ) . Clearly, m is a right invariant mean if and only if m ◦ R is a left invariant mean. Thus, G is amenable if and only if it has a right invariant mean. Co-amenability may be characterized by the following equivalent formulations,which were obtained by E. B´ e dos and L. Tuset in [2]. Theorem 3.3. ([2, Theorem 3.1])
For a locally compact quantum group G , thefollowing statements are equivalent:(a) G is co-amenable.(b) The canonical surjective homomorphism Λ : C u0 ( G ) → C ( G ) is an isomor-phism.(c) There exists a ∗ -character on the C ∗ -algebra C ( G ) .(d) There exists a net of unit vectors { ξ i } in L ( G ) such that lim i k W G ( ξ i ⊗ v ) − ( ξ i ⊗ v ) k = 0 , ∀ v ∈ L ( G ) . Remark 3.4.
Comparing Theorem 3.3 with [9, Theorem 2.3] , we easily see thatthe “amenability” for a Kac algebra in the Ng’s paper [8] is actually the co-amenability for its dual in the sense of Definition 3.1.
Now, we extend Bekka amenability and weakly Bekka amenability introducedin [9] to the general case.
Definition 3.5.
For any U ∈ Corep( G ) , we say that(a) U has WCP (weak containment property) if G ≺ U (equivalently, π G ≺ π U ,see Section 2.1 and 2.4). The WCP is actually the property ( A ) introduced in [9,Proposition and Definition 2.4] .(b) U is Bekka amenable if π G ≺ π U ⊤ (cid:13) U (equivalently, G ≺ U ⊤ (cid:13) U ), i.e., U ⊤ (cid:13) U has the WCP.(c) U is weakly Bekka amenable if there exists a positive functional M on L ( H U ) with M (id H U ) = 1 such that M [(id H U ⊗ ω )( α U ( T ))] = M ( T ) , EKKA-TYPE AMENABILITIES FOR UNITARY COREPRESENTATIONS 5 for any positive functional ω ∈ L ( G ) with ω ( ) = 1 and T ∈ L ( H U ) , where α U ( T ) := U ( T ⊗ ) U ∗ is called a coaction of G on L ( H U ) . Those M satisfying the above condition arecalled α U -invariant means . Remark 3.6. (a) Let G be a locally compact quantum group of Kac type, and U bean arbitrary finite dimensional unitary corepresentation of G . In [9, Proposition3.10] , C.-K. Ng proved that U is Bekka amenable. Clearly, so is U , since U isalso finite dimensional.(b) When G is actually a locally compact group G , its reduced C ∗ -algebraic quan-tum group C ( G ) is commutative. It can be obviously seen from the commutativ-ity that, for any two U, V ∈ Corep( G ) , we have U ⊤ (cid:13) V = V ⊤ (cid:13) U . So, we have U ⊤ (cid:13) U = U ⊤ (cid:13) U , which implies that U is Bekka amenable if and only if U is Bekkaamenable. In this case, Bekka amenability is in fact the amenability for unitaryrepresentations in Bekka’s paper [4] . Theorem 3.7. ([2, Theorem 5.2] and [9, Proposition and Definition 2.4])
Let G be a locally compact quantum group and consider U ∈ Corep( b G ) . Then thefollowing are equivalent:(a) U has the WCP.(b) There exists a state ψ on L ( H U ) such that ψ (id ⊗ ω )( U ) = ω ( ) , for ω ∈ L ( b G ) .(c) There exists a net { ξ i } of unit vectors in H U such that lim i k U ( ξ i ⊗ v ) − ( ξ i ⊗ v ) k = 0 , for all v ∈ L ( G ) . Corollary 3.8.
A locally compact quantum group G is co-amenable if and onlyif W G has the WCP as a unitary corepresentation of b G . Proof:
Since W G can be viewed as an element in Corep( b G ), the corollary easilyfollows from Theorem 3.3(d) and Theorem 3.7(c). (cid:3) The WCP is stable under some operations, for example, contra-gredient andtensor product.For any U ∈ Corep( G ), we denote by U the trivial unitary corepresentationid H U ⊗ of G on H U . Proposition 3.9. ([2, Proposition 5.3])
Suppose that G is a locally compact quan-tum group and consider U, V ∈ Corep( G ) .(a) If U has the WCP, then so does U .(b) If both of U and V have the WCP, then so does U ⊤ (cid:13) V .(c) If U ⊤ (cid:13) V or V ⊤ (cid:13) U has the WCP, then so does U . Next, we present a lemma and a proposition. These results are probably known.Since we have not found them or their proof explicitly stated in the literature,we give complete arguments for the benefit of the reader.
X. CHEN
Lemma 3.10.
Let U and V be two unitary corepresentations of G . Then onehas U ⊤ (cid:13) V ∼ = V ⊤ (cid:13) U .
Proof:
Since the unitary antipode R is a ∗ -anti automorphism, one has U ⊤ (cid:13) V = ( τ ⊗ τ ⊗ R )( U V )= ( τ ⊗ R )( V ) ( τ ⊗ R )( U ) = V U = (Σ ⊗ ) V U (Σ ⊗ )= (Σ ⊗ )( V ⊤ (cid:13) U )(Σ ⊗ ) . So, it follows that, for any ω ∈ L ∗ ( G ), we have thatΣ(id ⊗ ω )( U ⊤ (cid:13) V ) = (id ⊗ ω )( V ⊤ (cid:13) U )Σ , since Σ = id. This implies that the unitary Σ lies in Intw( U ⊤ (cid:13) V , V ⊤ (cid:13) U ). Hence,the Lemma holds. (cid:3) The following proposition is usually called the absorption principle , which isthe generalization of Fell’s absorption principle for locally compact groups. E.B´ e dos, R. Conti and L. Tuset in their paper [3] proved the analogue in algebraicquantum groups (see [3, Proposition 3.4]). Proposition 3.11.
Let G be a locally compact quantum group. For any U ∈ Corep( b G ) , one has that U ⊤ (cid:13) W G ∼ = U ⊤ (cid:13) W G . Proof:
Let U be an arbitrary unitary corepresentation of b G . Set T to be theimage of U on L ( H U ⊗ L ( G )).For any ω ∈ L ∗ ( b G ), one has T (id ⊗ ω )( U ⊤ (cid:13) W G ) = (id ⊗ id ⊗ ω )( U U ( W G ) )= (id ⊗ id ⊗ ω )(( W G ) U )= (id ⊗ id ⊗ ω )((id H U ⊗ ( W G ) U )= (id ⊗ ω )( U ⊤ (cid:13) W G ) T, where the second “=” comes from the pentagonal relation (see [1, DefinitionA.1]): U U ( W G ) = ( W G ) U for any U ∈ Corep( G ).The above calculation implies that T is a unitary intertwiner between U ⊤ (cid:13) W G and U ⊤ (cid:13) W G . Hence, the equivalence of U ⊤ (cid:13) W G and U ⊤ (cid:13) W G , as two elementsin Corep( b G ), is obtained. (cid:3) Corollary 3.12.
Let G be a locally compact quantum group. For any U ∈ Corep( b G ) , one has that W G ⊤ (cid:13) U ∼ = W G ⊤ (cid:13) U . EKKA-TYPE AMENABILITIES FOR UNITARY COREPRESENTATIONS 7
Proof:
For any U ∈ Corep( b G ), it is obvious that U is also in Corep( b G ). Byproposition 3.11, one has that U ⊤ (cid:13) W G ∼ = U ⊤ (cid:13) W G . Hence, since U ∼ = U , by Lemma 3.10, we have that W G ⊤ (cid:13) U ∼ = U ⊤ (cid:13) W G ∼ = U ⊤ (cid:13) W G ∼ = U ⊤ (cid:13) W G ∼ = W G ⊤ (cid:13) U . (cid:3) Corollary 3.13.
Let G be a locally compact quantum group. If W G is Bekkaamenable as a unitary corepresentation of b G , then W G is also Bekka amenable. Proof:
Consider W G as a unitary corepresentation of b G . If W G is Bekkaamenable, then W G ⊤ (cid:13) W G has WCP. Hence, combining Corollary 3.12 and Propo-sition 3.9(c), we have that W G has WCP as a unitary corepresentation of b G .Hence, by Proposition 3.9(a) and (b), both W G and W G ⊤ (cid:13) W G also have theWCP, i.e. W G is Bekka amenable. (cid:3) Using the results above and the concept of the WCP, we can get a characteri-zation of co-amenability for locally compact quantum groups.
Proposition 3.14.
Let G be a locally compact quantum group. The followingstatements are equivalent:(a) G is co-amenable.(b) W G is Bekka amenable as a unitary corepresentation of b G . Proof:
First, assume that G is co-amenable, that is, W G has the WCP byCorollary 3.8. Using the assertions (a) and (b) of Proposition 3.9, we know that W G has the WCP and so does W G ⊤ (cid:13) W G . Hence, by Definition 3.5(b), W G isBekka amenable as a unitary corepresentation of b G .Conversely, if W G is Bekka amenable, then W G ⊤ (cid:13) W G has the WCP. Consid-ering W G as a unitary corepresentation of b G , it follows from Proposition 3.11that W G ⊤ (cid:13) W G has the WCP. Consequently, by Proposition 3.9(c), we know that W G has the WCP. Therefore, using Corollary 3.8 again, we know that G is co-amenable. (cid:3) Corollary 3.15.
Let G be a locally compact quantum group. If G is co-amenable,then W G is Bekka amenable as a unitary corepresentation of b G . Proof:
It follows directly from Corollary 3.13 and Proposition 3.14. (cid:3)
In [9], using Bekka amenability of the fundamental multiplicative unitary, C.-K. Ng gave a characterization of “amenability” of a Kac algebra. Using ourterminology, we rewrite Ng’s proposition as follows.
X. CHEN
Proposition 3.16. ([9, Proposition 3.6])
Let G be a locally compact quantumgroup of Kac type. Then G is co-amenable if and only if W G is Bekka amenableas a unitary corepresentation of b G . As a direct consequence of Proposition 3.14 and Proposition 3.16, the followingcorollary implies that, in the Kac case, W G is Bekka amenable if and only if W G is Bekka amenable. Note that the equivalence of (a) and (b) in this corollary isin fact Proposition 3.16 proved by C.-K. Ng in [9, Proposition 3.6]. We list thesestatements here just for comparison with the other results. Corollary 3.17.
Let G be a locally compact quantum group of Kac type. Consider W G and W G as two unitary corepresentations of b G . The following statements areequivalent:(a) G is co-amenable.(b) W G is Bekka amenable.(c) W G is Bekka amenable. In the following, we focus on weak Bekka amenability of unitary corepresen-tations. Using this property, we give another characterization for amenability,and generalizes some results on weak Bekka amenability in Ng’s paper (see [9,Proposition 3.4]). Some proofs of these results below follow from similar linesof argument as that of [9, Proposition 3.4]. For completeness, we present theargument here.
Proposition 3.18.
Let G be a locally compact quantum group. The followingstatements are equivalent:(a) G is amenable.(b) the fundamental multiplicative unitary W b G of its dual group is weakly Bekkaamenable as an element in Corep( G ) .(c) every U ∈ Corep( G ) is weakly Bekka amenable. Proof:
To obtain that (a) implies (c), we first note that, by Remark 3.2 andamenability of G , there exists a right invariant mean m on G . Let U be anarbitrary unitary corepresentation of G . For any positive functional ω on L ( H U )with ω (id H U ) = 1, we can define a linear map Φ ω from L ( H U ) to C ( G ) byΦ ω ( T ) = ( ω ⊗ id) α U ( T ) , for any T ∈ L ( H U ) . Furthermore, one can easily show that Φ ω is a completely positive map such that∆ ◦ Φ ω = (Φ ω ⊗ id) ◦ α U and Φ ω (id H U ) = 1. Thus, we have that M = m ◦ Φ ω is an α U -invariant mean for U , and so U is weakly Bekka amenable. By arbitrarinessof the choice of U , the statement (c) holds.It is clear that (c) implies (b), since W b G can be viewed as a unitary corepre-sentation of G .To show that (b) implies (a), assume that W b G is weakly Bekka amenable, andlet ω be an α W b G -invariant mean. Hence, statement (a) follows from the fact thatthe restriction ω | L ∞ ( G ) is indeed a left invariant mean for G . (cid:3) EKKA-TYPE AMENABILITIES FOR UNITARY COREPRESENTATIONS 9
As in the Kac case, Bekka amenability is still stronger than weak Bekkaamenability in the general case.
Proposition 3.19.
Let G be a locally compact quantum group and U be anyunitary corepresentation of G . If U is Bekka amenable, then U is weakly Bekkaamenable. Proof: (a) If U is Bekka amenable, then, we know that U ⊤ (cid:13) U has the WCP.Consequently, by Theorem 3.7(c), there exists a net of unit vectors { ξ i } ⊂ H U ⊤ (cid:13) U such that, for any v ∈ L ( G ),lim i (cid:13)(cid:13) ( U ⊤ (cid:13) U )( ξ i ⊗ v ) − ξ i ⊗ v (cid:13)(cid:13) = lim i (cid:13)(cid:13) ( U ⊤ (cid:13) U ) ∗ ( ξ i ⊗ v ) − ξ i ⊗ v (cid:13)(cid:13) = 0 ( ∗ )Then the net of the vector states { ω ξ i ,ξ i } has a subnet weak ∗ -convergent to somepositive functional m ∈ L ( H U ⊤ (cid:13) U ) ∗ .For any unit vector v ∈ L ( G ) and T ∈ L ( H U ⊤ (cid:13) U ), one has m [(id H U ⊤ (cid:13) U ⊗ ω v,v )( α U ⊤ (cid:13) U ( T ))]= lim i ω ξ i ,ξ i [(id H U ⊗ ω v,v )( α U ⊤ (cid:13) U ( T ))]= lim i h ( T ⊗ id L ( G ) )( U ⊤ (cid:13) U ) ∗ ( ξ i ⊗ v ) , ( U ⊤ (cid:13) U ) ∗ ( ξ i ⊗ v ) i = lim i h ( T ⊗ id L ( G ) )( ξ i ⊗ v ) , ξ i ⊗ v i ( By Equation ( ∗ ))= lim i ω ξ i ,ξ i ( T ) k ξ k = m ( T ) . Because every ω ∈ L ( G ) is a linear combination of ω v,v ’s, the equalities aboveimply that m is an α U ⊤ (cid:13) U -invariant mean. Define the positive functional M on L ( H U ) by M ( T ) = m ( T ⊗ id H U ) for any T ∈ L ( H U ). Then, we can obtain weakBekka amenability of U by checking that M is indeed an α U -invariant mean asrequired. (cid:3) Finally, we conclude, from the results above, that for any locally compactquantum group G , the following relation holds:co-amenability of b G ⇔ Bekka amenability of W b G ⇒ Bekka amenabil-ity of W b G ⇒ weak Bekka amenability of W b G ⇔ amenability of G ,where “ ⇔ ” means“equal to” and “ ⇒ ” means “imply”. Acknowledgments.
The author sincerely appreciates Professor Chi-KeungNg (Nankai University) for valuable suggestions. Meanwhile, the author alsothanks the referees for valuable comments which improve this paper a lot. Theauthor are supported by the Post-doctoral Scientific Research Grant of ShandongUniversity (1090516300002).
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School of Mathematics and Statistics, Shandong University, Weihai, Shan-dong Province 264209, China.
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