Dilation theory for right LCM semigroup dynamical systems
aa r X i v : . [ m a t h . OA ] F e b DILATION THEORY FOR RIGHT LCM SEMIGROUPDYNAMICAL SYSTEMS
MARCELO LACA AND BOYU LI
Abstract.
This paper examines actions of right LCM semigroups by endomorphisms ofC*-algebras that encode an additional structure of the right LCM semigroup. We definecontractive covariant representations for these semigroup dynamical systems and provea generalized Stinespring’s dilation theorem showing that these representations can bedilated if and only if the map on the C ∗ -algebra is unital and completely positive. Thisgeneralizes earlier results about dilations of right LCM semigroups of contractions. Inaddition, we also give sufficient conditions under which a contractive covariant repre-sentation of a right LCM system can be dilated to an isometric representation of theboundary quotient. Introduction
A contractive representation of a semigroup P on a Hilbert space H is a semigrouphomomorphism T : P → B ( H ) such that k T p k ≤ p ∈ P . If p is not invertible in P the operator T p does not have to be isometric. For example, any contractive operator A ∈ B ( H ) gives rise to a contractive representation T : N → B ( H ) by T ( n ) = A n .Nevertheless, a celebrated theorem of Sz. Nagy’s shows that a contractive representationof N is always the compression of an isometric representation. In other words, there existsan isometry V on a larger Hilbert space K ⊃ H , such that A n = P H V n (cid:12)(cid:12) H ( n ∈ N ) . The operator V is called an isometric dilation of T . Nagy’s dilation theorem has beengeneralized in several directions. Examples include Ando’s dilation for commuting con-tractions [2], Brehmer’s regular dilation [3], and Frazho-Bunce-Popescu’s dilation of rowcontractions [14, 7, 29].In his seminal work on C ∗ -algebras of quasi-lattice ordered semigroups [26], Nica ab-stracts a notion of covariance (Nica-covariance) from the left-regular representation anduses it to define a sensible notion of universal semigroup C ∗ -algebra. The study of Nica-covariant representations transformed the study of semigroup C ∗ -algebras into a veryactive area, and guided its development until the inception of a vast generalization inwork of X. Li [24]. Early developments include the semigroup crossed product techniquesfrom [19], applications to Artin monoids [8, 9]; more recent ones include the analysis ofright LCM semigroups [6, 31]. Meanwhile, dilation theory also developed as an importanttool in connecting various types of semigroup representations, as, for example, dilationtheorems for lattice ordered semigroups [15, 20] and right LCM semigroups [21, 22].The study of semigroup representations and semigroup C ∗ -algebras is closely related tothe study of semigroup dynamical systems. For example, the Cuntz algebra O n can berealized as a crossed product of a certain semigroup dynamical system over a UHF algebraof type n ∞ . Moreover, as pointed out in [26] and formally established in [19], C ∗ -algebras ofquasi-lattice ordered semigroups also possess a crossed product structure, of a canonical Date : February 18, 2021.2010
Mathematics Subject Classification.
Key words and phrases. dilation, semigroup, dynamical system, right LCM. action of the semigroup on the commutative diagonal algebra. This construction alsomotivated the study of boundary quotients, an analogue of the Cuntz algebras in the realmof semigroup C ∗ -algebras. On the other hand, in the realm of non-self-adjoint operatoralgebras, semigroup dynamical systems play a central role in the construction of varioussemicrossed product algebras and the computation of their C ∗ -envelope [16, 13, 12, 23].Dilation theory is often a useful tool in these studies.In this paper, we consider a class of dynamical systems over cancellative right LCMsemigroups, which we call right LCM dynamical systems . In general, the semigroup action α for a semigroup dynamical system only encodes the semigroup multiplication via α p α q = α pq , and any additional structure of the semigroup is often lost. For example, even if thesemigroup has a lattice structure, there is no reason to expect that this structure willbe reflected by a semigroup action, although in some cases it is indeed automaticallypreserved, e.g. the action by endomorphisms that lead to the Bost-Connes algebra. Itwas the analysis of this specific example that led to the consideration of semigroup actionsthat “respect the lattice structure” [17, Definition 3], and our first step is to extend thisdefinition to actions by right LCM semigroups, see Definition 3.1 below.In the study of semigroup dynamical systems ( A , P, α ), an isometric covariant represen-tation is a pair ( π, V ) where π is a ∗ -representation of the C ∗ -algebra A , V is an isometricrepresentation of the semigroup P , and the P -action α is encoded by the covariance rela-tion: V ( p ) π ( a ) V ( p ) ∗ = π ( α p ( a )) . We can similarly define a contractive covariant representation to be a pair ( φ, T ) where φ isa unital ∗ -preserving linear map, which need not be multiplicative, and T is a contractiverepresentation of the semigroup P . Again, the P -action is encoded by the covariancerelation: T ( p ) φ ( a ) T ( p ) ∗ = φ ( α p ( a )) . Our main result, Theorem 5.1, states that a contractive covariant representation ( φ, T )can be dilated to an isometric covariant representation ( π, V ) if and only if φ is a unitalcompletely positive map. In the special case when the semigroup is trivial, i.e. P = { e } ,our theorem recovers the celebrated Stinespring’s dilation theorem for unital completelypositive maps.The paper is organized as follows. In Section 2 we give a brief overview of semigroupsand their representations. In Section 3 we define right LCM semigroup dynamical systemsand their isometric and covariant representations, Definitions 3.14 and 3.15. In order tostudy dilations on semigroup dynamical systems in section 4, we first define the notion ofkernel systems, Definition 4.4 and then develop a general framework to deal with dilationsof our semigroup dynamical systems, Theorem 4.9. This can be viewed as a generalizationof the Naimark dilation theorem for positive definite Toeplitz kernels, a major tool in thestudy of dilation theory of semigroup representations [30, 22]. In section 5 we prove ourmain result, Theorem 5.1, by first building a kernel system K from a contractive covariantrepresentation ( φ, T ), Definition 5.4, and then proving that this kernel system is positiveif and only if φ is a unital completely positive map, Proposition 5.10. The proof is thenconcluded by showing that when φ is unital completely positive, the minimal Naimarkdilation for K in an isometric covariant representation. One major motivation for thispaper comes from the dynamical systems arising from semigroup C ∗ -algebras and theirboundary quotients, and we explore such connections in section 6. As a highlight, wecharacterize all representations that can be dilated to a representation of the boundaryquotient, Corollary 6.7. Finally, in section 7, we explore some additional constructions ofright LCM semigroup dynamical systems and their dilation theory, yielding several newresults on dilating unital completely positive maps satisfying certain relations. ILATION AND SEMIGROUP DYNAMICAL SYSTEM 3 Preliminaries
Semigroups and representations.
A semigroup P is a set with an associativemultiplication. An element e ∈ P is called the identity if ex = xe = x for all x ∈ P .A semigroup with an identity is called a monoid. An element p ∈ P need not have aninverse; when it does, it is called a unit . The set of all units is denoted by P ∗ , which is agroup on its own.A semigroup P is called left-cancellative if for any p, x, y ∈ P with px = py , we must have x = y . We can similarly define the notion of right-cancellative, and we say a semigroupis cancellative if it is both left and right cancellative. It is often convenient to assumethat the semigroup P embeds inside a group G , in which case P must be cancellative.However, the converse is false: cancellative semigroups need not embed in groups, and itis often a challenging task to verify whether a semigroup can be embedded in a group.For example, it was only relatively recently that all Artin monoids were shown to embedin their corresponding Artin groups [27]. Throughout this paper, unless stated otherwise,we assume the semigroup P is cancellative and contains an identity e .Nica first defined and studied quasi-lattice ordered groups in [26]. Consider a semigroup P inside a group G with P ∩ P − = { e } . The semigroup P defines a partial order on G by x ≤ y when x − y ∈ P . This order is called a quasi-lattice order if, for any finite F ⊂ G with a common upper bound, there exists a least common upper bound. The pair ( G, P )is called a quasi-lattice ordered group , and we often refer to P as a quasi-lattice orderedsemigroup .Nica’s work has been extended to more general classes of semigroups, for example,right LCM semigroups [6], which are a large class of semigroups that closely resemblequasi-lattice ordered semigroups. A left-cancellative semigroup P is called a right LCMsemigroup if for any p, q ∈ P , either pP ∩ qP = ∅ or else pP ∩ qP = rP for some r ∈ P . Here, the element r can be seen as a least-common upper bound of p, q . Since wedo not preclude non-trivial units in P , it is possible that the choice of r is not unique.Nevertheless, for r, s ∈ P , rP = sP precisely when r = su for some unit u ∈ P ∗ . It is easyto see that quasi-lattice ordered semigroups, and more generally the weak quasi-latticeordered semigroups from [1], are right LCM, but the converse does not hold.Quasi-lattice ordered and right LCM semigroups cover a wide range of familiar semi-groups. Subsemigroups of R + , the free semigroup F + n , right-angled Artin monoids A +Γ are all quasi-lattice ordered, and hence right LCM. Finite-type Artin monoids are in factlattice ordered semigroups [4]. However, it is unknown whether all Artin monoids arequasi-lattice ordered semigroups inside their corresponding Artin groups. Nevertheless,since for right LCM semigroups the least common upper bound is only required for semi-group elements, we do know that Artin monoids are right LCM [4, Proposition 4.1], infact, they are weak quasi-lattice ordered.A contractive representation T of a semigroup P is a unital semigroup homomorphism T : P → B ( H ) such that each T ( p ) is a contraction. The representation is called iso-metric (resp. unitary) if each T ( p ) ∈ B ( H ) is an isometry (resp. a unitary operator). Anelementary argument shows that when u is invertible in P , then T ( u ) must be unitary.So the restriction of a contractive representation to the group of units P ∗ is a unitaryrepresentation.Just like with groups, an important representation of a semigroup is its left regularrepresentation λ : P → B ( ℓ ( P )) defined by λ ( p ) δ q = δ pq on the standard orthonormalbasis { δ p } p ∈ P . It follows from the left-cancellation that λ ( p ) maps the orthonormal basisto an orthonormal set, and thus λ is an isometric representation. The reduced semigroupC ∗ -algebra of P , often denoted as C ∗ λ ( P ) is the C ∗ -algebra generated by the image of λ . MARCELO LACA AND BOYU LI
One might be tempted to define the semigroup C ∗ -algebra as the universal C ∗ -algebrawith respect to isometric representations of the semigroup. Murphy proved that undersuch definition, the universal object for isometric representations of N is not nuclear,which is undesirable as the universal object for an abelian semigroup as simple as N . Theissue is resolved by requiring that the range projections commute. This was abstractedand generalized by Nica, leading to what is now known as the Nica-covariance condition.In terms of a right LCM semigroup P , an isometric representation V : P → B ( H ) is called Nica-covariant if for any p, q ∈ P , V ( p ) V ( p ) ∗ V ( q ) V ( q ) ∗ = ( V ( r ) V ( r ) ∗ , if pP ∩ qP = rP , if pP ∩ qP = ∅ One can easily verify that, for the left-regular representation λ , the range projection λ ( p ) λ ( p ) ∗ is the orthogonal projection onto the subspace ℓ ( pP ), and thus satisfies theNica-covariance condition. For a right LCM semigroup P , the universal C ∗ -algebra of theisometric Nica-covariant representations is called the semigroup C ∗ -algebra, which is oftendenoted as C ∗ ( P ). Example 2.1.
Consider the semigroup P = N , which is the free abelian semigroupgenerated by two generators e , e . An isometric representation V : N → B ( H ) is uniquelydetermined by its value on two commuting generators V i = V ( e i ), i = 1 ,
2. Here, V , V must commute because e , e commute.Since e P ∩ e P = ( e + e ) P , the Nica-covariance condition requires that V V ∗ V V ∗ = V V ( V V ) ∗ . One can easily check that this is equivalent to requiring that V and V are ∗ -commuting,i.e., V commutes with both V and V ∗ . Example 2.2.
Consider the free semigroup P = F + n , with n free generators e , · · · , e n .An isometric representation V : F + n → B ( H ) is uniquely determined by n non-commutingisometries V i = V ( e i ).Since e i P ∩ e j P = ∅ for all i = j , the Nica-covariance condition requires that V i V ∗ i are pairwise orthogonal projections. This is often characterized by a single condition P ni =1 V i V ∗ i ≤ I .2.2. Semigroup dynamical systems.
Suppose A is a unital C ∗ -algebra and P is asemigroup. A P -action on A is a map α that sends each p ∈ P to a ∗ -endomorphism α p : A → A that satisfies α p ◦ α q = α pq and α e = id . Throughout this paper, we alwaysassume that each α p is injective because this plays a crucial role in our constructions.The action α is called automorphic if each α p is a ∗ -automorphism of A . Whenever p isinvertible, α p must be a ∗ -automorphism.A semigroup dynamical system is a triple ( A , P, α ) where A is a unital C ∗ -algebra, P is a left-cancellative semigroup. When each α p is injective, we say that α is an injective P -action on A . For each p ∈ P , we denote A p = α p ( A ), which is a C ∗ -subalgebra of A . This subalgebra has a unit α p (1). Since α is a ∗ -endomorphism, α p (1) is always anorthogonal projection. Moreover, certain ordering on the semigroup is preserved by α p (1). Lemma 2.3. If ( A , P, α ) is a semigroup dynamical system, then α x (1) α y (1) = α y (1) for x, y, p ∈ P with y = xp .Proof. α x (1) α y (1) = α x (1) α x ( α p (1)) = α x ( α p (1)) = α y (1) . (cid:3) Remark 2.4. If P is a right LCM semigroup with P ∗ = { e } , then one can define a partialorder by x ≤ y whenever there exists a p ∈ P with y = xp . In this case Lemma 2.3 provesthat α x (1) ≥ α y (1) whenever x ≤ y . ILATION AND SEMIGROUP DYNAMICAL SYSTEM 5 If p ∈ P ∗ , then A p = A since α p is a ∗ -automorphism. If pP = qP , then p = qu forsome u ∈ P ∗ , in which case A p = A q .Semigroup dynamical systems occur naturally in many well-known C ∗ -algebraic con-structions. For example, the Cuntz algebra O n can be constructed from a semigroupdynamical system via a single ∗ -endomorphism α acting on a UHF algebra of type n ∞ .The single ∗ -endomorphism induces a N -action on the UHF algebra. An important classof semigroup dynamical systems that are central to this paper arises from consideringsemigroup C ∗ -algebras. Example 2.5.
Let P be a right LCM semigroup and V : P → B ( H ) be a universalNica-covariant isometric representation of P . The diagonal algebra is defined by D P = span { V ( p ) V ( p ) ∗ : p ∈ P } , and is a commutative C ∗ -algebra because the Nica-covariance condition implies that V ( p ) V ( p ) ∗ V ( q ) V ( q ) ∗ is either 0 or V ( r ) V ( r ) ∗ for some r ∈ P , and the set of such r only depends on the intersection pP ∩ qP . The semigroup P acts on D P naturally via α p : D P → D P , where α p ( x ) = V ( p ) xV ( p ) ∗ . We shall study this dynamical system inmore detail in section 6. 3. Right LCM Dynamical System
The LCM condition.
Recall that if p, q are two elements in a right LCM semigroup P and if pP ∩ qP = rP , then r behaves like a least common upper bound of p and q . In thispaper we focus on dynamical systems that respect this feature of right LCM semigroupsin a sense made precise in the following definition that generalizes [17, Definition 3]. Definition 3.1.
Let P be a right LCM semigroup. An injective P -action α on a C ∗ -algebra A respects the right LCM if each α p ( A ) is an ideal of A , and for any p, q ∈ P , α p (1) α q (1) = ( α r (1) , if pP ∩ qP = rP , if pP ∩ qP = ∅ . A right LCM dynamical system is a dynamical system ( A , P, α ) in which α respects theright LCM.Even though the choice of r such that rP = pP ∩ qP may not be unique, the conditionintroduces no ambiguity. The reason is that when rP = sP , then r = su for some unit u ∈ P ∗ , hence α u is a ∗ -automorphism so that α u (1) = 1 and α r (1) = α s (1). For each p ∈ P we denote E p := α p (1) which is clearly a projection and is the identity of theideal A p := α p ( A ) = E p A E p . The right LCM condition manifests itself in a right LCMdynamical system in many ways. One of them is through the intersection of ideals {A p } . Proposition 3.2.
Let ( A , P, α ) be a right LCM dynamical system. Then for every p, q ∈ P , A p ∩ A q = ( A r , if pP ∩ qP = rP { } , if pP ∩ qP = ∅ . Proof. If a ∈ A p ∩ A q , then aE p E q = aE q = a . If pP ∩ qP = ∅ , then E p E q = 0 and thus a = 0. If pP ∩ qP = rP , then E p E q = E r and thus a ∈ A r , proving A p ∩A q ⊂ A r . To provethe reverse inclusion, suppose r ∈ pP ∩ qP and write r = pp = qq for some p , q ∈ P ,so A r = α pp ( A ) ⊂ α p ( A ) = A p and similarly for q . This proves A r ⊆ A p ∩ A q . (cid:3) MARCELO LACA AND BOYU LI
Proposition 3.3.
Let P be a right LCM monoid. A dynamical system ( A , P, α ) is a rightLCM dynamical system if and only if for every p, q ∈ P , (3.1) A p A q := { ab : a ∈ A p , b ∈ A q } = ( A r , if pP ∩ qP = rP, { } , if pP ∩ qP = ∅ . Proof.
Suppose ( A , P, α ) is a right LCM dynamical system. For any a, b ∈ A , consider α p ( a ) α q ( b ) = α p ( a ) E p E q α p ( b ). When pP ∩ qP = ∅ , E p E q = 0 and α p ( a ) α q ( b ) = 0, andhence A p A q = { } . When pP ∩ qP = rP , E p E q = E r ∈ A r , and α p ( a ) E r α p ( b ) ∈ A r since A r is an ideal. Therefore, A p A q ⊆ A r . To see the other inclusion, take any α r ( a ) ∈ A r ,since E r = E p E q , we have α r ( a ) = α r ( a ) E p E q , where α r ( a ) E p ∈ A p and E q ∈ A q . Hence, A p A q = A r .To prove the converse, suppose ( A , P, α ) satisfies equation (3.1). Let p ∈ P and set q = e ∈ P . We have A e = A . Since pP ∩ eP = pP , we must have A p A ⊆ A p and AA p ⊆ A p . This implies that A p is an ideal. For any p, q ∈ P , it suffices to prove that E p = α p (1) satisfies the right LCM condition. If pP ∩ qP = ∅ , then E p E q ∈ A p A q = { } so that E p E q = 0. If pP ∩ qP = rP , then E p E q ∈ A p A q ⊆ A r . Therefore, E p E q = α r ( a )for some a ∈ A . Since r = pp = qq for some p , q ∈ P , A r ⊆ A p , A q . Hence, for any α r ( b ) ∈ A r , α r ( ab ) = E p E q α r ( b ) = α r ( b ) = α r ( b ) E p E q = α r ( ba ) . Since α r is injective, a must be the identity and E p E q = E r . Therefore, α respects theright LCM. (cid:3) Remark 3.4.
One has to be cautious that our definition of the product A p A q does notcorrespond to the product ideal of two ideals, which is usually defined to be the closure ofthe linear span of products and satisfiesspan { ab : a ∈ I, b ∈ J } = I ∩ J. In general, the product set { ab : a ∈ I, b ∈ J } is not an ideal, so it is crucial for Proposi-tion 3.3 that we take advantage of the right LCM condition of our semigroup dynamicalsystems.As an immediate corollary, we obtain the following factorization property. Corollary 3.5.
For a right LCM dynamical system and p, q ∈ P , A p A q = A p ∩ A q . Even though each α p is not a ∗ -automorphism, our injectivity assumption ensures thatit does have a left-inverse, defined on all of A . Proposition 3.6.
For each p ∈ P , let α − p : A p → A be the inverse of α p and define α p − : A → A by α p − ( a ) = α − p ( α p (1) a ) Then α p − is well defined, α p − ◦ α p = id , and α p α p − = Mult α p (1) is the multiplicationoperator by α p (1) .Proof. Since A p is an ideal, α p (1) a ∈ A p . Since α p is injective, α p − ( a ) is well defined.For any a ∈ A , α p − ( α p ( a )) = α − p ( α p (1) α p ( a )) = α − p ( α p ( a )) = a, and, α p ( α p − ( a )) = α p ( α − p ( α p (1) a )) = α p (1) a. (cid:3) The map α p − has many nice properties. ILATION AND SEMIGROUP DYNAMICAL SYSTEM 7
Proposition 3.7.
For a right LCM dynamical system ( A , P, α ) and for every p, q ∈ P ,and a ∈ A we have (1) α p − ( a ) = α − p ( aα p (1)) = α − p ( α p (1) aα p (1)) ; (2) α p − is a surjective ∗ -endomorphism on A ; and (3) α ( pq ) − ( a ) = α q − ( α p − ( a )) . Proof.
For (1), since α p ( A ) is an ideal of A , the elements α p (1) a and aα p (1) are in α p ( A ),and therefore, α p (1) a = α p (1) aα p (1) = aα p (1) . Since α p is injective, the inverses under α p of these three terms are all equal to α p − ( a ) = α − p ( α p (1) a ).For (2), it is easy to verify that α p − is a ∗ -preserving linear map since α p is. To see itis multiplicative, let a, b ∈ A ; then α p ( α p − ( ab )) = α p (1) ab On the other hand α p ( α p − ( a ) α p − ( b )) = α p (1) aα p (1) b = α p (1) ab, which gives the result because α p is injectiveFor (3), recall notice that α pq (1) = α pq (1) α p (1), and α p (1)( a ) = α p ( α p − ( a )), andcompute α ( pq ) − ( a ) = α − pq ( α pq (1) a )= α − q ( α − p ( α pq (1) α p (1) a ))= α − q ( α − p ( α p ( α q (1) α p − ( a ))))= α − q ( α q (1) α p − ( a ))= α q − ( α p − ( a )) . (cid:3) Example 3.8.
A right LCM semigroup P is called semi-lattice ordered if pP ∩ qP = ∅ for every p, q ∈ P . If a semi-lattice ordered monoid P acts by ∗ -automorphisms of A ,then α p (1) = 1 and A p = A for all p ∈ P , so the system ( A , P, α ) is always a right LCMdynamical system. Example 3.9.
Let X be a compact Hausdorff space and suppose that there exists acollection of clopen subsets { X p } p ∈ P and a P -action β on X such that each β p : X → X p isa homeomorphism from X onto X p . This induces a P -action α on the unital commutativeC ∗ -algebra C ( X ), where α p : C ( X ) → C ( X p ) ⊆ C ( X ) is given by α p ( f ) = f ◦ β − p The action α is an LCM action if X p satisfies X p ∩ X q ⊆ X r if pP ∩ qP = rP ,and X p ∩ X q = ∅ if pP ∩ qP = ∅ . Indeed, we have α p ( C ( X )) = C ( X p ) and thus α p ( C ( X )) α q ( C ( X )) = C ( X p ∩ X q ). Example 3.10.
Unfortunately, not all dynamical systems respect the right LCM. Forexample, the Cuntz algebra O n is closely related to the semigroup dynamical system( A , N , α ) where A is the UHF algebra of type n ∞ and the action α is determined by α ( a ) = e ⊗ a . This is not a right LCM dynamical system because the UHF algebra isa simple C ∗ -algebra, and so the proper subalgebra α ( A ) cannot be an ideal. MARCELO LACA AND BOYU LI
Dynamical systems over F + k and AF-algebras. It is clear that if ( A , F + k , α ) isan injective dynamical system, then the action α is uniquely determined by the endomor-phisms α , · · · , α k associated to the generators. We would like to use this to provide asimple characterization of right LCM semigroup dynamical systems over the free semi-group F + k . Proposition 3.11.
An injective semigroup dynamical system ( A , F + k , α ) is a right LCMdynamical system if and only if for each ≤ i ≤ k , α i ( A ) is an ideal of A , and for all ≤ i, j ≤ k with i = j , α i (1) α j (1) = 0 .Proof. First assume that α i ( A ) is an ideal of A for all 1 ≤ i ≤ k . We first prove that α w ( A ) is an ideal of A for every word w = w w · · · w n ∈ F + k . Since α i ( A ) is an ideal of A and α i is injective, we can define α i − ( x ) = α − i ( α i (1) x ) which is a left-inverse for α i . If a, b ∈ A , then α w ( a ) b = α w ( α w w ··· w n ( a )) α w (1) b = α w ( α w w ··· w n ( a )) α w ( α w − ( b ))= α w ( α w w ··· w n ( a ) α w − ( b )) . Letting b := α w − ( b ) we can use a similar argument to show that α w ( a ) b = α w ( α w w ··· w n ( a ) b )= α w w ( α w ··· w n ( a ) α w − ( b )) . Repeating this process, we see that α w ( a ) b ∈ α w ( A ). By the Proposition 3.7, the sameargument can be made for multiplication by b from the left. This proves that α w ( A ) is anideal.Now, for all 1 ≤ i, j ≤ k with i = j , α i (1) α j (1) = 0. For any two elements x, y ∈ F + k ,we write x = x x · · · x n and y = y y · · · y m . Without loss of generality, we may assumethat n ≤ m . There are two possibilities: if x F + k ∩ y F + k = ∅ , then it must be the case that y = xp for some p ∈ F + k and x F + k ∩ y F + k = y F + k . In this case, Lemma 2.3 proves that α x (1) α y (1) = α y (1).Otherwise, there must be an index 1 ≤ s ≤ n such that x t = y t for all 1 ≤ t < s but x s = y s . In this case, α x s (1) α y s (1) = 0, and thus for every a, b ∈ A , α x s ( a ) α y s ( b ) = α x s ( a ) α x s (1) α y s (1) α y s ( b ) = 0 . As a result, α x (1) α y (1) = 0.Therefore, the action α respects the right LCM and thus we have a right LCM dynamicalsystem. The converse follows easily from Proposition 3.3. (cid:3) In particular, the right LCM condition is easy to verify when k = 1 and the semigroup F + k is reduced to N . Corollary 3.12.
An injective semigroup dynamical system ( A , N , α ) is a right LCM dy-namical system if and only if α ( A ) is an ideal of A . Next, we construct a non-trivial example of a dynamical system over an AF-algebra.One may refer to [11] for the basic background on AF-algebras and Bratteli diagrams.
Example 3.13.
Consider the AF-algebra A with the following Bratteli diagram: ILATION AND SEMIGROUP DYNAMICAL SYSTEM 9 (3.2) 2 22 2222 · · · and notice that A is isomorphic to C ( X ) ⊗ M , where X is the Cantor set.Fix n momentarily and write 0 for the zero matrix in ⊕ n i =1 M . The map α : ⊕ n i =1 M →⊕ n +1 i =1 M defined by α ( a ) = a ⊕ ∗ -endomorphism of A . One can easily verify thatits image, corresponding to the upper branch of the Bratteli diagram, is an ideal of A .Therefore, the dynamical system ( A , N , α ) is a right LCM dynamical system. Here, themap α exploits the self-similar property of the given Bratteli diagram.One can also define α ( a ) = 0 ⊕ a and an F +2 action by sending each generator e i to α i . Since the image of α i are ideals and α (1) = 1 ⊕ α (1) = 0 ⊕ A , F +2 , α ) is a right LCM dynamical system by Proposition 3.11. We shall study dilationresults for such systems in Example 7.6.3.3. Covariant representations.
To study operator algebras associated with dynamicalsystems, it is useful to have a presentation in terms of a convenient notion of covariancethat encodes the dynamics. We define one that is suitable for right LCM dynamicalsystems.
Definition 3.14. An isometric covariant representation ( π, V ) of a right LCM dynamicalsystem ( A , P, α ) is given by(1) a unital ∗ -homomorphism π : A → B ( H ) and(2) an isometric representation V : P → B ( H )such that for every p ∈ P and a ∈ A , V ( p ) π ( a ) V ( p ) ∗ = π ( α p ( a )) . Definition 3.15. A contractive covariant representation ( φ, T ) of a right LCM dynamicalsystem ( A , P, α ) is given by(1) a unital ∗ -preserving linear map φ : A → B ( H ) and(2) a contractive representation T : P → B ( H )such that for every p ∈ P and a ∈ A , T ( p ) φ ( a ) T ( p ) ∗ = φ ( α p ( a )) . Remark 3.16.
There are several different notions one could have used to define contrac-tive covariant representations of a semigroup dynamical system. For instance, in the studyof semicrossed product algebras by abelian lattice ordered semigroups [12] (see also [28]),the requirement is usually φ ( a ) T ( p ) = T ( p ) φ ( α p ( a )) . The commutativity of the semigroup is required in this definition. For non-abelian semi-groups, one may also consider the covariance condition T ( p ) φ ( a ) = φ ( α p ( a )) T ( p ) . Remark 3.17.
If ( π, V ) is an isometric covariant representation, then V ( p ) ∗ π ( a ) V ( p ) = π ( α p − ( a )) for all p ∈ P and a ∈ A . Indeed, V ( p ) ∗ π ( a ) V ( p ) = V ( p ) ∗ V ( p ) π (1) V ( p ) ∗ π ( a ) V ( p )= V ( p ) ∗ π ( α p (1)) π ( a ) V ( p )= V ( p ) ∗ π ( α p (1) a ) V ( p )= V ( p ) ∗ π ( α p ( α − p ( α p (1) a ))) V ( p )= V ( p ) ∗ V ( p ) π ( α p − ( a )) V ( p ) ∗ V ( p ) = π ( α p − ( a )) . However, the analogous property may fail for general contractive covariant representationsbecause T ( p ) ∗ T ( p ) may not be the identity and the map φ may not be multiplicative. Remark 3.18.
If the action α is ∗ -automorphic, then α p (1) = 1 = V ( p ) V ( p ) ∗ for all p ∈ P . So every isometric covariant representation ( π, V ) is in fact a unitary representationof P . Definition 3.19.
Let ( φ, T ) be a contractive covariant representation of the right LCMdynamical system ( A , P, α ) on a Hilbert space H . An isometric covariant dilation of ( φ, T )is an isometric covariant representation ( π, V ) on a Hilbert space K containing H , suchthat(1) φ ( a ) = P H π ( a ) (cid:12)(cid:12)(cid:12) H for every a ∈ A , and(2) T ( p ) = P H V ( p ) (cid:12)(cid:12)(cid:12) H for every p ∈ P .In other words, ( φ, T ) is the compression of ( π, V ) to the subspace H of K .Let ( π, V ) be an isometric covariant representation of a right LCM dynamical system( A , P, α ) on B ( K ). Suppose H ⊆ K is co-invariant for V (this means that H is invariantfor all the V ( p ) ∗ with p ∈ P ). Then we can build a contractive covariant pair ( φ, T ) bycompressing ( π, V ) to the corner of H . That is, we define φ : A → B ( H ) and T : P → B ( H )by φ ( a ) := P H π ( a ) (cid:12)(cid:12)(cid:12)(cid:12) H and T ( p ) := P H V ( p ) (cid:12)(cid:12)(cid:12)(cid:12) H . The operators π ( a ) and V ( p ), when written as 2 × H ⊕ H ⊥ have the form π ( a ) = (cid:20) φ ( a ) ∗∗ ∗ (cid:21) and V ( p ) = (cid:20) T ( p ) 0 ∗ ∗ (cid:21) where the 0 in the upper right hand corner for V ( p ) reflects the fact that H is invariantfor V ( p ) ∗ . Therefore, keeping track of the upper left corner of the matrix, which is theonly relevant one for the compression, we compute (cid:20) φ ( α p ( a )) ∗∗ ∗ (cid:21) = π ( α p ( a ))= V ( p ) π ( a ) V ( p ) ∗ = (cid:20) T ( p ) 0 ∗ ∗ (cid:21) (cid:20) φ ( a ) ∗∗ ∗ (cid:21) (cid:20) T ( p ) ∗ ∗ ∗ (cid:21) = (cid:20) T ( p ) φ ( a ) T ( p ) ∗ ∗∗ ∗ (cid:21) , where we used the covariance of ( π, V ) in the second line. Hence, after compressing to H , we get T ( p ) φ ( a ) T ( p ) ∗ = φ ( α p ( a )). Moreover, one can easily verify that φ is a unital ∗ -preserving linear map (not necessarily multiplicative) and T is a representation of P . ILATION AND SEMIGROUP DYNAMICAL SYSTEM 11
Therefore, we have showed that the compression of an isometric covariant representation( π, V ) to a subspace that is co-invariant for V is a contractive covariant representation( φ, T ), of which the given ( π, V ) is obviously a covariant isometric dilation.The main focus of this paper is the reverse process: When does a given contractivecovariant representation ( φ, T ) of a right LCM dynamical system ( A , P, α ) have an iso-metric covariant dilation ( π, V ) ? A moment’s thought reveals that to be realized as thecompression of a ∗ -homomorphic representation π , the given φ would have to be a unitalcompletely positive map to begin with. As it turns out, this is all we need to assumein order to guarantee there exists an isometric covariant dilation. Before we prove this,in Theorem 5.1, we need to develop the necessary machinery about dilations in the nextsection. 4. Naimark Dilation on Semigroup Dynamical Systems
The goal of this section is to prove a generalization of Naimark’s dilation theorem validfor semigroup dynamical systems. Throughout the section, we only require that P be aleft-cancellative semigroup. In particular we do not assume that the semigroup is rightLCM or even if it is, that the dynamical system respects the LCM structure.A Toeplitz kernel for a left-cancellative semigroup P is a map K : P × P → B ( H ) suchthat K ( e, e ) = I, K ( p, q ) = K ( q, p ) ∗ , and K ( rp, rq ) = K ( p, q )for all p, q, r ∈ P . A Toeplitz kernel is positive definite if for any choice of p , · · · , p n ∈ P , the operator matrix [ K ( p i , p j )] is positive. The original Naimark dilation establishesthat positive definite Toeplitz kernels arise from certain isometric representations. Thefollowing generalization of Naimark’s theorem is due to Popescu. Theorem 4.1 (Theorem 3.2 of [30]) . Let P be a left-cancellative unital semigroup and let K be a unital kernel for P on the Hilbert space H . Then K is a positive definite Toeplitzkernel if and only if there exists a (minimal) isometric representation V : P → B ( K ) onsome Hilbert space K ⊃ H , such that for every p, q ∈ P , K ( p, q ) = P H V ( p ) ∗ V ( q ) (cid:12)(cid:12)(cid:12) H and span { V ( p ) h : p ∈ P, h ∈ H} = K . The minimal dilation is unique up to unitary equivalence.
Kernel systems.
We first define and study kernel systems associated to semigroupdynamical systems. These prove to be an essential tool in dilation. The kernel systemdefined here can be seen as a strengthened version of positive definite Toeplitz kernels onsemigroups, as in the traditional Naimark dilation.When ( A , P, α ) is a semigroup dynamical system, we define A p,q = { α p (1) aα q (1) : a ∈A} to be the ‘off diagonal’ corner of the C ∗ -algebra A corresponding to p, q ∈ P . One caneasily verify that if a ∈ A p,q , then a ∗ ∈ A q,p . Remark 4.2.
Notice that when ( A , P, α ) is a right LCM dynamical system, then A p,q = α p ( A ) ∩ α q ( A ) = α p ( A ) α q ( A ). This is because in this case α p ( A ) , α q ( A ) are ideals so thatfor any a ∈ A , α p (1) aα q (1) ∈ α p ( A ) ∩ α q ( A ). Conversely, if a ∈ α p ( A ) ∩ α q ( A ), then a = α p (1) aα q (1) ∈ A p,q . Lemma 4.3.
Suppose ( A , P, α ) is a semigroup dynamical system. Then α r ( A p,q ) ⊆ A rp,rq for every p, q, r ∈ P . Moreover, α r ( A p,q ) = A rp,rq for every p, q, r ∈ P if and only if α r ( A e,e ) is hereditary for every r ∈ P .Proof. For the first assertion, take a = α p (1) bα q (1) ∈ A p,q and notice that α r ( a ) = α rp (1) α r ( b ) α rq (1) ∈ A rp,rq . If equality holds for every r, p, q , then, then in particular, α r ( A ) = α r ( A e,e ) = A r,r = α r (1) A α r (1), hence α r ( A ) is hereditary by [25, Lemma 4.1]. Conversely, assume α r ( A ) ishereditary for every r ; then α r ( A p,q ) = α rp (1) α r ( A ) α rq (1) = α rp (1) α r (1) A α r (1) α rq (1) = A rp,rq , and equality holds. (cid:3) We use e for the identity element in the semigroup, 1 for the unit of the C ∗ -algebra A ,and I for the identity operator on a Hilbert space. Definition 4.4.
Let ( A , P, α ) be a semigroup dynamical system and defineΛ ( A ,P,α ) = { ( p, a, q ) : p, q ∈ P, a ∈ A p,q } . A kernel system for ( A , P, α ) on a Hilbert space H is a map K : Λ ( A ,P,α ) → B ( H ). We saythat K is(1) unital if K ( e, , e ) = I ,(2) Hermitian if K ( p, a, q ) ∗ = K ( q, a ∗ , p ),(3) Toeplitz if K ( p, a, q ) = K ( rp, α r ( a ) , rq ) (notice that ( rp, α r ( a ) , rq ) ∈ Λ ( A ,P,α ) byLemma 4.3),(4) linear if K ( p, a, q ) + λK ( p, b, q ) = K ( p, a + λb, q ) for every a, b ∈ A p,q and λ ∈ C ,(5) positive if for every choice p , · · · , p n ∈ P and a , · · · , a n ∈ A with a i ∈ A α p i (1),we have [ K ( p i , a ∗ i a j , p j )] ≥ , (6) bounded if for any p , · · · , p n ∈ P and a , · · · , a n ∈ A with a i ∈ A α p i (1), and forevery a ∈ A , [ K ( p i , a ∗ i a ∗ aa j , p j )] ≤ k a k [ K ( p i , a ∗ i a j , p j )] , (7) continuous if for every p, q ∈ P , and every sequence ( a n ) in A p,q that converges to a ∈ A p,q , then K ( p, a n , q ) also converges to K ( p, a, q ) (in norm). Lemma 4.5.
Every linear positive kernel system K is bounded.Proof. Let p , · · · , p n ∈ P , and for each i = 1 , , . . . n choose a i ∈ A α p i (1). For each a ∈ A ,define b i = ( k a k I − a ∗ a ) / a i , which is again in A α p i (1). The positivity of K implies thatthe operator matrix [ K ( p i , b ∗ i b j , p j )] = [ K ( p i , a ∗ i ( k a k I − a ∗ a ) a j , p j )]is positive definite. The linearity of K implies that[ K ( p i , a ∗ i ( k a k I − a ∗ a ) a j , p j )] = k a k [ K ( p i , a ∗ i a j , p j )] − [ K ( p i , a ∗ i a ∗ aa j , p j )] ≥ , which gives the desired inequality, proving that K is bounded. (cid:3) Lemma 4.6. If K is a unital, Hermitian, Toeplitz, linear, positive kernel system, then k K ( p, a, q ) k ≤ k a k , for all p, q ∈ P and a ∈ A p,q , and thus K is continuous.Proof. Fix p, q ∈ P and a ∈ A p,q . Let a = α p (1) and a = a , the positivity assumptionensures that (cid:20) K ( p, α p (1) , p ) K ( p, a, q ) K ( q, a ∗ , p ) K ( q, a ∗ a, q ) (cid:21) ≥ . By Lemma 4.5, K is also bounded so that K ( q, a ∗ a, q ) ≤ k a k K ( q, α q (1) , q ) = k a k K ( e, I, e ) = k a k I. Also note that K ( p, α p (1) , p ) = K ( e, I, e ) = I because K is Toeplitz. Therefore, (cid:20) K ( p, α p (1) , p ) K ( p, a, q ) K ( q, a ∗ , p ) k a k I (cid:21) ≥ (cid:20) K ( p, α p (1) , p ) K ( p, a, q ) K ( q, a ∗ , p ) K ( q, a ∗ a, q ) (cid:21) ≥ ILATION AND SEMIGROUP DYNAMICAL SYSTEM 13
This can only happen when K ( q, a ∗ , p ) K ( p, a, q ) = K ( p, a, q ) ∗ K ( p, a, q ) ≤ k a k I (by usingfor example [10, Lemma 14.13]). Therefore, k K ( p, a, q ) k ≤ k a k as desired. (cid:3) It is often easier to verify the properties of a kernel system on a dense subset of A ; thefollowing proposition shows that it is also enough. Proposition 4.7.
Suppose ( A , P, α ) is a semigroup dynamical system and let A be aunital dense ∗ -subalgebra of A that is invariant under α . Then every unital, Hermit-ian, Toeplitz, linear, positive kernel system K on ( A , α, P ) can be extended to a unital,Hermitian, Toeplitz, linear, positive kernel system K on ( A , α, P ) .Proof. If a n ∈ A p,q converges to a ∈ A p,q , linearity and Lemma 4.6 give k K ( p, a n , q ) − K ( p, a m , q ) k = k K ( p, a n − a m , q ) k ≤ k a n − a m k . Therefore, K ( p, a n , q ) is a Cauchy sequence that converges in B ( H ). Define K ( p, a, q ) :=lim n →∞ K ( p, a n , q ). One can easily verify that K is unital, Hermitian, Toeplitz (sincethe α p are continuous ∗ -endomorphisms), and linear. For positivity, pick p , · · · , p n ∈ P and a , · · · , a n ∈ A with a i ∈ A α p i (1). For each i , pick a sequence a i,n in A α p i (1) thatconverges to a i . Then K ( p i , a ∗ i,n a j,n , p j ) converges to K ( p i , a ∗ i a j , p j ), and since the matrix[ K ( p i , a ∗ i,n a j,n , p j )] is positive definite so is [ K ( p i , a ∗ i a j , p j )]. This proves K is positive. (cid:3) Naimark dilation for kernel systems.
Naimark dilation is a powerful tool in thestudy of dilation theory on semigroups because it explicitly constructs isometric repre-sentations of the semigroup via positive definite kernels [30, 20, 22]. With the additionalC ∗ -algebra A in our definition of the kernel system, we aim to establish an extended ver-sion of Naimark dilation theorem that explicitly constructs both a representation of theC ∗ -algebra A and an isometric representation of the semigroup P .The usual way of constructing a kernel is by compressing an isometric covariant rep-resentation to a subspace. The next proposition shows that this also works for kernelsystems. We emphasize that only the weaker covariance condition is involved in the pro-cess. Proposition 4.8.
Let ( π, V ) be an isometric representation of a semigroup dynami-cal system ( A , α, P ) on a Hilbert space K that satisfies the (weak) covariance condition π ( α p ( a )) V ( p ) = V ( p ) π ( a ) . Suppose H is a subspace of K and define K : Λ ( A ,P,α ) → B ( H ) by K ( p, a, q ) = P H V ( p ) ∗ π ( a ) V ( q ) (cid:12)(cid:12)(cid:12)(cid:12) H . Then K satisfies conditions (1) to (6) in Definition 4.4.Proof. For condition (1), notice that K ( e, , e ) = P H V ( e ) ∗ π (1) V ( e ) (cid:12)(cid:12)(cid:12)(cid:12) H is equal to I because V is an isometric representation with V ( e ) = I and π (1) = I .For condition (2), simply compute K ( p, a, q ) ∗ = P H ( V ( p ) ∗ π ( a ) V ( q )) ∗ (cid:12)(cid:12)(cid:12)(cid:12) H = P H V ( q ) ∗ π ( a ∗ ) V ( p ) (cid:12)(cid:12)(cid:12)(cid:12) H = K ( q, a ∗ , p )For condition (3), V ( rp ) ∗ π ( α r ( a )) V ( rq ) = V ( p ) ∗ V ( r ) ∗ π ( α r ( a )) V ( r ) V ( q ) = V ( p ) ∗ V ( r ) ∗ V ( r ) π ( a ) V ( q )cancelling V ( r ) ∗ V ( r ) = I and compressing down to H gives K ( rp, α r ( a ) , rq ) = K ( p, a, q ).For condition (4), V ( p ) ∗ π ( a + λb ) V ( q ) = V ( p ) ∗ π ( a ) V ( q ) + λV ( p ) ∗ π ( b ) V ( q ) , again, compressing down to H gives K ( p, a + λb, q ) = K ( p, a, q ) + λK ( p, b, q ). For condition (5), fix p , · · · , p n ∈ P and a , · · · , a n ∈ A with a i ∈ A α p i (1), and definean 1 × n operator matrix R to be R := (cid:2) π ( a ) V ( p ) · · · π ( a n ) V ( p n ) (cid:3) . Then, R ∗ R = [ V ( p i ) ∗ π ( a ∗ i a j ) V ( p j )] ≥ H proves that [ K ( p i , a ∗ i a j , p j )] ≥ (cid:3) The extended Naimark dilation theorem tackles the converse.
Theorem 4.9.
Let K : Λ ( A ,P,α ) → B ( H ) be a kernel system that satisfies conditions (1) to(5) of Definition 4.4. Then there exists a Hilbert space K ⊇ H , an isometric representation V : P → B ( K ) and a ∗ -homomorphism π : A → B ( K ) satisfying V ( p ) π ( a ) = π ( α p ( a )) V ( p ) for p, q ∈ P, a ∈ A , such that K ( p, a, q ) = P H V ( p ) ∗ π ( a ) V ( q ) (cid:12)(cid:12)(cid:12)(cid:12) H for ( p, a, q ) ∈ Λ ( A ,P,α ) . Moreoever, ( π, V ) can be taken to be minimal in the sense that K = span { π ( a ) V ( p ) h : p ∈ P, a ∈ A α p (1) , h ∈ H ⊂ K} . Such a minimal dilation ( π, V ) is unique up to unitary equivalence.Proof. Consider the discrete space X := { ( p, a ) : p ∈ P, a ∈ A α p (1) } , and write δ p,a ⊗ h for the (elementary) function that equals h at ( p, a ) and is zero everywhere else. Let K := C c ( X ; H ) = span { δ p,a ⊗ h : p ∈ P, a ∈ A α p (1) , h ∈ H} . If ( p, a ) and ( q, b ) are in X , then b ∗ a ∈ α q (1) A α p (1) = A q,p , so we may define h δ p,a ⊗ h, δ q,b ⊗ k i := h K ( q, b ∗ a, p ) h, k i which can be extended uniquely to a sesquilinear form on K . This sesquilinear form ispositive definite because for every k = P ni =1 δ p i ,a i ⊗ h i , h k, k i = n X i =1 n X j =1 h K ( p j , a ∗ j a i , p i ) h i , h j i = * [ K ( p j , a ∗ j a i , p i )] h ... h n , h ... h n + ≥ , where the last inner product is in H n . Define k k k = h k, k i / . The sesquilinear form h· , ·i is a pre-inner product that satisfies the Cauchy-Schwarz inequality |h k, h i| ≤ k k kk h k . Ifwe now let N := { k ∈ K : k k k = 0 } = { k ∈ K : h k, h i = 0 for all h ∈ K } , we see that N is a linear subspace of K and that h· , ·i becomes an inner product on thequotient K / N . The completion of K / N with respect to the associated norm will bedenoted K . On the image of K / N in this completion the inner product is given by h k + N , h + N i = h k, h i and the norm is k k + N k = h k + N , k + N i / . Since h δ e, ⊗ h, δ e, ⊗ k i = h K ( e, , e ) h, k i = h h, k i , the Hilbert space H naturally embeds in K via h → δ e, ⊗ h . ILATION AND SEMIGROUP DYNAMICAL SYSTEM 15
Notice that if b ∈ A α q (1), then α p ( b ) ∈ A α pq (1), so that ( pq, α p ( b )) ∈ X and we maydefine V ( p ) : K → K by V ( p ) δ q,b ⊗ h = δ pq,α p ( b ) ⊗ h, for p, q ∈ P and b ∈ A α q (1). The operator V ( p ) is isometric because for every linearcombination k = P ni =1 δ p i ,a i ⊗ h i , k V ( p ) k k = k n X i =1 δ pp i ,α p ( a i ) ⊗ h i k = n X i =1 n X j =1 h K ( pp j , α p ( a j ) ∗ α p ( a i ) , pp i ) h i , h j i = n X i =1 n X j =1 h K ( pp j , α p ( a ∗ j a i ) , pp i ) h i , h j i = n X i =1 n X j =1 h K ( p j , a ∗ j a i , p i ) h i , h j i = k k k ;where we have used the Toeplitz condition in the last line.Noticing now that if b ∈ A α q (1) and a ∈ A , then ab ∈ A α q (1), so that ( q, ab ) ∈ X , wedefine a map π ( a ) : K → K by first letting π ( a )( δ q,b ⊗ h ) = δ q,ab ⊗ h and then extending it by linearity. In order to show that π ( a ) is bounded, we take k = P ni =1 δ p i ,a i ⊗ h i and compute k π ( a ) k k = k n X i =1 δ p i ,aa i ⊗ h i k = n X i =1 n X j =1 h K ( p j , a ∗ j a ∗ aa i , p i ) h i , h j i≤ k a k n X i =1 n X j =1 h K ( p j , a ∗ j a i , p i ) h i , h j i = k a k k k k , where the inequality holds because the kernel K is assumed to be bounded.Since V ( p ) and π ( a ) are both bounded on K and leave N invariant, they determinebounded linear operators on K , which we also denote by V ( p ) and π ( a ). We now verifythat ( π, V ) has the desired properties. Claim 1 . V : P → B ( K ) is an isometric representation.We have proved each V ( p ) is an isometry. Let p, q ∈ P ; then V ( p ) V ( q ) δ r,a ⊗ h = δ pqr,α p ( α q ( a )) ⊗ h = δ pqr,α pq ( a ) ⊗ h = V ( pq ) δ r,a ⊗ h. Therefore, V ( pq ) = V ( p ) V ( q ). That V ( e ) = I is obvious from the definition. Claim 2 . π : A → B ( K ) is a unital ∗ -homomorphism.It is clear that π (1) = I . To prove π is a homomorphism, pick any a, b ∈ A . It sufficesto show that π ( a ) π ( b ) = π ( ab ) on a set that has dense linear span, so choose ( r, c ) ∈ X and h ∈ H ; then π ( a ) π ( b ) δ r,c ⊗ h = δ r,abc ⊗ h = π ( ab ) δ r,c ⊗ h. It remains to show that π ( a ∗ ) = π ( a ) ∗ , and for this it suffices to prove that for every pairof elementary functions δ p,b ⊗ h and δ q,c ⊗ k in K , we have h π ( a ∗ ) δ p,b ⊗ h, δ q,c ⊗ k i = h π ( a ) ∗ δ p,b ⊗ h, δ q,c ⊗ k i To verify this we compute h π ( a ∗ ) δ p,b ⊗ h, δ q,c ⊗ k i = h δ p,a ∗ b ⊗ h, δ q,c ⊗ k i = h K ( q, c ∗ a ∗ b, p ) h, k i = h δ p,b ⊗ h, δ q,ac ⊗ k i = h δ p,b ⊗ h, π ( a ) δ q,c ⊗ k i . Claim 3 . K ( p, a, q ) = P H V ( p ) ∗ π ( a ) V ( q ) (cid:12)(cid:12)(cid:12)(cid:12) H .For any h, k ∈ H , which embed in K as δ e, ⊗ h, δ e, ⊗ k , h V ( p ) ∗ π ( a ) V ( q ) δ e, ⊗ h, δ e, ⊗ k i = h π ( a ) V ( q ) δ e, ⊗ h, V ( p ) δ e, ⊗ k i = h δ q,a ⊗ h, δ p,α p (1 ⊗ k i = h K ( p, α p (1) a, q ) h, k i = h K ( p, a, q ) h, k i Here, we use the fact that a ∈ A p,q = α p (1) A α q (1) and thus α p (1) a = a. Claim 4 . V ( p ) π ( a ) = π ( α p ( a )) V ( p ).Take any elementary function δ b,r ⊗ h ∈ K . Then V ( p ) π ( a ) δ b,r ⊗ h = δ α p ( ab ) ,pr ⊗ h = δ α p ( a ) α p ( b ) ,pr ⊗ h = π ( α p ( a )) V ( p ) δ b,r ⊗ h. Since V ( p ) π ( a ) and π ( α p ( a )) V ( p ) are bounded linear operators, the equality also holdsfor every vector in K . Claim 5 . ( π, V ) is minimal.Since π ( a ) V ( p ) h = δ a,p ⊗ h we havespan { π ( a ) V ( p ) h : p ∈ P, a ∈ A α p (1) , h ∈ H} =span { δ a,p ⊗ h : p ∈ P, a ∈ A α p (1) , h ∈ H} , which is clearly dense in K . Claim 6 . ( π, V ) is unique up to unitary equivalence.Suppose e π and e V is another minimal dilation on a Hilbert space e K and consider x = X i ∈ F e π ( a i ) e V ( p i ) h i , y = X i ∈ F e π ( a i ) e V ( p i ) k i . Their e K inner product is given by h x, y i = X i,j ∈ F h e V ( p i ) ∗ e π ( a ∗ i a j ) e V ( p j ) h j , k i i . Since K ( p, a, q ) = P H V ( p ) ∗ π ( a ) V ( q ) (cid:12)(cid:12)(cid:12)(cid:12) H for all ( p, a, q ) ∈ Λ ( A ,P,α ) , h e V ( p i ) ∗ e π ( a ∗ i a j ) e V ( p j ) h j , k i i = h K ( p i , a ∗ i a j , p j ) h j , k i i . Therefore, h x, y i = X i,j ∈ F h K ( p i , a ∗ i a j , p j ) h j , k i i . ILATION AND SEMIGROUP DYNAMICAL SYSTEM 17
A similar argument can be applied to the two vectors
U x = X i ∈ F π ( a i ) V ( p i ) h i , U y = X i ∈ F π ( a i ) V ( p i ) k i , proving that h U x, U y i = h x, y i for all x, y ∈ span { e π ( a ) e V ( p ) h : p ∈ P, a ∈ A α p (1) , h ∈ H} .Therefore, U can be extended to a unitary on its closure e K . It is routine to check that e π ( a ) = U ∗ π ( a ) U and e V ( p ) = U ∗ V ( p ) U for all a ∈ A and p ∈ P . This establishes that theminimal dilation ( π, V ) is unique up to unitary equivalence and completes the proof. (cid:3) Notice that Theorem 4.9 only involves the weaker covariance condition V ( p ) π ( a ) = π ( α p ( a )) V ( p ), and that, in principle, the minimal dilation ( π, V ) may not satisfy thestronger covariance condition V ( p ) π ( a ) V ( p ) ∗ = π ( α p ( a )). A major difficulty is that theconstruction is not explicit about the vectors V ( p ) ∗ δ q,b ⊗ h . However, in the particularcase when V ( p ) V ( p ) ∗ = π ( α p (1)) holds for all p ∈ P , then we can recover the covariancecondition, as the following proposition shows. Proposition 4.10.
Suppose π : A → B ( K ) is a unital ∗ -homomorphism and V : P →B ( K ) is an isometric representation. Then the following are equivalent: (1) For every p ∈ P and a ∈ A , V ( p ) π ( a ) V ( p ) ∗ = π ( α p ( a )) . (2) For every p ∈ P and a ∈ A , V ( p ) π ( a ) = π ( α p ( a )) V ( p ) . and V ( p ) V ( p ) ∗ = π ( α p (1)) .Proof. The conditions in (2) follow from (1) by multiplying V ( p ) on the right hand sideand setting a = 1. To see the converse, assume (2) holds and multiply V ( p ) ∗ on the righton both sides, V ( p ) π ( a ) V ( p ) ∗ = π ( α p ( a )) V ( p ) V ( p ) ∗ = π ( α p ( a )) π ( α p (1))= π ( α p ( a )) . (cid:3) Remark 4.11.
The connection between the covariance relations V ( p ) π ( a ) = π ( α p ( a )) V ( p )and V ( p ) π ( a ) V ( p ) ∗ = π ( α p ( a )) is further explored in [23], where it is shown that whenthe semigroup is abelian, the former relation can be dilated to the latter one. Remark 4.12.
Even though our main focus is on right LCM dynamical systems, ourversion of generalized Naimark dilation can be applied to a much wider range of semigroupdynamical systems.Recall the semigroup dynamical system arising from the Cuntz algebra O k . Let S , · · · , S k be isometries that generate O k , and for any word w = w w · · · w n ∈ F + k , let | w | = n bethe length of the word w and S w = S w S w · · · S w n . Recall that the core A = span { S µ S ∗ ν : | µ | = | ν |} of O k is isomorphic to the UHF algebra of type k ∞ . One can define the ac-tion α ( x ) = e ⊗ x on the UHF core, which is the same as α ( x ) = S xS ∗ . We havea semigroup dynamical system ( A , N , α ), where the N -action α is induced by the single ∗ -endomorphism α via α n := α n . For every m, n ∈ N , one can explicitly compute thecorner A n,m = α n (1) A α m (1) as A n,m = span { S µ S ∗ ν : | µ | = | ν | , µ = 1 · · · | {z } n µ ′ , ν = 1 · · · | {z } m ν ′ } = span { S n S µ ′ S ∗ ν ′ S ∗ m : | µ ′ | + n = | ν ′ | + m } Let T , · · · , T k be contractions in B ( H ) such that P ki =1 T i T ∗ i = I . For any w ∈ F + k , wedenote T w in a similar way as S w . One can define a kernel system K by K ( n, S n S µ ′ S ∗ ν ′ S ∗ m , m ) = T µ ′ T ∗ ν ′ . This kernel is from the compression of the identity map of A and the isometric representa-tion V ( n ) = S n . Therefore, it does satisfy all the conditions in Definition 4.4. This is notsurprising due to the well-known dilation theorem of Popescu, [30, Theorem 3.2], wherebythe T i can be dilated to Cuntz isometries.5. Covariant Pairs and Dilation
We turn our attention back to right LCM dynamical systems. The goal of this section isto prove the following result, which can be seen as an equivariant version of Stinespring’stheorem for right LCM dynamical systems.
Theorem 5.1.
Let ( φ, T ) be a contractive covariant representation of a right LCM dy-namical system ( A , P, α ) . Then there exists an isometric covariant dilation ( π, V ) of ( φ, T ) if and only if φ is a unital completely positive map. Moreover, the dilation ( π, V ) can betaken to be minimal and the minimal dilation is unique up to unitary equivalence. Remark 5.2.
Consider the trivial case where P = { e } . Then Theorem 5.1 is simplyStinespring’s dilation theorem which states that every unital completely positive map canbe dilated to a ∗ -homomorphism.For nontrivial P , Stinespring’s theorem only gives a ∗ -homomorphic dilation π of theunital completely positive map φ . Surprisingly, it also allows us to construct an isometricdilation V alongside, such that ( π, V ) satisfies the covariance condition.The proof is divided into two parts. First we construct a kernel system K from a contrac-tive covariant representation ( φ, T ) and show, in a series of lemmas and propositions, thatthe kernel system K enjoys all the nice properties from Definition 4.4 whenever φ is unitalcompletely positive. This allows us to invoke Theorem 4.9 to obtain a ∗ -homomorphism π of the C ∗ -algebra and an isometric representation V of the semigroup. Then the proof isconcluded by verifying that ( π, V ) satisfies the covariance condition from Definition 3.14.We begin by constructing kernel systems from contractive covariant representations. Lemma 5.3.
Let ( φ, T ) be a contractive covariant representation of the injective rightLCM dynamical system ( A , P, α ) . Suppose that rP = sP = pP ∩ qP . Then T ( p − r ) φ ( α − r ( a )) T ( q − r ) ∗ = T ( p − s ) φ ( α − s ( a )) T ( q − s ) ∗ Proof. If rP = sP = pP ∩ qP , then r = su for some invertible element u ∈ P , so we have T ( p − r ) φ ( α − r ( a )) T ( q − r ) ∗ = T ( p − s ) T ( u ) φ ( α − u ( α − s ( a ))) T ( u ) ∗ T ( q − s ) ∗ = T ( p − s ) φ ( α − s ( a )) T ( q − s ) ∗ . (cid:3) Definition 5.4.
The kernel system K associated with a contractive covariant representa-tion ( φ, T ) is the function on Λ ( A ,P,α ) defined by K ( p, a, q ) := ( T ( p − r ) φ ( α − r ( a )) T ( q − r ) ∗ if rP = pP ∩ qP = ∅ pP ∩ qP = ∅ . Notice K is well-defined because of Lemma 5.3. Lemma 5.5.
Suppose that a ∈ A s for some s ∈ pP ∩ qP . Then K ( p, a, q ) = T ( p − s ) φ ( α − s ( a )) T ( q − s ) ∗ . ILATION AND SEMIGROUP DYNAMICAL SYSTEM 19
Proof.
Let pP ∩ qP = rP ; since s ∈ rP we have s = rt for some t ∈ P . We have, T ( p − s ) φ ( α − s ( a )) T ( q − s ) ∗ = T ( p − r ) T ( t ) φ ( α − s ( a )) T ( t ) ∗ T ( q − r ) ∗ = T ( p − r ) φ ( α t ( α − s ( a ))) T ( q − r ) ∗ = T ( p − r ) φ ( α − r ( a )) T ( q − r ) ∗ = K ( p, a, q ) . (cid:3) Lemma 5.6.
Let p, q ∈ P and r ∈ qP . Let a ∈ A p A r ⊆ A p A q . Then K ( p, a, q ) = K ( p, a, r ) T ( q − r ) ∗ . Proof.
In the case when pP ∩ rP = ∅ , a ∈ A p A r = { } and thus a = 0, and either sidebecomes 0.Assume otherwise that pP ∩ rP = sP ⊂ pP ∩ qP . Then s ∈ pP ∩ qP , and thus byLemma 5.5, K ( p, a, q ) = T ( p − s ) φ ( α − s ( a )) T ( q − s ) ∗ = T ( p − s ) φ ( α − s ( a )) T ( r − s ) ∗ T ( q − r ) ∗ = K ( p, a, r ) T ( q − r ) ∗ . (cid:3) Proposition 5.7.
Let K be the kernel system associated with ( φ, T ) . Then, K is unital,Hermitian, Toeplitz, and linear.Proof. The kernel K is unital because K ( e, , e ) = T ( e ) φ (1) T ( e ) ∗ = I . To see K isHermitian, consider first the case when pP ∩ qP = ∅ ; then K ( p, a, q ) ∗ = 0 = K ( q, a ∗ , p ).Next consider the case pP ∩ qP = rP and use the fact that φ and α r are ∗ -maps: K ( p, a, q ) ∗ = (cid:0) T ( p − r ) φ ( α − r ( a )) T ( q − r ) ∗ (cid:1) ∗ = T ( q − r ) φ ( α − r ( a )) ∗ T ( p − r ) ∗ = T ( q − r ) φ ( α − r ( a ∗ )) T ( p − r ) ∗ = K ( q, a ∗ , p ) . To see that K is Toeplitz, assume first pP ∩ qP = ∅ ; then it is clear that rpP ∩ rqP = r · ( pP ∩ qP ) = ∅ , and K ( p, a, q ) = 0 = K ( rp, α r ( a ) , rq ). Assume next pP ∩ qP = sP ; then rpP ∩ rqP = rsP , and K ( rp, α r ( a ) , rq ) = T (( rp ) − rs ) φ ( α − rs ( α r ( a ))) T (( rq ) − rs ) ∗ = T ( p − s ) φ ( α − s ( a )) T ( q − s ) ∗ = K ( p, a, q ) . Finally, the linearity of K follows from that of φ and α r . (cid:3) Lemma 5.8.
Suppose p , · · · , p n ∈ P have a least common multiple r ∈ T ni =1 p i P , and let a , · · · , a n ∈ A r . If φ is completely positive, then the operator matrix [ K ( p i , a ∗ i a j , p j )] ispositive definite.Proof. For each i , write r = p i q i for some q i ∈ P and a i = α r ( b i ) for b i ∈ A . Let b i = α − r ( a i ). By Lemma 5.5, K ( p i , a ∗ i a j , p j ) = T ( p − i r ) φ ( α − r ( a ∗ i a j )) T ( p − j r ) ∗ = T ( p − i r ) φ ( b ∗ i b j ) T ( p − j r ) ∗ . Let A = [ φ ( b ∗ i b j )], B = [ b ∗ i b j ], and D be the n × n diagonal matrix with diagonal entries T ( p − i r ). Then [ K ( p i , a ∗ i a j , p j )] = DAD ∗ = Dφ ( n ) ( B ) D ∗ . Since B is obviously positive and φ is completely positive, the matrix [ K ( p i , a ∗ i a j , p j )] ispositive definite. (cid:3) Lemma 5.9.
Let F = { p , · · · , p n } ⊂ P and for each subset W ⊆ F define E W,F = (cid:16) Y p ∈ W E p (cid:17)(cid:16) Y p ∈ F \ W ( I − E p ) (cid:17) . Then (1) E W,F is an orthogonal projection. (2)
For each p ∈ F \ W and a ∈ A p , aE W,F = E W,F a = 0 . (3) For any W = W ⊆ F , E W ,F E W ,F = 0 . (4) P W ⊆ F E W,F = I .Proof. Since { E p : p ∈ F } is a family of commuting projections, their products (andproducts of their orthogonal complements I − E q ) are projections. For a ∈ A p we have a ( I − E p ) = ( I − E p ) a = 0 and thus aE W,F = E W,F a = 0 whenever p / ∈ W . In particular,for any W = W ⊆ F , there exists p ∈ F that is an element for exactly one of W , W .Without loss of generality, assume p ∈ W but p / ∈ W . We have E W ,F E p = E W ,F , but E p E W ,F = 0. Therefore, E W ,F E W ,F = 0. Finally, I = Q p ∈ F ( E p + ( I − E p )). Expandingthe product derives the desired equality. (cid:3) Proposition 5.10. K is positive if and only if φ is completely positive.Proof. Notice that K ( e, a, e ) = φ ( a ) for every a ∈ A . Assume that K is positive. Givenany positive n × n operator matrix A = [ a i,j ] ∈ M n ( A ), we can find B = [ b i,j ] ∈ M n ( A )with A = BB ∗ . We have φ ( A ) = φ ( BB ∗ ) = n X k =1 [ φ ( b i,k b ∗ j,k )] n × n = n X k =1 [ K ( e, b i,k b ∗ j,k , e )] n × n ≥ , proving that φ is completely positive.Conversely, assume that φ is completely positive. Let F = { p , · · · , p n } ⊂ P and a i ∈ A p i . We need to prove that the operator matrix [ K ( p i , a ∗ i a j , p j )] is positive definite.By Lemma 5.9(4) and the linearity of K ,[ K ( p i , a ∗ i a j , p j )] = X W ⊆ F [ K ( p i , a ∗ i E W,F a j , p j )] . Hence it suffices to prove that for all W ⊆ F ,[ K ( p i , a ∗ i E W,F a j , p j )] ≥ . For each W ⊆ F and each j / ∈ W we have E W,F a j = 0. Therefore, the entries of thematrix [ K ( p i , a ∗ i E W,F a j , p j )] vanish outside the | W | × | W | submatrix S W = [ K ( p i , a ∗ i E W,F a j , p j )] i,j ∈ W . If T p i ∈ W p i P = ∅ , then Q p i ∈ W E p i = 0 so that E W,F = 0, in which case S W is thezero matrix. Otherwise, by the right LCM condition, T p i ∈ W p i P = rP for some r ∈ P and c i := a i E W,F ∈ A r . Since E W,F is a projection, c ∗ i c j = a ∗ i E W,F a j . Therefore, S W =[ K ( p i , c ∗ i c j , p j )] ≥ (cid:3) We are now ready to prove Theorem 5.1.
ILATION AND SEMIGROUP DYNAMICAL SYSTEM 21
Proof of Theorem 5.1.
Assume first φ has a dilation; then it is a compression of a ∗ -homomorphism, and thus must be unital completely positive.Conversely, assume that φ is unital completely positive. Define a kernel system K as inDefinition 5.4. Proposition 5.10 proves that K is positive. By Proposition 5.7 and Theorem4.9, there exists a ∗ -homomorphism π : A → B ( K ) and an isometric representation V : P → B ( K ) on some Hilbert space K ⊃ H , such that φ ( a ) = P H π ( a ) (cid:12)(cid:12)(cid:12)(cid:12) H and T ( p ) = P H V ( p ) (cid:12)(cid:12)(cid:12)(cid:12) H for all a ∈ A and p ∈ P . Moreover, by Theorem 4.9, ( π, V ) satisfies the weaker covari-ance relation V ( p ) π ( a ) = π ( α p ( a )) V ( p ). To see ( π, V ) is in fact an isometric covariantrepresentation, by Proposition 4.10, it suffices to prove that V ( p ) V ( p ) ∗ = π ( α p (1)) for all p ∈ P .We claim that for every δ q,b ⊗ h ∈ K ,(5.1) V ( p ) ∗ δ q,b ⊗ h = ( δ p − r,α p − ( b ) ⊗ T ( q − r ) ∗ h, if pP ∩ qP = rP , if pP ∩ qP = ∅ . Pick any δ s,c ⊗ k ∈ K ; then h V ( p ) ∗ δ q,b ⊗ h, δ s,c ⊗ k i = h δ q,b ⊗ h, V ( p ) δ s,c ⊗ k i = h δ q,b ⊗ h, δ ps,α p ( c ) ⊗ k i = h K ( ps, α p ( c ) ∗ b, q ) h, k i . Assume first pP ∩ qP = rP . Then h δ p − r,α p − ( b ) ⊗ T ( q − r ) ∗ h, δ s,c ⊗ k i = h K ( s, c ∗ α p − ( b ) , p − r ) T ( q − r ) ∗ h, k i = h K ( ps, α p ( c ) ∗ b, r ) T ( q − r ) ∗ h, k i , where we have used the Toeplitz property of K in the last line. By Lemma 5.6 we get h δ p − r,α p − ( b ) ⊗ T ( q − r ) ∗ h, δ s,c ⊗ k i = h K ( ps, α p ( c ) ∗ b, r ) T ( q − r ) ∗ h, k i = h K ( ps, α p ( c ) ∗ b, q ) h, k i = h V ( p ) ∗ δ q,b ⊗ h, δ s,c ⊗ k i . Therefore, V ( p ) ∗ δ q,b ⊗ h = δ p − r,α p − ( b ) ⊗ T ( q − r ) ∗ h , proving the first case in (5.1). Assumenext pP ∩ qP = ∅ . Then psP ∩ qP = ∅ , hence A ps A q = { } , which implies that α p ( c ) ∗ b = 0.Thus h V ( p ) ∗ δ q,b ⊗ h, δ s,c ⊗ k i = 0 for arbitrary δ s,c ⊗ k ∈ K . Hence V ( p ) ∗ δ q,b ⊗ h = 0,proving the second case in (5.1) and completing the proof of the claim.In order to conclude that V ( p ) V ( p ) ∗ = π ( α p (1)), since both sides are bounded linearoperators, and since the dilation is minimal, it suffices to show that V ( p ) V ( p ) ∗ δ q,b ⊗ h equals π ( α p (1)) δ q,b ⊗ h = δ q,α p (1) b ⊗ h for all ( q, b ) ∈ X and h ∈ H .Assume first pP ∩ qP = rP . Then equation (5.1) shows that V ( p ) V ( p ) ∗ δ q,b ⊗ h = V ( p ) δ p − r,α p − ( b ) ⊗ T ( q − r ) ∗ h = δ r,α p ( α p − ( b )) ⊗ T ( q − r ) ∗ h = δ r,α p (1) b ⊗ T ( q − r ) ∗ h. Now let δ s,c ⊗ k ∈ K and compute, h δ r,α p (1) b ⊗ T ( q − r ) ∗ h, δ s,c ⊗ k i = h K ( s, c ∗ α p (1) b, r ) T ( q − r ) ∗ h, k i = h K ( s, c ∗ α p (1) b, q ) h, k i = h δ q,α p (1) b ⊗ h, δ s,c ⊗ k i . where we have applied Lemma 5.6 in the second equality. So in this case we have V ( p ) V ( p ) ∗ δ q,b ⊗ h = δ r,α p (1) b ⊗ T ( q − r ) ∗ h = δ q,α p (1) b ⊗ h =: π ( α p (1))( δ q,b ⊗ h ) . Assume now that pP ∩ qP = ∅ . Equation (5.1) shows that V ( p ) ∗ δ q,b ⊗ h = 0 andhence V ( p ) V ( p ) ∗ δ q,b ⊗ h = 0. On the other hand, ( q, b ) ∈ X implies α q (1) b = b so α p (1) b = α p (1) α q (1) b = 0. Recall now that, by definition, h δ q, ⊗ h, δ r,a ⊗ k i = h K ( r, , q ) h, k i , for every δ r,a ⊗ k ∈ K , and since K ( r, , q ) = 0, we conclude that in the present case π ( α p (1))( δ q,b ⊗ h ) = 0 = δ q,α p (1) b ⊗ h . Hence V ( p ) V ( p ) ∗ = π ( α p (1)).We know from Theorem 4.9 that the dilation satisfies the weak covariance conditionand now that we have verified V ( p ) V ( p ) ∗ = π ( α p (1)) we may use Proposition 4.10 toconclude that ( π, V ) is an isometric covariant representation. The minimality of ( π, V )and its uniqueness are established by Theorem 4.9, completing the proof. (cid:3) Dilations on Right LCM Semigroups
Nica-covariance and dilation.
The main motivation of this paper comes fromthe recent development of dilation theory on right LCM semigroups in [22] where it isshown that a contractive representation T : P → B ( H ) admits an isometric Nica-covariantdilation if and only if( ⋄ ) X U ⊆ F ( − | U | T ( ∨ U ) T ( ∨ U ) ∗ ≥ F ⊂ P. In trying to interpret this dilation result, we noticed the key role played by the semi-group dynamical system ( D P , P, α ) based on the diagonal C*-algebra D P := span { E p = V ( p ) V ( p ) ∗ : p ∈ P } of the semigroup C ∗ -algebra. Recall that D P := span { E p = V ( p ) V ( p ) ∗ : p ∈ P } is commutative by the Nica-covariance condition and that P actson D P by α p ( x ) = V ( p ) xV ( p ) ∗ . The system ( D P , P, α ) is the template we used for thedefinition of right LCM dynamical systems. Indeed, Nica-covariance implies that α p (1) α q (1) = E p E q = ( E r = α r (1) , if pP ∩ qP = rP , if pP ∩ qP = ∅ . Using this one can easily check that α p ( D P ) is an ideal in D P , showing that ( D P , P, α ) isa right LCM dynamical system in the sense of Definition 3.1. Proposition 6.1.
Let φ : D P → B ( H ) be a unital ∗ -preserving linear map. Then thefollowing are equivalent: (1) φ is positive (2) φ is completely positive. (3) For each finite F ⊂ P and W ⊆ F , φ (cid:18) Y f ∈ W E f (cid:19)(cid:18) Y f ∈ F \ W ( I − E f ) (cid:19) ≥ . (4) For each finite F ⊂ P φ (cid:18) Y f ∈ F ( I − E f ) (cid:19) ≥ . Do we need to assume φ to be contractive? ILATION AND SEMIGROUP DYNAMICAL SYSTEM 23
Proof.
Recall that D P is a commutative C ∗ -algebra, so (1) and (2) are equivalent.For a finite subset F ⊂ P and W ⊆ F , define E W,F = (cid:18) Y p ∈ W E p (cid:19)(cid:18) Y p ∈ F \ W ( I − E p ) (cid:19) . For fixed F , one can apply Lemma 5.9 to show that the family { E W,F } W ⊆ F consists oforthogonal projections with mutually orthogonal ranges that sum up to I . An element x ∈ span { E p : p ∈ F } can therefore be written as x = P W ⊆ F λ W,F E W,F , and it is positiveif and only if λ W,F ≥ W ⊆ F such that E W,F = 0. Therefore, φ is positive ifand only if φ ( E W,F ) ≥ F ⊂ P and W ⊆ F , which proves the equivalencebetween (1) and (3).It is easy to see that (3) implies (4) by taking W = ∅ . To see the converse notice that,by Nica-covariance, either Q f ∈ W E f = 0, in which case φ ( E W,F ) = 0 and (3) holds, or else Q f ∈ W E f = E r for rP = T f ∈ W f P . In the latter case, for each f ∈ F \ W , E r E f = ( E s , if rP ∩ f P = sP , if rP ∩ f P = ∅ . Therefore, V ( r ) ∗ ( I − E f ) = ( ( I − E r − s ( f ) ) V ( r ) ∗ , if rP ∩ f P = s ( f ) PV ( r ) ∗ , if rP ∩ f P = ∅ . Next let F = { r − s ( f ) : s ( f ) P = rP ∩ f P, f ∈ F \ W, and rP ∩ f P = ∅} . We have E r Y f ∈ F \ W ( I − E f ) = V ( r ) Y f ∈ F ( I − E f ) V ( r ) ∗ . = α r ( Y f ∈ F ( I − E f )) . By (4), φ ( Q f ∈ F ( I − E f )) ≥
0. Therefore, φ ( E W,F ) = φ ( α r ( Y f ∈ F ( I − E f ))) = T ( r ) φ ( Y f ∈ F ( I − E f )) T ( r ) ∗ ≥ , so (3) holds in this case too. (cid:3) For each contractive representation T : P → B ( H ) one can define φ T ( E p ) = T ( p ) T ( p ) ∗ .Since the projections E p are linearly independent this can be extended by linearity to allof span { E p = V ( p ) V ( p ) ∗ : p ∈ P } . It is not immediate that φ T can be extended to acontractive positive map on D P . The following lemma gives a criterion for when this ispossible. Lemma 6.2.
Suppose T : P → B ( H ) is a contractive representation, let φ T ( E p ) := T ( p ) T ( p ) ∗ and extend φ T linearly to span { E p = V ( p ) V ( p ) ∗ : p ∈ P } . The following areequivalent (1) The map φ T extends to a positive map on D P = span { E p = V ( p ) V ( p ) ∗ : p ∈ P } . (2) For each finite F ⊂ P and W ⊆ F , φ T (cid:18) Y f ∈ W E f (cid:19)(cid:18) Y f ∈ F \ W ( I − E f ) (cid:19) ≥ . (3) For each finite F ⊂ P φ T (cid:18) Y f ∈ F ( I − E f ) (cid:19) ≥ . Proof.
The equivalence among (2) and (3) follows in the same way as the proof of Proposi-tion 6.1. It is obvious that (1) implies (2) since each (cid:18) Q f ∈ W E f (cid:19)(cid:18) Q f ∈ F \ W ( I − E f ) (cid:19) ≥ x = P p ∈ F λ p E p with k x k = 1. By Lemma 5.9, x = X W ⊆ F E W,F x = X W ⊆ F X p ∈ F λ p E W,F E p = X W ⊆ F ( X p ∈ W λ p ) E W,F . Let µ W = P p ∈ W λ p . Since { E W,F } is a collection of pairwise orthogonal projections, k x k = max W ⊂ F,E
W,F =0 | µ W | = 1 . To prove φ T can be extended by continuity to D P , it suffices to prove that k φ T ( x ) k ≤ φ T is contractive. This is equivalent to proving that the 2 × (cid:20) I φ T ( x ) φ T ( x ) ∗ I (cid:21) is positive definite. Since (cid:20) I φ T ( x ) φ T ( x ) ∗ I (cid:21) = X W ⊆ F (cid:20) φ T ( E W,F ) µ W φ T ( E W,F ) µ W φ T ( E W,F ) φ T ( E W,F ) (cid:21) and each φ T ( E W,F ) is positive by (2), this operator matrix is always positive. (cid:3)
As an application of our Theorem 5.1, we recover the following dilation theorem from[22].
Theorem 6.3. [22, Theorem 3.9]
Let T : P → B ( H ) be a contractive unital representationof a right LCM semigroup. The following are equivalent: (1) T has an isometric Nica-covariant dilation; (2) For any finite set F ⊂ P , X U ⊆ F ( − | U | T ( ∨ U ) T ( ∨ U ) ∗ ≥ . Proof.
Suppose (1) holds. We may assume that the dilation V is minimal and hence H is co-invariant. Let E f := V f V ∗ f . From the inclusion-exclusion principle, for every finite F ⊂ P , Y f ∈ F ( I − E f ) = X U ⊆ F ( − | U | E ∨ U , where we set E ∨ U = 0 when T p ∈ U pP = ∅ , and E ∨ U = E r when T p ∈ U pP = rP . That (1)implies (2) is now easy to see because P U ⊆ F ( − | U | T ( ∨ U ) T ( ∨ U ) ∗ is the compression of Q f ∈ F ( I − E f ) to the co-invariant subspace H .Suppose now that T is a contractive unital representation of P such that (2) holds anddefine φ ( E r ) := T ( r ) T ( r ) ∗ . Then condition (3) in Lemma 6.2 also holds. By condition (1)in Lemma 6.2 we can extend φ to a contractive representation of D P , which we also denoteby φ , such that ( φ, T ) is a contractive covariant pair. Since T satisfies (2), Proposition 6.1implies that φ is positive, hence completely positive. Theorem 5.1 does the rest. (cid:3) ILATION AND SEMIGROUP DYNAMICAL SYSTEM 25
Nica spectrum and boundary quotient.
The diagonal D P is a unital commuta-tive C ∗ -algebra, which is isomorphic to C (Ω P ) for a compact Hausdorff space Ω P . Thespace Ω P is called the Nica-spectrum of P , and one can refer to [26, 18, 9] for more detaileddescriptions. The action α by injective endomorphisms of D P induces an action ˆ α of P bysurjective maps of Ω P to itself. A compact subset K ⊂ Ω P is called α -invariant if both K and its complement are invariant, i.e. ˆ α p ( K ) ⊆ K and ˆ α p (Ω P \ K ) ⊆ Ω P \ K . Notice thatin the case of a group one of the inclusions implies the other, but for semigroup actions,both inclusions have to be required. Proposition 6.4.
Let K ⊂ Ω P be a closed ˆ α -invariant subset of Ω P . Then ( C ( K ) , P, α ) is also a right LCM dynamical system.Proof. Let Ω
P,p = ˆ α p (Ω P ) and K p = ˆ α p ( K ). Since K and its complement are invariantfor α , K p ∩ K q ⊂ Ω P,p ∩ Ω P,q ∩ K p, q ∈ P. If pP ∩ qP = ∅ , then Ω P,p ∩ Ω P,q = ∅ and also K p ∩ K q . If pP ∩ qP = rP , thenΩ P,p ∩ Ω P,q ∩ K = Ω P,r ∩ K = K r ⊆ K p ∩ K q . and thus K p ∩ K q = K r . Therefore, A p = α p ( C ( K )) = C ( K p ) and A p A q ⊆ C ( K p ∩ K q ),which becomes { } when pP ∩ qP = ∅ and becomes C ( K r ) when pP ∩ qP = rP . (cid:3) Observe that C (Ω P ) = D P = span { E p = V ( p ) V ( p ) ∗ } . Thus, for every contractiverepresentation ( φ, T ) of the dynamical system, φ ( E p ) = φ ( α p (1)) = T ( p ) T ( p ) ∗ . This relation remains true when we consider ( C ( K ) , P, α ) for any invariant subset K . Theorem 6.5.
Let K be a closed invariant subset of Ω P and ( φ, T ) be a contractivecovariant representation of ( C ( K ) , P, α ) . Then the following are equivalent: (1) There exists an isometric covariant dilation ( π, V ) of ( φ, T ) . (2) φ is positive. (3) For each finite F ⊆ P , X U ⊆ F ( − | U | T ( ∨ U ) T ( ∨ U ) ∗ ≥ . Proof.
That (1) implies (2) is easy to see, and (2) ⇐⇒ (3) follows from covariance andProposition 6.1. Suppose now (2) holds. Since C ( K ) is commutative φ is completelypositive, hence (1) holds by Theorem 5.1. (cid:3) An important invariant subspace of Ω P is the boundary spectrum ∂ Ω P , which consistsof the closure of the set of maximal elements in Ω P [5, Definition 5.1], see also [18].The boundary quotient corresponding to the boundary spectrum can be characterized viafoundation sets. By definition, a finite set F ⊂ P is a foundation set if for all p ∈ P ,there exists f ∈ F such that f P ∩ pP = ∅ . Let I ∞ be the ideal of D P generated by theprojections Q f ∈ F ( I − E f ) for every foundation set F . Then C ( ∂ Ω P ) = D P /I ∞ . The boundary quotient C ∗ -algebra Q ( P ) is the universal C ∗ -algebra generated by Nica-covariant representations V with the additional requirement that(6.1) Y f ∈ F ( I − V f V ∗ f ) = 0 for every foundation set F. This can also be realized as the cross-product ∂ D P ⋊ P . For more detailed studies of theboundary quotient C ∗ -algebras, one may refer to [18, 9, 31]. Example 6.6.
Consider the free semigroup F + n . A Nica-covariant representation V isuniquely determined by its value on the n generators V , · · · , V n . In this case, the Nica-covariance condition is equivalent to saying that the generating isometries V , · · · , V n haveorthogonal ranges, or in short, n X i =1 V i V ∗ i ≤ I. The universal C ∗ -algebra (or the semigroup C ∗ -algebra for F + n ) is thus the Toeplitz-Cuntzalgebra T O n . However, the boundary quotient requires further that P ni =1 V i V ∗ i = I , andthe boundary quotient semigroup C ∗ -algebra for F + n is the Cuntz algebra O n .A contractive representation of the boundary quotient dynamical system ( C ( ∂ Ω P ) , P, α )is uniquely determined by a contractive representation T : P → B ( H ) with the additionalrequirement that for every foundation set F ⊂ P , X U ⊆ F ( − | U | T ( ∨ U ) T ( ∨ U ) ∗ = 0 . As a corollary of Theorem 6.3, we establish a condition for a contractive representationto be dilated to an isometric representation of the boundary quotient.
Corollary 6.7.
Let T : P → B ( H ) be a contractive representation of a right LCM semi-group P such that for each foundation set F ⊂ P , X U ⊆ F ( − | U | T ( ∨ U ) T ( ∨ U ) ∗ = 0 . Then, the following are equivalent: (1)
There exists a Nica covariant isometric dilation V : P → B ( K ) satisfying (6.1)(2) For each finite F ⊆ P , X U ⊆ F ( − | U | T ( ∨ U ) T ( ∨ U ) ∗ ≥ . Dilation on C (Ω P ) ⊗ A In this section we build more examples of right LCM dynamical systems, starting fromautomorphic actions. Throughout we fix a right LCM semigroup P with trivial unit group P ∗ = { e } , so that whenever pP ∩ qP = rP , the choice of r is unique. As before, Ω P denotesits Nica-spectrum and ∂ Ω P denotes the boundary of Ω P .7.1. A dynamical system on C (Ω P ) ⊗ A . Suppose β is a ∗ -automorphic P -action onthe unital C*-algebra A . The system ( A , P, β ) is not necessarily a right LCM dynamicalsystem, in fact it cannot be when there exist elements p, q such that pP ∩ qP = ∅ , becausethen β p ( A ) β q ( A ) = A 6 = { } . Our first goal is to construct a right LCM dynamical systemon the C ∗ -algebra e A := C (Ω P ) ⊗ A of continuous A -valued functions on Ω P , and thengive conditions under which unital completely positive maps on A can be lifted to e A .Recall the natural right LCM dynamical system ( C (Ω) , P, α ) where α p ( V q V ∗ q ) = V pq V ∗ pq ,and define e α on e A by f α p ( f ⊗ a ) = α p ( f ) ⊗ β p ( a ) . In other words, e α = α ⊗ β , which is clearly a P -action on e A by ∗ -endomorphisms. Lemma 7.1.
The semigroup dynamical system ( e A , P, e α ) is a right LCM dynamical system. ILATION AND SEMIGROUP DYNAMICAL SYSTEM 27
Proof.
Since β p is a ∗ -automorphism, the image f α p ( e A ) is α p ( C (Ω P )) ⊗A . Since ( C (Ω P ) , P, α )is a right LCM dynamical system, by Proposition 3.3, α p ( C (Ω P )) α q ( C (Ω P )) = ( α r ( C (Ω P )) , if pP ∩ qP = rP { } , if pP ∩ qP = ∅ . This relation is preserved by tensor products, and another application of Proposition 3.3shows that ( e A , P, e α ) is a right LCM dynamical system. (cid:3) Next, given a unital completely positive map φ : A → B ( H ) and a contractive repre-sentation T : P → B ( H ), we hope to construct a covariant representation ( e φ, T ) for theright LCM dynamical system ( e A , P, e α ). Note that φ and T do not need to satisfy anycovariance relation by themselves. Recall C (Ω P ) ∼ = D P is the closed linear span of theprojections E p = V p V ∗ p and define e φ ( E p ⊗ a ) = T ( p ) φ ( β − p ( a )) T ( p ) ∗ . For every finite F ⊂ P , we define a map φ F : A → B ( H ) as follows. If a subset U ⊂ F has an upper bound, we let s U be the least upper bound of U , so that s U P = T p ∈ U pP and s U is uniquely determined because we are assuming P has no nontrivial units. If U has no upper bound, so that T p ∈ U pP = ∅ , then by convention we say s u = ∞ and we let T ( ∞ ) = 0. Then we define φ F ( a ) = X U ⊆ F ( − | U | T ( s U ) φ ( β − s U ( a )) T ( s U ) ∗ . Proposition 7.2.
Suppose β is a ∗ -automorphic P -action on A , φ : A → B ( H ) is a unitalcompletely positive map and T : P → B ( H ) is a contractive representation. Let e φ and φ F be as above. The pair ( e φ, T ) can be extended to a contractive covariant representation of ( e A , P, e α ) with unital completely positive e φ if and only if φ F is completely positive for allfinite F ⊂ P .Proof. For each finite subset F ⊂ P and a ∈ A we have φ F ( a ) = X U ⊆ F ( − | U | T ( s U ) φ ( β − s U ( a )) T ( s U ) ∗ = X U ⊆ F ( − | U | e φ ( E s U ⊗ a )= e φ ( E F ⊗ a ) . where E F = P U ⊆ F ( − | U | E s U = Q p ∈ F ( I − E p ) is an orthogonal projection in C (Ω P ). If( e φ, T ) is a contractive covariant representation with e φ a unital completely positive map,then φ F must be completely positive.For the converse, we first show that for every p, q ∈ P and a ∈ A , e φ ( f α p ( E q ⊗ a )) = e φ ( E pq ⊗ β p ( a ))= T ( p ) T ( q ) φ ( β − pq ( β p ( a )) T ( q ) ∗ T ( p ) ∗ = T ( p ) T ( q ) φ ( β − q ( a )) T ( q ) ∗ T ( p ) ∗ = T ( p ) e φ ( E q ⊗ a ) T ( p ) ∗ . Therefore, for any x ∈ span { E q ⊗ a : q ∈ P, a ∈ A} , e φ ( f α p ( x )) = T ( p ) e φ ( x ) T ( p ) ∗ . Next, for each W ⊆ F we define φ W,F ( a ) = e φ (cid:16)(cid:16) Y e ∈ W E e Y f ∈ F \ W ( I − E f ) (cid:17) ⊗ a (cid:17) . We claim that φ W,F is completely positive. If T e ∈ W eP = ∅ , we have Q e ∈ W E e = 0 so theclaim is trivially satisfied. Otherwise, T e ∈ W eP = rP for some r ∈ P , and Q e ∈ W E e = E r .Define F = { r − s : sP = rP ∩ f P, f ∈ F \ W } . Then, for each f ∈ F \ W , either f P ∩ rP = ∅ and E r ( I − E f ) = E r , or f P ∩ rP = sP so that E r ( I − E f ) = E r − E s = α r ( I − E r − s ).Therefore, φ W,F ( a ) = e φ ( α r ( Y p ∈ F ( I − E p )) ⊗ a )= e φ ( f α r ( E F ⊗ β − r ( a )))= T ( r ) φ F ( β − r ( a )) T ( r ) ∗ . Since φ F is completely positive, composing with a ∗ -automorphism β − r and conjugatingwith a contraction T ( r ) yields another completely positive map φ W,F . This completes theproof of the claim.Let C (Ω P ) = span { E p } , which is dense in C (Ω P ). We first extend e φ to f A = C (Ω P ) ⊗A . This extension is clearly unital because e φ ( I ⊗
1) = T ( e ) φ (1) T ( e ) ∗ = I . To prove that e φ is completely positive, pick any x , · · · , x n ∈ f A . Since C (Ω P ) = span { E p } , one canfind a finite subset F ⊂ P and elements { a f,i } f ∈ F, ≤ i ≤ n ⊂ A , such that x i = X f ∈ F E f ⊗ a f,i . For each W ⊆ F , let E W,F = Q e ∈ W E e Q f ∈ F \ W ( I − E f ). From Lemma 5.9, { E W,F } W ⊆ F are orthogonal projections and they are pairwise orthogonal. Moreover, for each f ∈ F ,the projection E f decomposes as E f = X f ∈ W E W,F . Therefore, rearranging and combining terms, we can rewrite x i = X W ⊆ F E W,F ⊗ a W,F,i for some a W,F,i ∈ A , and thus x ∗ i x j = X W ⊆ F E W,F ⊗ a ∗ W,F,i a W,F,j . It follows that the matrix[ e φ ( x ∗ i x j )] = X W ⊆ F [ e φ ( E W,F ⊗ a ∗ W,F,i a W,F,j )]= X W ⊆ F [ φ W,F ( a ∗ W,F,i a W,F,j )]is positive because φ W,F is completely positive for each W . This shows that e φ is unitalcompletely positive on f A . Since completely positive maps are also completely contractive, e φ can be extended by continuity to a unital completely positive map on e A . (cid:3) When we apply Theorem 5.1 to ( e φ, T ) we obtain the following strengthening of Theo-rem 6.3. ILATION AND SEMIGROUP DYNAMICAL SYSTEM 29
Theorem 7.3.
Suppose β is a ∗ -automorphic P -action on the unital C ∗ -algebra A . Let φ : A → B ( H ) be a unital completely positive map on A and let T : P → B ( H ) be acontractive representation of the right LCM semigroup P . The following are equivalent: (1) There exist a ∗ -homomorphism π : A → B ( K ) and an isometric representation V : P → B ( K ) on a Hilbert space K ⊃ H such that for a, b ∈ A and p, q ∈ P , if pP ∩ qP = rP , then (7.1) V ( p ) π ( β − p ( a )) V ( p ) ∗ V ( q ) π ( β − q ( b )) V ( q ) ∗ = V ( r ) π ( β − r ( ab )) V ( r ) ∗ ; and if pP ∩ qP = ∅ , then V ( p ) and V ( q ) have orthogonal ranges.Moreover, π and V are dilations for φ and T , respectively, and H is co-invariantfor V . (2) For each finite F ⊂ P , the map φ F : A → B ( H ) defined by φ F ( a ) = X U ⊆ F ( − | U | T ( s U ) φ ( β − s U ( a )) T ( s U ) ∗ is completely positive.Proof. Assume first that (2) holds and let ( e φ, T ) be the contractive covariant representationof the system ( e A , P, e α ) from Proposition 7.2. The map e φ is unital and completely positive,therefore, by Theorem 5.1, ( e φ, T ) admits an isometric covariant dilation ( e π, V ) on B ( K )for K ⊃ H in which H is co-invariant for V . Define π : A → B ( K ) by π ( a ) = e π ( I ⊗ a ). Onecan easily verify that π is a ∗ -homomorphism that is a dilation for φ and that e π ( E p ⊗ a ) = V ( p ) π ( β − p ( a )) V ( p ) ∗ . Notice that E p ⊗ a · E q ⊗ b = ( E r ⊗ ab, if pP ∩ qP = rP , if pP ∩ qP = ∅ . If pP ∩ qP = rP , then V ( p ) π ( β − p ( a )) V ( p ) ∗ V ( q ) π ( β − q ( b )) V ( q ) ∗ = e π ( E p ⊗ a ) e π ( E q ⊗ b )= e π ( E r ⊗ ab )= V ( r ) π ( β − r ( ab )) V ( r ) ∗ , and if pP ∩ qP = ∅ , then e π ( E p ⊗ e π ( E q ⊗
1) = V ( p ) V ( p ) ∗ V ( q ) V ( q ) ∗ = 0so that V ( p ) and V ( q ) have orthogonal ranges.Conversely, assume now (1) holds. Then π is a ∗ -homomorphism and thus unital com-pletely positive. Therefore, one can extend π and V to obtain a contractive covariant rep-resentation ( e π, V ) of ( e A , P, e α ). From the construction, e π ( E p ⊗ a ) = V ( p ) π ( β − p ( a )) V ( p ) ∗ ,and thus by equation (7.1), e π is multiplicative. Therefore, e π is a ∗ -homomorphism of e A .For each finite F ⊂ P , we have e π F ( a ) = e π (( Y f ∈ F ( I − E f )) ⊗ a ) = X U ⊆ F ( − | U | V ( s U ) π ( a ) V ( s U ) ∗ . which shows that e π F is a (non-unital) ∗ -homomorphism of A . Since π and V are dilationsof φ and T and H is co-invariant for V , projecting to the corner of H obtains φ F ( a ), whichis thus completely positive. (cid:3) Remark 7.4.
In the special case when A = C , a unital completely positive map φ on A is uniquely determined by φ ( x ) = xI . In this case, condition (2) is reduced to that inTheorem 6.3.One can use a similar technique to obtain a strengthened version of Corollary 6.7. Corollary 7.5.
Suppose that in addition to the conditions in Theorem 7.3, we also havethat, (7.2) X U ⊆ F ( − | U | T ( s U ) φ ( β − s U ( a )) T ( s U ) ∗ = 0 for every (finite) foundation set F ⊂ P . Let V be the resulting dilation from Theorem 7.3.Then (7.3) X U ⊆ F ( − | U | V ( s U ) π ( β − s U ( a )) V ( s U ) ∗ = 0 for every (finite) foundation set F ⊂ P .Proof. Notice first that the kernel of the quotient C (Ω P ) ⊗ A → C ( ∂ Ω P ) ⊗ A is the idealgenerated by E ⊗ a where the projection E is associated to a foundation set. Hence,condition (7.2) ensures that the map ˜ φ from the pair ( ˜ φ, T ) obtained in Proposition 7.2factors through this quotient. This gives a pair ( ˙˜ φ, T ) for the LCM system on the quotient˙˜ A := C ( ∂ Ω P ) ⊗ A .The equivalent conditions in Theorem 7.3 ensure that ˜ φ is completely positive and henceso is ˙˜ φ . Therefore, we can dilate ( ˙˜ φ, T ) to an isometric covariant representation ( ρ, V ) ofthe ˙˜ A system. Letting π ( a ) := ρ ( I ⊗ a ) for a ∈ A gives a pair ( π, V ) satisfying the requiredequation (7.3). (cid:3) Examples from F + k .Example 7.6. Let us consider the case when P = F +2 . Suppose A is any unital C ∗ -algebrawith a unital completely positive map φ : A → B ( H ) and two ∗ -automorphisms β , β .Suppose that T , T ∈ B ( H ) are two contractions such that for all a ∈ A , φ ( a ) = T φ ( β − ( a )) T ∗ + T φ ( β − ( a )) T ∗ We would like to claim that φ, T can be dilated to a ∗ -homomorphism π : A → B ( K ) andisometries V , V ∈ B ( K ) such that π ( a ) = V π ( β − ( a )) V ∗ + V π ( β − ( a )) V ∗ We first build C ( ∂ Ω P ) ⊗ A as an inductive limit of the following system: let A = A ,and A n +1 = A n ⊕ A n with connecting map ϕ n : A n → A n +1 by ϕ n ( a ) = a ⊕ a . One canverify that the inductive limit e A ∼ = C ( X ) ⊗ A , where X is the Cantor set.Define two ∗ -endomorphisms α , α on e A by α ( a ) = β ( a ) ⊕ α ( a ) = 0 ⊕ β ( a )for a ∈ A n . This extends to an F +2 -action on e A , by sending generator e i to α i , i = 1 ,
2. ByProposition 3.11, the resulting dynamical system is a right LCM dynamical system sinceranges of α i are both ideals in e A and their ranges are orthogonal to one another. A AA AAAA α α α α α α Let φ = φ : A → B ( H ), and recursively define φ n +1 : A n +1 = A n ⊕ A n → B ( H ) by φ n +1 ( a ⊕ b ) = T φ n ( β − ( a )) T ∗ + T φ n ( β − ( b )) T ∗ . ILATION AND SEMIGROUP DYNAMICAL SYSTEM 31
Since φ is unital completely positive, the map a T i φ ( β − i ( a )) T ∗ i is completely positive, φ is also a unital completely positive map. Inductively, each φ n is a unital completelypositive map. Notice that φ ( ϕ ( a )) = φ ( a ⊕ a )= T φ ( β − ( a )) T ∗ + T φ ( β − ( a )) T ∗ = φ ( a ) , and inductively, φ n +1 ( ϕ n ( a )) = φ n ( a ). A A A · · ·B ( H ) B ( H ) B ( H ) · · · ϕ ϕ φ = φ φ φ The maps φ n can be extended to a unital completely positive map e φ on the inductivelimit e A . Moreover, for each a ∈ A n , φ n +1 ( α ( a )) = φ n +1 ( β ( a ) ⊕
0) = T φ n ( a ) T ∗ , and, φ n +1 ( α ( a )) = φ n +1 (0 ⊕ β ( a )) = T φ n ( a ) T ∗ . We have for all a ∈ e A , e φ ( α i ( a )) = T i e φ ( a ) T ∗ i . We can view ( T , T ) as a contractive representation T of F +2 by sending generator e i to T i .The resulting pair ( e φ, T ) is a contractive covariant representation of ( e A , F +2 , α ). Therefore,by Theorem 5.1, it dilates to an isometric covariant representation ( e π, V ) on some Hilbertspace K ⊃ H . Define π : A → B ( K ) by π ( a ) = e π ( a ) for all a ∈ A = A ⊂ e A . We have π ( a ) = e π ( a )= e π ( a ⊕ a )= e π ( α ( β − ( a ))) + e π ( α ( β − ( a )))= V e π ( β − ( a )) V ∗ + V e π ( β − ( a )) V ∗ = V π ( β − ( a )) V ∗ + V π ( β − ( a )) V ∗ . Therefore, the pair ( φ, T ) gets dilated to the pair ( π, V ), and the relation φ ( a ) = T φ ( β − ( a )) T ∗ + T φ ( β − ( a )) T ∗ , is preserved as π ( a ) = V π ( β − ( a )) V ∗ + V π ( β − ( a )) V ∗ . One should also notice that by setting a = I , one gets V V ∗ + V V ∗ = I . Therefore V and V have orthogonal ranges.One can easily generalize Example 7.6 to F + k and obtain the following corollary. Corollary 7.7.
Let φ : A → B ( H ) be a unital completely positive map on a unital C ∗ -algebra A . Let β , · · · , β n be ∗ -automorphisms of A . Suppose that T , · · · , T k ∈ B ( H ) arecontractions such that for any a ∈ A , φ ( a ) = k X i =1 T i φ ( β − i ( a )) T ∗ i . Then there exists a ∗ -homomorphism π : A → B ( K ) on a Hilbert space K ⊃ H andisometries V , · · · , V k ∈ B ( K ) with orthogonal ranges, such that π ( a ) = k X i =1 V i π ( β − i ( a )) V ∗ i and π, V are dilations of φ, T , and H is co-invariant for V . Example 7.8.
Let φ : C → B ( H ) by φ ( x ) = xI and β i be identity on A . Let T , · · · , T k ∈B ( H ) such that P ki =1 T i T ∗ i = I . Then it is clear that φ ( x ) = k X i =1 T i φ ( x ) T ∗ i . Corollary 7.7 implies that we can dilate ( φ, T ) to a ∗ -homomorphism π : C → B ( K ) andisometries V , · · · , V k ∈ B ( K ) with orthogonal ranges, and π ( x ) = k X i =1 V i π ( x ) V ∗ i . But π ( x ) = xI is the only choice for π , and thus P ki =1 V i V ∗ i = I . This is preciselyPopescu’s dilation of row contractions. Example 7.9.
Let φ : M k → M k be the map φ ( E i,j ) = δ i,j E i,j . In other words, φ mapsa matrix A to its diagonal. One can easily verify that φ is unital completely positive mapbut φ is not multiplicative for any k ≥
2. Let β i be the identity map on M k for each i ,and define T i = E i, . Then for each A ∈ M k , φ ( A ) = diag( A , · · · , A kk ), and one cancheck that φ ( A ) = k X i =1 T i φ ( A ) T ∗ i . Corollary 7.7 implies that we can dilate φ, T i into a ∗ -homomorphism π : M k → B ( K ) andisometries V i with orthogonal ranges, such that for each A ∈ M k , π ( A ) = k X i =1 V i π ( A ) V ∗ i . Example 7.10.
Let X be a compact Hausdorff space and µ be a probability measure on X . One can define a unital completely positive map φ : C ( X ) → C by φ ( f ) = R X f dµ .Pick t , · · · , t k ∈ C such that P ki =1 | t i | = 1, and thus φ ( f ) = k X i =1 t i φ ( f ) t i . Corollary 7.7 implies that there exists a ∗ -homomorphism π : C ( X ) → B ( K ) andisometries V , · · · , V k ∈ B ( K ) with orthogonal ranges. Here, K = C ⊕ K and with respectto such decomposition, π ( f ) = (cid:20)R X f ( x ) dµ ∗∗ ∗ (cid:21) , and, V i = (cid:20) t i ∗ ∗ (cid:21) . ILATION AND SEMIGROUP DYNAMICAL SYSTEM 33
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