C*-envelopes of semicrossed products by lattice ordered abelian semigroups
aa r X i v : . [ m a t h . OA ] J a n C*-ENVELOPES OF SEMICROSSED PRODUCTS BY LATTICEORDERED ABELIAN SEMIGROUPS
ADAM HUMENIUK
Abstract.
A semicrossed product is a non-selfadjoint operator algebra en-coding the action of a semigroup on an operator or C*-algebra. We provethat, when the positive cone of a lattice ordered abelian group acts on a C*-algebra, the C*-envelope of the associated semicrossed product is a full cornerof a crossed product by the whole group. By constructing a C*-cover thatitself is a full corner of a crossed product, and computing the Silov ideal, weobtain an explicit description of the C*-envelope. This generalizes a result ofDavidson, Fuller, and Kakariadis from Z n + to the class of all lattice orderedabelian groups. Introduction
Preliminaries.
A semicrossed product is a non-selfadjoint generalization ofthe crossed product of a C*-algebra by a group. A crossed product B ⋊ G en-codes the action of a group G on a C*-algebra B , by embedding both into a largerC*-algebra in which the G -action is by unitaries. Built similarly, a semicrossedproduct of a (possibly non-selfadjoint) operator algebra A by an abelian semigroup P encodes a given action of P on A by completely contractive endomorphisms.First introduced by Arveson in [1], and first formally studied by Peters in [35] inthe case P = Z + , subsequent work on semicrossed products has focused on conju-gacy problems [3, 8, 10–12, 20, 23] and their C*-envelopes [6, 9, 22, 24, 25, 31]. For acomplete survey of the history of semicrossed products, and a thorough discussionof the conjugacy problem, we recommend Davidson, Fuller, and Kakariadis’ treat-ment in [7]. For a given action of P on A , there are multiple associated semicrossedproducts A × F P , depending on what family of admissible representations F of P one considers. Generally, we have distinct unitary, isometric, and contractive semi-crossed products A × un P , A × is P , and A × P , which satisfy universal propertiesfor “covariant” contractive/isometric/unitary representations of P with respect to A . Our main question of interest is: If P is a generating subsemigroup of an abeliangroup G , can the C*-envelope of a semicrossed product A × F P be realized as acrossed product B ⋊ G by G , for some G -C*-algebra B ⊇ A ? If the action of P on A is by automorphisms, then A × is P = A × un P , and the P action extends to ∗ -automorphisms of the C*-envelope C ∗ e ( A ). It follows that C ∗ e ( A × is P ) ∼ = C ∗ e ( A ) ⋊ G Date : January 22, 2020.2010
Mathematics Subject Classification.
Key words and phrases. dynamical systems of operator algebras, crossed product, semicrossedproduct, Nica-covariant, C*-envelope.Author supported by NSERC Alexander Graham Bell Canada Graduate Scholarship-Doctoral. is a crossed product [6, Theorem 3.3.1]. If G = P − P , and P acts on a C*-algebra A by ∗ -monomorphisms, then(1) C ∗ e ( A × un P ) ∼ = ˜ A ⋊ G is a crossed product for a certain unique minimal C*-algebra ˜ A ⊇ A whose G -action extends the action of P , called the minimal automorphic extension of A .Kakariadis and Katsoulis [24, Theorem 2.6] established (1) in the case P = Z + .Laca [26] showed how to build the automorphic dilation ˜ A in general, and from thisDavidson, Fuller, and Kakariadis establish (1) in [6, Theorem 3.2.3].Parrott’s example [34, Chapter 7] of three commuting contractions without asimultaneous isometric dilation, shows that the dilation theory of representationsof any semigroup at least as complicated as Z is intractable. To make progress,we need to restrict our class of representations F if we wish a nice dilation theoryfor A × F P . Of interest are lattice ordered abelian groups ( G, P ). These are pairsconsisting of a subsemigroup P of a group G , where the induced ordering g ≤ h ⇐⇒ h − g ∈ P makes G a lattice. In the lattice ordered setting, one studies the more tractable classof Nica-covariant representations, first introduced by Nica in [33]. Nica-covariance isa ∗ -commutation type condition which ensures a nice dilation theory. For instance,Li [28, 29] showed that every Nica-covariant representation of P has an isometricdilation.In the Nica-covariant setting, for injective C*-systems (1) holds with A × nc P inplace of A × un P . For non-injective systems, it is not possible to embed A × nc P into any crossed product B ⋊ G via inclusions A ⊆ B and P ⊆ G , because sucha system has no faithful unitary covariant pairs. The best one can do is embed A × nc P into a full corner of a crossed product. For a lattice ordered abelian group( G, P ) and an action of P on a C*-algebra A , one expects to prove(2) C ∗ e ( A × nc P ) ∼ = p ( B ⋊ G ) p, is a full corner of a crossed product of some G -C*-algebra B . Here A embeds into B non-unitally, and p is the projection coming from the unit in A . In the case( G, P ) = ( Z n , Z n + ), the result (2) was established in the case n = 1 by Kakariadisand Katsoulis [22, 24], and extended to general n ≥ G -C*-algebra B was intwo stages. First, one builds a bigger C*-algebra B ⊇ A which has an injective P -action dilating the P -action on A . This is accomplished by a tail-adding technique.Then one takes the minimal automorphic dilation B := ˜ B .We establish that (2) holds for any lattice ordered abelian group ( G, P ), when A is a C*-algebra (Corollary 3.14). Our approach differs from Davidson, Fuller, andKakariadis’ construction for P = Z n + . First, we define a notion of a Nica-covariantautomorphic dilation of A , which is a certain G -C*-algebra B with a non-unitalembedding A ⊆ B . This definition is meant to capture a sufficient set of conditionsto get a completely isometric embedding A × nc P ⊆ p ( B ⋊ G ) p, with p = 1 A . When the dilation B is minimal , this is a C*-cover. Then, we showthat the Silov ideal in such a cover has the form p ( I ⋊ G ) p , for a unique maximum *-ENVELOPES OF SEMICROSSED PRODUCTS 3 G -invariant ideal I ⊳ B not intersecting A . Upon taking a quotient by the Silovideal, C ∗ e ( A × nc P ) ∼ = ( p + I ) (cid:18) BI ⋊ G (cid:19) ( p + I )is a full corner of a crossed product. Then, it suffices to show any C*-algebra A with P -action has at least one minimal Nica-covariant automorphic dilation. Webuild one via a direct product construction (Definition 3.4).A semicrossed product is a special instance of the tensor algebra of a C*-correspondence [19, 31] (when P = Z + ) or product system [13, 14, 16–18, 37]. Kat-soulis and Kribs [25] showed that the C*-envelope of the tensor algebra of a C*-correspondence X is the associated Cuntz-Pimsner algebra O X , a generalization ofthe usual crossed product. In [15], Dor-On and Katsoulis extend this result andshow that the C*-envelope of the Nica tensor algebra N T X associated to a prod-uct system X over P coincides with a certain co-univeral quotient of the associatedCuntz-Nica-Pimsner algebra N O X considered by Carlsen et al. [5], and also coin-cides with an associated covariance algebra A × X P defined by Sehnem [36]. Ourresult shows further that, when this product system arises from a C*-dynamicalsystem, this same C*-envelope has the structure of a corner of a crossed product,and so is Morita equivalent to a crossed product.Before proceeding, we should also direct the reader to the extensive literature onC*-algebras associated to semigroups and semigroup dynamical systems, including[21, 26, 27, 30, 32, 33, 38, 39]. Following Nica [33], Laca and Raeburn [27] demon-strated for quasi-lattice ordered ( G, P ) that the universal C*-algebra C ∗ ( G, P ) forNica-covariant representations of P has the structure of a semigroup crossed prod-uct. Interestingly, we will see (Remark 3.7) that our direct product construction ofan automorphic dilation reduces to Laca-Raeburn’s in the case where P acts on C trivially.1.2. Structure of This Paper.
Throughout this section, (
G, P ) is a lattice or-dered abelian group, and P acts on a C*-algebra A by ∗ -endomorphisms. In Section2, we review the construction of the semicrossed product, and necessary backgroundon ordered groups and C*-envelopes. Section 3 contains our main results. We definethe notion of a minimal Nica-covariant automorphic dilation , construct a canonicalsuch dilation we call the product dilation , and show that any such dilation alwaysyields a C*-cover of the Nica-covariant semicrossed product A × nc P via full cor-ner of a crossed product (Proposition 3.8). We show the Silov ideal arises froma unique maximum G -invariant boundary ideal in any such C*-cover in Theorem3.12, and hence show that the C*-envelope of A × nc P is a full corner of a crossedproduct (Corollary 3.14). In two immediate applications, we show that Theorem3.12 reduces to the known result (1) for A × nc P in the injective case (Proposition3.15), and we compute the unique maximum boundary ideal in the product dilationin the case P = Z + (Proposition 3.16).Section 4 is devoted to explicitly computing the Silov ideal in the C*-cover arisingfrom the product dilation for any Nica-covariant semicrossed product A × nc P . Wedo so by describing a unique maximum G -invariant boundary ideal I in the productdilation B . Then C ∗ e ( A × nc P ) ∼ = p A (cid:18) BI ⋊ G (cid:19) p A ADAM HUMENIUK is a full corner by p A := 1 A + I . Using the explicit construction of I from Section 4,in Section 5 we show that the G -C*-algebra B/I in the case P = Z n + is equivariantly ∗ -isomorphic to the construction given by Davidson, Fuller, and Kakariadis in [6,Section 4.3]. So, our description of the C*-envelope reduces to the known resultwhen P = Z n + . In Section 6, we give some applications both of Theorem 3.12 andthe explicit description of I from Section 4. In particular, for totally ordered groups( G, P ) which are direct limits of ordered subgroups (
G, P ) = S λ ( G λ , P λ ), such as Q = S n Z /n !, we have C ∗ e ( A × nc P ) = lim −→ λ C ∗ e ( A × nc P λ )naturally, as long as P acts on A by surjections. This result is sharp and fails fornon-totally ordered groups and non-surjective actions. Acknowledgements.
The author would like to thank Kenneth Davidson andMatthew Kennedy for their helpful comments and useful discussions.2.
Background
In this paper, a (discrete, unital) semigroup P is a set equipped with an as-sociative binary operation, and we require that P contains a two-sided identityelement. We are primarily interested in abelian semigroups. In the abelian setting,we will always denote the semigroup operation by + and the identity element by0. A semigroup homomorphism is a function between semigroups preserving thesemigroup operations and the identity.If A is a C*-algebra, an ideal I ⊳ A always means a closed, two-sided ideal.We make frequent use of the following two inductivity properties of ideals in C*-algebras. Firstly, if A = [ λ ∈ Λ A λ ∼ = lim −→ λ ∈ Λ A λ is an internal direct limit of C*-subalgebras A λ , and I ⊳ A is an ideal, then I = [ λ ∈ Λ I ∩ A λ . In particular, I = 0 if and only if I ∩ A λ = 0 for all λ ∈ Λ. Secondly, if { I λ | λ ∈ Λ } is a family of ideals in A that is directed under inclusion, then I := S λ ∈ Λ I λ is alsoan ideal in A .Let P be a semigroup. An (operator algebra) dynamical system ( A, α, P )over P consists of an operator algebra A and a semigroup action α of P on A bycompletely contractive algebra endomorphisms. That is, there is a distinguished(unital) semigroup homomorphism p α p : P → End( A ) . We do not require the α p to be automorphisms. We say ( A, α, P ) is injective (resp.surjective, automorphic) if each α p is injective (resp. surjective, bijective). When A has an identity 1 A and each α p is unital, we call ( A, α, P ) a unital dynamicalsystem. If A is a C*-algebra, and hence each α p is an ∗ -endomorphism, then( A, α, P ) is a
C*-dynamical system . *-ENVELOPES OF SEMICROSSED PRODUCTS 5 Let G be an abelian group. A subsemigroup P ⊆ G is a positive cone if P ∩ ( − P ) = { } , and a spanning cone if in addition G = P − P . Any positivecone P ⊆ G induces a partial order on G by defining g ≤ h ⇐⇒ h − g ∈ P. This ordering respects the group operation +. A lattice ordered abelian group ( G, P ) consists of an abelian group G and a spanning cone P ⊆ G such that thepartial order ≤ induced by P on G makes G into a lattice. That is, for any g, h ∈ G ,the { g, h } has a least upper bound g ∨ h and a greatest lower bound g ∧ h . If ( G, P )is a lattice ordered abelian group, we also refer to P as a lattice ordered abeliansemigroup . Example 2.1.
The pair ( Z n , Z n + ) forms a lattice ordered abelian group. Here, adynamical system ( A, α, Z n + ) consists of a choice of n commuting completely con-tractive endomorphisms of A , which we usually just write as α , . . . , α n ∈ End( A ). Example 2.2.
Any totally ordered group (
G, P ) is automatically lattice ordered.For instance, ( Q , Q + ) and ( R , R + ) are both totally ordered groups. If P ⊆ Z n isthe set of elements larger than (0 , . . . ,
0) in the lexicographic ordering of Z n , then( Z n , P ) is totally ordered, and the induced ordering is lexicographic.A representation T : P → B ( H ) is a (unital) semigroup homomorphism, andwe usually write T ( p ) = T p . The representation T is contractive/isometric/unitarywhenever each T p is contractive/isometric/unitary. If ( G, P ) is a lattice orderedgroup, a contractive representation T : P → B ( H ) is Nica-covariant if whenever p, q ∈ P satisfy p ∧ q = 0, we have T p T ∗ q = T ∗ q T p , so T p and T q not only commute,but ∗ -commute [33]. If V : P → B ( H ) is an isometric representation, V is Nica-covariant if and only if V p V ∗ p V q V ∗ q = V p ∨ q V ∗ p ∨ q . That is, the range projections of the V p ’s give a lattice homomorphism P → proj( H ). A representation T of Z n + is Nica-covariant if and only if the genera-tors T , . . . , T n ∗ -commute, and in this case we can find a simultaneous dilationto isometries V , . . . , V n , which yield an isometric Nica-covariant representation V that dilates T [6, Theorem 2.5.10]. More generally, for any lattice ordered abeliansemigroup P , any contractive Nica-covariant representation T : P → B ( H ) has anisometric Nica-covariant co-extension [28].Let ( A, α, P ) be a dynamical system over an abelian semigroup P . A covariantpair ( π, T ) : ( A, P ) → B ( H )consists of a ∗ -homomorphism π : A → B ( H ), and a representation T : P → B ( H ),such that, if a ∈ A and p ∈ P ,(3) π ( a ) T p = T p π ( α p ( a )) . We say ( π, T ) is unitary, isometric, contractive, or Nica-covariant when T is. Let F be a sufficiently well behaved family of representations of P on Hilbert space (cf.[6, Definition 2.1.1]). For our purposes, F is always one of “un” (unitary representa-tions), “is” (isometric representations), “c” (contractive representations), or when P is a lattice ordered abelian semigroup, “nc” (Nica-covariant representations).The semicrossed product A × F α P ADAM HUMENIUK is an operator algebra defined by the following universal property [6, Section 3.1].There is a covariant pair ( i, v ) : (
A, P ) → A × F α P such that whenever ( π, T ) :( A, P ) → B ( H ) is a covariant pair with T ∈ F , there is a unique completelycontractive homomorphism(4) π × T : A × F α P → B ( H )with ( π × T ) ◦ i = π and ( π × T ) ◦ v = T . Concretely, A × F α P is densely spannedby formal monomials v p a , for p ∈ P and a ∈ A , which satisfy the relation( v p a ) · ( v q b ) = v p + q α q ( a ) b. Indeed, one can construct A × F α P by starting with the algebraic tensor product C [ P ] ⊙ A, defining a multiplication relation( δ p ⊗ a ) · ( δ q ⊗ b ) = δ p + q ⊗ ( α q ( a ) b ) , (Associativity here requires commutativity in P !) and completing in the universaloperator algebra norm defined by k X k := sup n(cid:13)(cid:13)(cid:13) ( π × T ) ( n ) ( X ) (cid:13)(cid:13)(cid:13) (cid:12)(cid:12)(cid:12) ( π, T ) is a covariant pair with T ∈ F o , for any X ∈ M n ( C [ P ] ⊙ A ). Here π × T : δ p ⊗ a T p π ( a ) defines a homomorphismon C [ P ] ⊙ A . When the action α is clear, we usually just write A × F P . We do notomit F , because it is standard that A × α P := A × c α P always denotes the contractive semicrossed product.We are primarily interested in the Nica-covariant semicrossed product A × nc P ,when ( G, P ) is a lattice ordered abelian group. By [6, Proposition 4.2.1] and [29], A × nc P is also universal for isometric Nica-covariant pairs, and these completelynorm the algebra A × nc P . In fact, there is a distinguished isometric Nica-covariantpair for any pair ( A, P ). Let π : A → B ( H ) be any completely isometric represen-tation. Define a pair (˜ π, V ) : ( A, P ) → B ( H ⊗ ℓ ( P )) by˜ π ( a )( x ⊗ δ p ) = α p ( a ) x ⊗ δ p , V q ( x ⊗ δ p ) = x ⊗ δ p + q . Then (˜ π, V ) is an isometric Nica-covariant pair, and we call ˜ π × V : A × nc P → B ( H )the Fock representation (induced by π ) for A × nc P . By [6, Theorem 4.2.9], anyFock representation is completely isometric. This is a key tool which makes it easyto prove A × nc P embeds completely isometrically into a crossed product.Let G be an abelian group and ( B, β, G ) a C*-dynamical system over G . In thispaper, we use the nonstandard convention that the crossed product B ⋊ β G is theuniversal C*-algebra generated by monomials u g a, g ∈ G, a ∈ B satisfying u g a · u h b = u g + h β h ( a ) b , or when B is unital, u ∗ g au g = β g ( a ) . Usually one takes the convention that u g au ∗ g = β g ( a ). This backwards conventionis only valid because G is abelian, so g u ∗ g defines a unitary representation of G .Clearly this construction is isomorphic to the usual crossed product, so we lose no *-ENVELOPES OF SEMICROSSED PRODUCTS 7 generality. What we gain is an alignment with the semicrossed covariance relations(3) and (4). Indeed B ⋊ β G ∼ = B × un β G ∼ = B × is β G is also a semicrossed product, and a C*-algebra with the obvious ∗ -structure. Aswith semicrossed products, we usually write B ⋊ G when the action β is clear.Generally, for any dynamical system ( A, α, P ), the semicrossed product A × F α P is a (non-selfadjoint) operator algebra, even when A is a C*-algebra. Let A beany operator algebra. A C*-cover ϕ : A → B for A consists of a C*-algebra B ,and a unital completely isometric homomorphism ϕ such that B = C ∗ ( ϕ ( A )). TheC*-envelope C ∗ e ( A ) is a co-universal or terminal C*-cover ι : A → C ∗ e ( A ). That is,whenever ϕ : A → B is a C*-cover, there is a ∗ -homomorphism π : B → C ∗ e ( A )such that A BC ∗ e ( A ) ϕι π commutes. The homomorphism π is necessarily unique and surjective. The C*-envelope exists and is unique up to a ∗ -homomorphism fixing A [34, Theorem15.16]. In fact, it can be produced from any C*-cover. If ϕ : A → B is a C*-cover,a boundary ideal I ⊳ B is an ideal such that the quotient map q : B → B/I iscompletely isometric on ϕ ( A ). There is a unique maximum boundary ideal in B for A called the Silov ideal , and qϕ : A → B → B/I is a C*-envelope for A [2].3. Main Results
Let (
G, P ) be a lattice ordered abelian group, and let (
A, α, P ) be a unitalC*-dynamical system over P . Our goal is to embed the Nica-covariant semicrossedproduct A × nc α P into a crossed product B ⋊ β G . Here, A should be a C*-subalgebraof B and the action β of G on A should extend or dilate the action α of P . Write B ⋊ β G = span { u g b | g ∈ G, b ∈ B } . We might hope to embed A × nc P in B ⋊ G via a map of the form ι × u , where ι : A → B is some unital ∗ -monomorphism that intertwines α and β . However, thisis impossible whenever any α p has kernel. Indeed, if a ∈ ker α p ⊆ A is nonzero,then we would require ι ( a ) = 0, but0 = ι ( α p ( a )) = u ∗ p ι ( a ) u p . This is impossible when u p is unitary.In the non-injective case, the best we can do is embed A × nc P into a corner of B ⋊ G . We do this by taking a nonunital embedding ι : A → B . Then, p A := ι (1 A )is a projection in B . Consequently, up A : p u p p A defines an isometric representation of P in the corner p A ( B ⋊ G ) p A . The followingdefinition is meant to capture a set of sufficient conditions for ( ι, up A ) to be a Nica-covariant covariant pair, and give an embedding ι × up A : A × nc P → p A ( B ⋊ G ) p A (Proposition 3.8). ADAM HUMENIUK
Definition 3.1.
Let (
G, P ) be a lattice ordered abelian group. Suppose (
A, α, P )is a C*-dynamical system over P . An automorphic dilation ( B, β, G ) is a C*-dynamical system (
B, β, G ) together with(1) a ∗ -monomorphism ι : A → B , such that(2) for all a, b ∈ A and p ∈ P , ι ( a ) β p ( ι ( b )) = ι ( aα p ( b )) . (And, by taking adjoints, β p ( ι ( A )) ι ( b ) = ι ( α p ( a ) b ).) Moreover, ( B, β, G ) is a
Nica-covariant automorphic dilation if in addition(3) for all a, b ∈ A and g, h ∈ G , we have β g ( ι ( a )) β h ( ι ( b )) = β g ∧ h ( ι ( α g − g ∧ h ( a ) α h − g ∧ h ( b ))) . We call an automorphic dilation (
B, β, G ) minimal if(4) ι ( A ) generates B as a G -C*-algebra, i.e. B = C ∗ [ g ∈ G β g ι ( A ) . We are primarily concerned with minimal Nica-covariant automorphic dilations,which satisfy all of (1)-(4). Note that if the automorphic dilation (
B, β, G ) is bothminimal and Nica-covariant then property (3) above implies that X g ∈ G β g ι ( A )is a ∗ -subalgebra, and hence B = X g ∈ G β g ι ( A ) . We are also mostly concerned with the unital case.
Remark 3.2.
Let (
G, P ) be a lattice ordered abelian group, and let (
A, α, P ) bea unital C*-dynamical system. Suppose (
B, β, G ) is an automorphic C*-dynamicalsystem, with a (possibly nonunital) ∗ -monomorphism ι : A → B . Setting p A := ι (1 A ), it is straightforward to check that properties (2) and (3) in Definition 3.1 areequivalent to:(2’) For all a ∈ A and p ∈ P , p A β p ( ι ( a )) = ι ( α p ( a )) (= β p ( ι ( a )) p A ) , and(3’) for all g, h ∈ G , β g ( p A ) β h ( p A ) = β g ∧ h ( p A ) , respectively.The reason we assign property (3) the name “Nica-covariant” is because in theunital case, the identity β g ( p A ) β h ( p A ) = β g ∨ h ( p A ) ensures that the isometric semi-group representation p u p p A ∈ p A ( B ⋊ G ) p A is Nica-covariant. Indeed,( u p p A )( u p p A ) ∗ = u p p A u ∗ p = β − p ( p A ) . So if (3) holds, β − p ( p A ) β − q ( p A ) = β ( − p ) ∧ ( − q ) ( p A ) = β − p ∨ q ( p A ) = ( u p ∨ q p A )( u p ∨ q p A ) ∗ . *-ENVELOPES OF SEMICROSSED PRODUCTS 9 In a minimal Nica-covariant automorphic dilation, the projections β p ( p A ) for p ∈ P are central and in fact form an approximate identity. Lemma 3.3.
Let ( G, P ) be a lattice ordered abelian group. Suppose ( A, α, P ) is aC*-dynamical system, with ( B, β, G ) a minimal Nica-covariant automorphic dila-tion. Considering P (a lattice) as a directed set, the net ( β p ( p A )) p ∈ P is an increasing approximate identity for B , consisting of central projections.Proof. Suppose p ≤ q in P . Then β p ( p A ) β q ( p A ) = β p ∧ q ( p A ) = β p ( p A ) . Therefore β p ( p A ) ≤ β q ( p A ), since both are projections. These projections arecentral, because for an element of the form β g ι ( a ), where a ∈ A and g ∈ G , we have β p ( p A ) β g ( ι ( a )) = β p ∧ g ι ( α g − p ∧ g ( a )) = β g ( ι ( a )) β p ( p A ) . Here, we have used property (3) in Definition 3.1 and the fact that each α p fixes1 A . Further, when p ≥ g , we have p ∧ g = g and the same computation shows β p ( p A ) β g ι ( a ) = β g ι ( a ) . Thus ( β p ( p A )) p ∈ P is an approximate identity for β g ι ( A ), which commutes with β g ι ( A ). Since the automorphic dilation ( B, β, G ) is minimal, B = X g ∈ G β g ι ( A )) . Thus each β p ( p A ) is central. Since the net ( β p ( p A )) p is norm-bounded, a standard ε/ B . (cid:3) The key observation is that any C*-dynamical system over a lattice orderedabelian semigroup admits a minimal Nica-covariant automorphic dilation. In fact,we can build one with an infinite product construction.
Definition 3.4.
Let (
G, P ) be a lattice ordered abelian group, and (
A, α, P ) a C*-dynamical system. We construct a minimal Nica-covariant automorphic dilation asfollows. Define the ∗ -monomorphism ι : A → Y G A by ι ( a ) g = ( α g ( a ) , g ∈ P, , g P. Throughout, [ x ] g or simply x g always denotes the g ’th element of a tuple x ∈ Q G A .Then, G acts on Q G A by the “left-shift” β : G → End( Q G A ), where[ β g ( x )] h = x h + g . Set B := C ∗ [ g ∈ G β g ι ( A ) = X g ∈ G β g ι ( A ) . Then, (
B, β, G ) is a minimal Nica-covariant automorphic dilation of (
A, α, P ), whichwe call the product dilation of (
A, α, P ). Note that (
B, β, G ) is an automorphic dilation because, for a, b ∈ A and p ∈ P ,[ ι ( a ) β p ι ( b )] g = [ ι ( aα p ( b ))] g = ( α g ( a ) α g + p ( b ) , g ≥ , , else.To see that the dilation is Nica-covariant, for p, q ∈ P we also have[ β p ι ( a ) β q ι ( b )] g = [ β p ∧ q ι ( α p − p ∧ q ( a ) α q − p ∧ q ( b ))] g = ( α g + p ( a ) α g + q ( b ) , g ≥ − p and g ≥ ( − q ) , , else,and g ≥ − p, − q if and only if g ≥ ( − p ) ∨ ( − q ) = − ( p ∧ q ). The product dilation isminimal because we’ve chosen B = P g ∈ G β g ι ( A ). Remark 3.5.
While finalizing this paper, the author was made aware that theproduct dilation defined here was defined first by Zahmatkesh for totally orderedabelian groups in [39], and for general lattice ordered abelian groups in [38]. InProposition 3.8, we prove that a full corner of the crossed product associated to theproduct dilation is a C*-cover of the semicrossed product A × nc P . Zahmatkeshproves in [38] that this same full corner is the universal C*-algebra associated toNica-Toeplitz covariant representations of ( A, α, P ). Example 3.6.
It is most instructive to consider the product dilation in the case(
G, P ) = ( Z , Z + ). For simplicity, also assume A is unital. Here, we embed A in Q Z A via ι ( a ) := ( . . . , , , a, α ( a ) , α ( a ) , . . . ) , the “ a ” occurring at index 0. Then, Z acts on Q Z A by the backwards bilateralshift β . This is an automorphic dilation, because p A = ( . . . , , , , , , . . . ) and p A βι ( a ) = ( . . . , , , α ( a ) , α ( a ) , . . . ) = ια ( a ) . Remark 3.7.
When A = C and P acts trivially, the product dilation ( B, β, G )is the C*-algebra B P that Laca and Raeburn define in [27, Section 2]. In termsof their notation, 1 = p A , and for p ∈ P , 1 p = β − p ( p A ). Nica-covariance of thedilation B P is seen in Equation (1.2) in [27].As promised, the Nica-covariant semicrossed product A × nc P embeds into thecrossed product of any Nica-covariant automorphic dilation. Proposition 3.8.
Let ( G, P ) be a lattice ordered abelian group. Let ( A, α, P ) bea unital C*-dynamical system. Suppose ( B, β, G ) is a Nica-covariant automorphicdilation of ( A, α, P ) , with ∗ -embedding ι : A → B . With p A = ι (1 A ) , there is acompletely isometric homomorphism ϕ = ι × up A : A × nc α P → B ⋊ β G, where ( up A ) p = u p p A . Moreover, if ( B, β, P ) is a minimal automorphic dilation,then C ∗ ( ϕ ( A × nc α P )) = p A ( B ⋊ β G ) p A is a full corner of B ⋊ β G .Proof. As shown after Remark 3.2, Nica-covariance of the dilation (
B, β, G ) ensuresthat up A : P → p A ( B ⋊ G ) p A is an isometric Nica-covariant representation of P . *-ENVELOPES OF SEMICROSSED PRODUCTS 11 Further, because p A = ι (1 A ), ι maps A into p A ( B ⋊ G ) p A . The pair ( ι, up A ) iscovariant, as for a ∈ A and p ∈ P , ι ( a ) u p p A = u p β p ι ( a ) p A = u p ι ( α p ( a )) = u p p A ι ( α p ( a )) . By the universal property, there exists a completely contractive homomorphism ϕ = ι × up A : A × nc P → p A ( B ⋊ G ) p A ⊆ B ⋊ G. In fact, ϕ is completely isometric. Fix any faithful nondegenerate representation π : B → B ( H ). As G is abelian, the left regular representation U × ˜ π : B ⋊ G → B ( H ⊗ ℓ ( G ))is faithful. Then H ⊗ ℓ ( P ) ⊆ H ⊗ ℓ ( G ) is a π ( B ) and U ( P )-invariant subspace.Let σ := π ◦ ι | H ⊗ ℓ ( P ) : A → B ( H ⊗ ℓ ( P )) , and V := U | H ⊗ ℓ ( P ) : P → B ( H ⊗ ℓ ( P )) . Then, it is immediate that ( σ, V ) is a Nica-covariant covariant pair for (
A, P ),and by definition, σ × V is the Fock representation of A × nc P on H ⊗ ℓ ( P ).By [6, Theorem 4.2.9], the Fock representation is completely isometric. Let κ : B ( H ⊗ ℓ ( G )) → B ( H ⊗ ℓ ( P )) be the compression map. The diagram A × nc P B ⋊ GB ( H ⊗ ℓ ( P )) B ( H ⊗ ℓ ( G )) ϕσ × V π × Uκ commutes. As the vertical maps are complete isometries, and κ is a completecontraction, it follows that ϕ is completely isometric, as claimed.Now, suppose ( B, β, G ) is minimal. We claim that the corner p A ( B ⋊ G ) p A is fulland generated by ϕ ( A × nc P ). This is a full corner, because p A ( B ⋊ G ) p A contains A ⊆ p A Bp A , and as A generates B as a G -C*-algebra, the ideal A generates in B ⋊ G is everything. By minimality, B = X g ∈ G β g ι ( A ) , so B is densely spanned by monomials x = u g β h ι ( a )) for a ∈ A and g, h ∈ G . Givensuch a monomial, as p A is central in B , p A xp A = p A u g p A β h ι ( a ) p A = p A u g p A u ∗ h ι ( a ) u h p A = ( u g − p A ) ∗ ( u g + p A )( u h p A ) ∗ ι ( a )( u h p A ) . Here, since P is a spanning cone we have written g = g + − g − , where g ± ∈ P .Thus, x ∈ C ∗ ( ι, up A ) and p A ( B ⋊ G ) p A ⊆ C ∗ ( ι, up A ). Conversely, since ( ι, up A )is a Nica-covariant isometric pair, by [6, Proposition 4.2.3], C ∗ ( ι, up A ) is denselyspanned by monomials y = ( u p p A ) ι ( a )( u q p A ) ∗ , for a ∈ A and p, q ∈ P . Given p ∈ P , we have p A u p p A = u p β p ( p A ) p A = u p β p ∧ ( p A ) = u p p A , and by takingadjoints p A u ∗ p = p A u ∗ p p A . Then, for such a monomial y , we find y = u p p A ι ( a ) p A u ∗ q = p A u p p A ι ( a ) p A u ∗ q p A = p A yp A . This proves C ∗ ( ι, up A ) = p A ( B ⋊ G ) p A , as desired. (cid:3) Proposition 3.8 asserts that p A ( B ⋊ G ) p A is a C*-cover of A × nc P . To findthe C*-envelope C ∗ e ( A × nc P ), it suffices to describe the Silov ideal. We will show(Theorem 3.12) that the Silov ideal arises as a corner of a crossed product I ⋊ G ,where I ⊳ B is some G -invariant ideal in B . Remark 3.9.
It is routine to check that if (
B, β, G ) is a C*-dynamical system,and
I ⊳ B is a β -invariant ideal, then ( B/I, ˙ β, G ) is also a C*-dynamical system.Here ˙ β g ( b + I ) := β g ( b ) + I is well defined, by invariance of I . The quotient map q : B → B/I is G -equivariant.Further, suppose ( G, P ) is a lattice ordered abelian group, and (
B, β, G ) is anautomorphic dilation of (
A, α, P ) with inclusion ι : A → B . If I ⊳ B is a β -invariant A -boundary ideal (meaning ι ( A ) ∩ I = 0), then ( B/I, ˙ β, G ) is also anautomorphic dilation of ( A, α, P ), because qι is faithful on A . Moreover, if ( B, β, G )is Nica-covariant or minimal, then so too is (
B/I, ˙ β, G ), which easily follows fromequivariance of q .The following lemma summarizes that under reasonable hypotheses we can “com-mute” taking quotients with either taking corners or crossed products. Lemma 3.10.
The following are true. (i)
Suppose C is a C*-algebra, and p ∈ C is a projection. Let J ⊳ pCp be anideal. If K = h J i C = CJC is the ideal J generates in C , then J = pKp .And, there is a canonical isomorphism pCpJ ∼ = ˙ p (cid:18) CK (cid:19) ˙ p, where ˙ p = p + K . (ii) Suppose ( B, β, G ) is an automorphic C*-dynamical system over an abeliangroup G . Let I ⊳ B be a G -invariant ideal. Then the natural map B ⋊ G → ( B/I ) ⋊ G induces an isomorphism B ⋊ β GI ⋊ β G ∼ = BI ⋊ ˙ β G. Proof. (i) Since J ⊆ K , certainly J = pJp ⊆ pKp . Conversely, for any term ajb ,for a, b ∈ C and j ∈ J ⊆ pAp , the product p ( ajb ) p = pa ( pjp ) bp = ( pap ) j ( pbp )lies in J , since J ⊳ pCp . Thus pKp = J . Restricting the quotient map C → C/K gives a ∗ -homomorphism with range ˙ p ( C/K ) ˙ p and kernel K ∩ pCp = pKp = J , sothe stated isomorphism follows.(ii) This follows because G is abelian, and hence an exact group [4, Theorem5.1.10]. (cid:3) Recall that when G is an abelian group, the compact dual group b G has a naturalgauge action γ on any crossed product B ⋊ G , which satisfies γ χ ( u g b ) = χ ( g ) u g b. Consequently, there is a faithful expectation E b G : B ⋊ G → B ⋊ G *-ENVELOPES OF SEMICROSSED PRODUCTS 13 with range B , defined by the formula E b G ( x ) = Z b G γ χ ( x ) dχ. Here dχ denotes integration against Haar measure. Lemma 3.11.
Suppose ( B, β, G ) is an automorphic C*-dynamical system over anabelian group G . Let J ⊳ B ⋊ β G be an ideal. Then J is invariant under the gaugeaction of b G if and only if J = I ⋊ β G , where I = J ∩ B ⊳ B is a β -invariant idealin B .Proof. Since b G acts diagonally on the spanning monomials u g b in B ⋊ G , any idealof the form I ⋊ G is b G -invariant. Conversely, let J ⊳ B be b G -invariant. Then I := J ∩ B ⊳ B is a G -invariant ideal, since the action β is implemented by unitariesin B ⋊ G . Then, I ⊆ J implies I ⋊ G ⊆ J .Conversely, as in Lemma 3.10.(ii), there is a canonical onto ∗ -homomorphism π : B ⋊ G → ( B/I ) ⋊ G with kernel I ⋊ G . Given x ∈ J , because J is closedand b G -invariant E b G ( x ∗ x ) ∈ J ∩ B = I and hence π ( E b G ( x ∗ x )) = 0. Since π is b G -equivariant, we find 0 = π ( E b G ( x ∗ x )) = E b G ( π ( x ∗ x )) . As the expectation E b G is faithful, π ( x ) = 0 and x ∈ ker π = I ⋊ G . Therefore J = I ⋊ G . (cid:3) We can now identify the Silov ideal in C ∗ ( ϕ ( A × nc P )) = p A ( B ⋊ G ) p A , for anyminimal Nica-covariant automorphic dilation ( B, β, G ). Theorem 3.12.
Let ( G, P ) be a lattice ordered abelian group, and let ( A, α, P ) be a unital C*-dynamical system over P . Suppose ( B, β, G ) is any minimal Nica-covariant automorphic dilation of ( A, α, P ) , with ∗ -embedding ι : A → B . Then,there is a unique maximum β -invariant ι ( A ) -boundary ideal I ⊳ B . Further, if p A = ι (1 A ) ∈ B and ϕ = ι × up A : A × nc α P → B ⋊ β G is the completely isometricembedding from Proposition 3.8, then p A ( I ⋊ β G ) p A ⊳ p A ( B ⋊ β G ) p A = C ∗ ( ϕ ( A × nc α P )) is the Silov ideal for A × nc α P . Consequently C ∗ e ( A × nc α P ) ∼ = ˙ p A (cid:18) BI ⋊ ˙ β G (cid:19) ˙ p A is a full corner of a crossed product.Proof. Let ϕ = ι × up A : A × nc P → B ⋊ G be the C*-cover from Proposition3.8. Let J ⊳ p A ( B ⋊ G ) p A be the Silov ideal for A × nc P . Since ϕ ( A × nc P ) =span { u p ι ( a ) | p ∈ P, a ∈ A } is invariant under the gauge action of b G , it follows that J is also b G -invariant. Let K = ( B ⋊ G ) J ( B ⋊ G ) be the ideal J generates in theentire crossed product B ⋊ G . Since J is b G -invariant, so too is K . By Lemma 3.11,we have K = I ⋊ G for some β -invariant I ⊳ B . And, by Lemma 3.10.(i), J = p A Kp A = p A ( I ⋊ G ) p A . Because ι ( A ) ⊆ p A ( B ⋊ G ) p A , we also find I ∩ ι ( A ) = K ∩ ι ( A ) = p A ( K ∩ ι ( A )) p A = J ∩ ι ( A ) = 0 , since J does not intersect ϕ ( A × nc P ) ⊇ ι ( A ). Therefore I is a β -invariant boundaryideal. By Lemma 3.10, we have a canonical isomorphism C ∗ e ( A × nc P ) ∼ = p A ( B ⋊ G ) p A p A ( I ⋊ G ) p A ∼ = ˙ p A (cid:18) BI ⋊ ˙ β G (cid:19) ˙ p A . To see that I is maximum, suppose that R ⊳ B is any β -invariant ι ( A )-boundaryideal. Then p A ( R ⋊ G ) p A ⊳ p A ( B ⋊ G ) p A . By Lemma 3.10 again, p A ( B ⋊ G ) p A p A ( R ⋊ G ) p A ∼ = ( p A + R ) (cid:18) BR ⋊ G (cid:19) ( p A + R ) . Then by Remark 3.9, (
B/R, ˙ β, G ) is a minimal Nica covariant automorphic dilation.By Proposition 3.8, ( p A + R )(( B/R ) ⋊ G )( p A + R ) is a C*-cover for A × nc P . Bydefinition of the C*-envelope, there is an onto ∗ -homomorphism p A ( B ⋊ G ) p A p A ( R ⋊ G ) p A ∼ = ( p A + R ) (cid:18) BR ⋊ G (cid:19) ( p A + R ) → C ∗ e ( A × nc P ) ∼ = p A ( B ⋊ G ) p A p A ( I ⋊ G ) p A , which fixes A × nc P . It follows that p A ( R ⋊ G ) p A ⊆ p A ( I ⋊ G ) p A . Upon intersectingwith B , in which p A is central, we find p A R ⊆ p A I . Since R and I are β -invariant,and ( β g ( p A )) g ∈ G is an approximate identity in B , by Lemma 3.3, it follows that R ⊆ I . Indeed, for x ∈ R , xβ g ( p A ) = β g ( β − g ( x ) p A )lies in β g ( Rp A ) ⊆ β g ( I ) ⊆ I , and converges as a net indexed by g ∈ G to x ∈ R . (cid:3) Corollary 3.13.
Suppose ( A, α, P ) is a unital C*-dynamical system over a latticeordered abelian semigroup P . If ( B, β, G ) is a minimal Nica-covariant automorphicdilation of ( A, α, P ) , then the C*-cover ϕ : A × nc α P → p A ( B ⋊ β G ) p A is a C*-envelope if and only if B contains no nontrivial β -invariant boundary idealsfor A . Corollary 3.14.
Let ( G, P ) be a lattice ordered abelian group. The C*-envelope C ∗ e ( A × nc α P ) is a full corner of a crossed product of a minimal Nica-covariantautomorphic extension of ( A, α, P ) .Proof. To apply Theorem 3.12, it is enough to note that (
A, α, P ) has at least oneminimal Nica-covariant automorphic dilation. The product dilation (
B, β, G ) fromDefinition 3.4 suffices. Then C ∗ e ( A × nc α P ) ∼ = ˙ p A (cid:18) BI ⋊ G (cid:19) ˙ p A , and by Remark 3.9, ( B/I, ˙ β, G ) is itself a minimal Nica-covariant automorphicdilation. (cid:3) When (
A, α, P ) is an injective C*-dynamical system, we recover a known re-sult that the C*-envelope of A × nc P is a crossed product of a certain minimalautomorphic extension of A . Proposition 3.15. [6, Theorem 4.2.12]
Let ( G, P ) be a lattice ordered abeliangroup, and ( A, α, P ) an injective C*-dynamical system. Then C ∗ e ( A × nc α P ) ∼ = ˜ A ⋊ ˜ α G, *-ENVELOPES OF SEMICROSSED PRODUCTS 15 where ( ˜ A, ˜ α, G ) is an automorphic C*-dynamical system (unique up to equivariant ∗ -isomorphism) satisfying A ⊆ ˜ A and ˜ α p | A = α p for p ∈ P .Proof. Let (
B, β, G ) be the product dilation for (
A, α, P ). Then B ⊆ Q G A . Let c ( G, A ) := { x ∈ Y G A | lim g k x g k = 0 } ⊳ Y G A. Here, by writing “lim g ∈ G ”, we are considering G as a directed set in its orderinginduced by P , and thinking of G -tuples as nets. We will show that I := B ∩ c ( G, A ) ⊂ B is the maximum G -invariant boundary ideal in B . It is easy to check it is a β -invariant ideal. Because the action α is injective, each α p is isometric. So, if a ∈ ι − ( I ), we have0 = lim g ∈ G k ι ( a ) g k = lim p ∈ P k α p ( a ) k = lim p ∈ P k a k = k a k , hence a = 0. Note that the second equality holds because P is a cofinal subset of G . This proves ι ( A ) ∩ I = 0, so I is a G -invariant boundary ideal.Suppose J ⊳ B is any other β -invariant boundary ideal. Let x ∈ J ⊆ Q G A . Let ε >
0. Because B is a minimal dilation, we can choose an element of the form y = X g ∈ F β − g ι ( a g ) , where F ⊆ G is finite, and a g ∈ A , and k y − x k < ε . Since J is a β -invariantideal, p A β ∨ F ( x ) is in J . However, since ( B, β, G ) is a Nica-covariant automorphicdilation, p A β ∨ F ( y ) = X g ∈ F p A β ∨ F − g ι ( a g )= X g ∈ F ια ∨ F − g ( a g )= ι X g ∈ F α ∨ F − g ( a g ) is in ι ( A ). Since J is an ι ( A )-boundary ideal, the projection A → B → B/J isinjective, and so isometric. Therefore k p A β ∨ F ( y ) k = k p A β ∨ F ( y ) + J k≤ k p A β ∨ F ( y ) − p A β ∨ F ( x ) k≤ k y − x k < ε. Since [ p A β ∨ F y ] p = y p + ∨ F for p ∈ P , it follows that g ≥ ∨ F implies k y g k < ε , andalso k x g k ≤ k y g k + k x − y k < ε . This proves that x ∈ c ( G, A ), so J ⊆ I , and I isthe maximum β -invariant boundary ideal in B .By Theorem 3.12, C ∗ e ( A × nc α P ) ∼ = ( p A + I ) (cid:18) BI ⋊ ˙ β G (cid:19) ( p A + I ) . However, p A = 1 Q A modulo c ( G, A ), because p ≥ p A ] g = 1. It followsthat p A + I is a two-sided identity 1 B/I , and the C*-envelope is just the crossed product (
B/I ) ⋊ ˙ β G . By Remark 3.9, Nica-covariance of the dilation ( B/I, ˙ β, G ),with unital embedding η = q I ι : A → B → B/I , implies that, for p ∈ P and a ∈ A , β p η ( a ) = ˙ p A β p ( η ( a )) = η ( α p ( a )) . So, ˙ β p η = ηα p , which when we identify A ∼ = η ( A ) ⊆ B/I , reads ˙ β p | A = α p . Sincethe automorphic dilation ( B/I, ˙ β, G ) is minimal, it also follows easily that BI = [ p ∈ P ˙ β − p η ( A ) . Thus (
B/I, ˙ β, G ) is a minimal automorphic extension of ( A, α, P ). Such an exten-sion is unique up to an equivariant isomorphism fixing A , since if˜ A = [ p ∈ P ˜ α − p ( A ) ⊇ A, with G -action ˜ α extending α , the map ˙ β − p η ( a ) ˜ α − p ( a ) extends to an equivariant ∗ -isomorphism B/I ∼ = ˜ A . (cid:3) In the proof of Proposition 3.15, we showed the maximum β -invariant boundaryideal was B ∩ c ( G, A ). In the case (
G, P ) = ( Z , Z + ), this result generalizes readilyto the non-injective case.Recall that if A is a C*-algebra and I ⊳ A is an ideal, then I ⊥ := { a ∈ A | b ∈ I = ⇒ ab = 0 } ⊆ A is also an ideal, and satisfies I ∩ I ⊥ = 0. If π : A → B is a ∗ -homomorphism and a ∈ (ker π ) ⊥ , then k π ( a ) k = k a + ker π k = k a + (ker π ) ⊥ ∩ ker π k = k a k . This shows π | (ker π ) ⊥ is always isometric. Proposition 3.16.
Let ( A, α, Z + ) be a unital C*-dynamical system over Z + , andlet ( B, β, Z ) be its product dilation. The unique maximum β -invariant boundaryideal for ι ( A ) in B is I = B ∩ c ( Z , (ker α ) ⊥ ) . Consequently, C ∗ e ( A × nc α P ) ∼ = ˙ p A (cid:18) BI ⋊ ˙ β G (cid:19) ˙ p A . Proof.
Because (ker α ) ⊥ is an ideal in A , it follows easily that I is a β -invariantideal in B . Suppose a ∈ A with ι ( a ) ∈ I ⊆ c ( Z , (ker α ) ⊥ ). Then, each α n ( a ) ∈ (ker α ) ⊥ . Because α is isometric on (ker α ) ⊥ , one sees that k α n ( a ) k = k a k by aneasy induction on n , and so0 = lim n →∞ k ι ( a ) n k = lim n →∞ k α n ( a ) k = lim n →∞ k a k = k a k . Thus ι ( A ) ∩ I = 0.Suppose J ⊳ B is any β -invariant boundary ideal for A . The same argument asin the proof of Proposition 3.15 shows that all tuples in J vanish at + ∞ . So, itsuffices to let x ∈ J and prove each x g ∈ (ker α ) ⊥ . If b ∈ ker α , then ι ( b ) = ( . . . , , , b, , , . . . ) . *-ENVELOPES OF SEMICROSSED PRODUCTS 17 So, β g ( x ) ι ( b ) = ( . . . , , , x g b, , . . . ) = ι ( x g b ) ∈ ι ( A ) ∩ J = 0 . Since ι is injective, x g b = 0, and this proves each x g ∈ (ker α ) ⊥ . So, J ⊆ I .Therefore I is the maximum G -invariant boundary ideal in B , and Theorem 3.12applies. (cid:3) Example 3.17.
Proposition 3.16 does not generalize so readily to the case P = Z n + .Consider the unital C*-dynamical system ( A, α, Z ), where A = C and the actionis determined by generators by α ( a, b, c ) = ( a, c, c ) ,α ( a, b, c ) = ( c, b, c ) . (This is the unitization of the nonunital system ( C ⊕ C , α , Z ), where α ( a, b ) =( a,
0) and α ( a, b ) = (0 , b ).) Reviewing Proposition 3.16, we might expect B ∩ c ( Z , R ⊥ α ) = { b ∈ B | b g ∈ R ⊥ α and lim g ∈ Z b g = 0 } , for R α = ker α ∩ ker α , to be the maximum invariant boundary ideal. However,this fails to even be a boundary ideal, since here R α = 0, and for any element x = ( a, b, ∈ A , the tuple ι ( x ) = . . . 0 0 0 0 · · · a, b,
0) (0 , b,
0) (0 , b, · · · a, ,
0) 0 0 · · · a, ,
0) 0 . . .... ... ... lies in A ∩ c ( Z , R ⊥ α ) = ι ( A ) ∩ c ( Z , A ). A correct description of the Silov ideal inthe case P = Z n + is more complicated, and follows as described in Section 4. SeeSection 5 for more discussion in the case P = Z n + .4. Explicit Computation of the Silov Ideal
Throughout this section, let (
G, P ) be a lattice ordered abelian group, and let(
A, α, P ) be a unital C*-dynamical system. And, let (
B, β, G ) be the productdilation for (
A, α, P ), with inclusion ι : A → B ⊆ Q G A , as in Definition 3.4. ByTheorem 3.12, B contains a unique maximum ideal I which is both β -invariant andan A -boundary ideal (does not intersect ι ( A )). In this section, we will explicitlydescribe I . The following construction of I was inspired both by the construction in[6, Section 4.3], and the construction of Sehnem’s covariance algebra in [36, Section3.1]. Definition 4.1.
Define the following ideals.(1) Given a finite subset F ⊆ G , let K F := \ g ∈ Fg ker α g ∨ ⊳ A be the ideal of elements vanishing under the action of any strictly positivepart of an element in F . (2) For F ⊆ G finite, let J F := K ⊥ F ⊳ A be the annihilator of K F .(3) For F ⊆ G finite, define I F := { b ∈ B | b g ∈ J F − g for all g ∈ G } ⊳ B. (4) Finally, set I := [ F ⊆ G finite I F ⊳ B. It is straightforward to check that if F ⊆ F ′ are finite subsets of G , then K F ⊇ K F ′ , and hence J F ⊆ J F ′ . Consequently I F ⊆ I F ′ , so { I F | F ⊆ G finite } is adirected system of ideals, and so I is indeed an ideal in B . Further, it’s just asstraightforward to show that for any g ∈ G , and any finite F ⊆ G , that β g ( I F ) = I F − g . It follows that I = S F I F is a β -invariant ideal. Theorem 4.2.
The ideal
I ⊳ B from Definition 4.1 is the unique maximum β -invariant boundary ideal in the product dilation ( B, β, G ) . Consequently, C ∗ e ( A × nc α P ) ∼ = ˙ p A (cid:18) BI ⋊ ˙ β G (cid:19) ˙ p A is a full corner of a crossed product. For clarity, we break the proof of Theorem 4.2 into lemmas. Our first lemma isa verification that I is indeed a β -invariant boundary ideal. Lemma 4.3.
The ideal I satisfies ι ( A ) ∩ I = 0 .Proof. Since I = S F I F is an inductive union of ideals, it suffices to prove I F ∩ ι ( A ) = 0 for every finite F ⊆ G . Suppose for a contradiction that there is somefinite F ⊆ G , and some nonzero a ∈ A \ { } with ι ( a ) ∈ I F . By definition of I ,0 = a = ι ( a ) ∈ J F = K ⊥ F = (cid:18) \ g ∈ F g ker α g ∨ (cid:19) ⊥ . Since a = 0 and K F ∩ K ⊥ F = 0, a is not in K F . So, there is a g ∈ F with g ∨ > α g ∨ ( a ) = 0.Set a = α g ∨ ( a ) = 0. Since ι ( a ) ∈ I F , it will follow that ι ( a ) = ια g ∨ ( a )lies in I F , where F := { h − g ∨ | h ∈ F , h g } ⊂ F − g ∨ . Because g − g ∨ F − g ∨ \ F , we also have | F | < | F | strictly. Butthen, because a = 0 and ι ( a ) ∈ I F , we may repeat the same argument to finda nonzero a ∈ A and an F ⊆ G , with ι ( a ) ∈ I F , and | F | < | F | . Continuingrecursively, we find an infinite sequence | F | > | F | > | F | > · · · of finite subsets of G , and each I F n ∩ ι ( A ) = 0. This is absurd, since eventuallysuch a sequence must terminate at ∅ , and I ∅ = 0. This proves I F ∩ ι ( A ) = 0. *-ENVELOPES OF SEMICROSSED PRODUCTS 19 To prove that ι ( a ) ∈ I F as needed in the paragraph above, it suffices to notethat for any p ∈ P , that K F − g ∨ − p = \ h ∈ F h − g ∨ − p ker α ( h − g ∨ − p ) ∨ ⊇ \ k ∈ F k − p ker α ( k − p ) ∨ = K F − p . This is because if h ∈ F with h − g ∨ − p
0, then0 < ( h − g ∨ − p ) ∨ ≤ ( h − g ∨ ∨ h ∨ g ∨ − g ∨ . Therefore h ∨ g = g and h g , so in fact h − g ∨ ∈ F . Knowing this, for any p ∈ P , we have ι ( a ) p = α g ∨ p ( a ) ∈ I F − g ∨ − p ⊆ I F − p , proving ι ( a ) ∈ I F . (cid:3) To prove Theorem 4.2, it will be very helpful to identify B as a direct limit overcertain finite subsets of G . Definition 4.4. [6, Section 4.2] Let (
G, P ) be a lattice ordered abelian group. Asubset F ⊆ G is a grid if F is finite and closed under ∨ .Since any finite subset F of G is contained in a grid, found by appending alljoins of finite subsets of F , the set of all grids in G is directed under inclusion and G = S { F ⊆ G grid } . Lemma 4.5.
The product dilation B is an internal direct limit B = [ F ⊆ G grid B F of C*-subalgebras B F := X g ∈ F β − g ι ( A ) . Proof.
In fact, for any minimal Nica-covariant automorphic dilation (
B, β, G ) (Def-inition 3.1), we have B = X g ∈ G β g ι ( A ) = [ F ⊆ G grid B F . And, B F are always ∗ -subalgebras, because all maps involved are ∗ -linear, and themultiplication formula β − g ι ( a ) β − h ι ( b ) = β − ( g ∨ h ) ι ( α g ∨ h − g ( a ) α h − g ∨ h ( b )) , for g, h ∈ G and a, b ∈ A , implies B F is multiplicatively closed when F is ∨ -closed.So, we need only show each B F is norm closed, and this is where we use theconstruction of the product dilation. We will use induction on | F | . Certainly B ∅ = 0 is closed. Fix a nonempty grid F ⊆ G , and suppose whenever F ′ ⊆ G is a grid with | F ′ | < | F | , that B F ′ ⊆ B is closed. Choose a convergent sequence x n ∈ B F , and write x n = X g ∈ F β − g ι ( a gn ) , a gn ∈ A. Since F is finite, F contains a minimal element g . By minimality of g , we have[ x n ] g = a g n . Then, k a g n − a g m k ≤ k x n − x m k , so the sequence a g n is Cauchy, and has a limit a g ∈ A . Then, y n := x n − β − g ι ( a g n ) = X g ∈ F \{ g } β − g ι ( a gn )is a Cauchy sequence in B F \{ g } . As F \ { g } is a grid of smaller size then F , y n has a limit y ∈ B F \{ g } ⊆ B F . But then x n = β − g ι ( a g n ) + y n converges to β − g ι ( a g ) + y ∈ B F . So, B F is closed, finishing the induction. (cid:3) The next lemma offloads a technical step in the proof of Theorem 4.2. The pointis that, when F ⊆ G is any grid, and a ∈ A , the entries of the tuple ι ( a ) ∈ B ⊆ Q G A are realized by an element of B F for “large enough” g ∈ G . Lemma 4.6.
Let F ⊆ G be a grid. Then there are integers c g ∈ Z , such thatwhenever a ∈ A , and h ≥ g for at least one element g ∈ F , we have ι ( a ) h = α h ( a ) = X g ∈ F c g · β − g ια g ( a ) h . Proof.
It will be enough to find integers c g such that, for any g ∈ F , c g = 1 − X h ∈ Fh From Lemma 4.3, we already know I is a β -invariant ι ( A )-boundary ideal. So, it remains to prove I is maximum among such ideals. Suppose R ⊳ B is the maximum G -invariant boundary ideal for A , from Theorem 3.12. Then I ⊆ R , but we wish to prove I = R . Since B = S { B F | F grid } is a directlimit (Lemma 4.5), and ideals in a C*-algebra are inductive, R ⊆ I if and only if R ∩ B F ⊆ I ∩ B F for every grid F ⊆ G .We will prove R ∩ B F ⊆ I ∩ B F for every grid F by induction on | F | . This isimmediate when | F | = 0, since B ∅ = 0. Suppose now that | F | > F ′ isany grid with | F ′ | < | F | , then R ∩ B F ′ ⊆ I ∩ B F ′ . Choose any element x = X g ∈ F β − g ι ( a g ) ∈ R ∩ B F . Pick a minimal element g ∈ F . In fact, since R is β -invariant we are free totranslate so that g = 0 is minimal in F . By Lemma 4.6 applied to the grid F \ { } ,we can find an element y = X g ∈ F \{ } c g · β − g ια g ( a )such that if h ≥ g for any g ∈ F \ { } , then y h = ι ( a ) h . Writing z = y + X g ∈ F \{ } β − g ι ( a g )= X g ∈ F \{ } β − g ι ( a g + c g α g ( a )) ∈ B F \{ } , we find that x h = z h , whenever h dominates a nonzero element of F . Otherwise, if h g for all g ∈ F \ { } , we have x h = ι ( a ) h and z h = 0.We will show that x − z lies in I F ⊆ I ⊆ R . We have(5) [ x − z ] h = ( α h ( a ) , h ≥ h g for all g ∈ F \ { } , , else . So, suppose p ∈ P , with p g for all g ∈ F \ { } . Let b ∈ K F − p = \ g ∈ F \{ } ker α ( g − p ) ∨ . Then it follows from (5) that β p ( x − z ) ι ( b ) = β p ( x ) ι ( b ) = ι ( α p ( a ) b ) , which, since x ∈ R , and R is β -invariant, lies in ι ( A ) ∩ R = 0. Since ι is injective, α p ( a ) b = 0. This proves α p ( a ) = [ x − z ] p ∈ J F − p = K ⊥ F − p , so indeed x − z ∈ I F ⊆ I ⊆ R .As x and x − z are in R , z = x − ( x − z ) ∈ R ∩ B F \{ } . By inductive hypothesis, since | F \ { }| < | F | , we conclude z ∈ I ∩ B F \{ } . Since x − z ∈ I , we find x = z + ( x − z ) lies in I ∩ B F , completing the induction. (cid:3) Recall that an ideal I in a C*-algebra A is essential if it intersects every nonzeroideal of A , or equivalently if I ⊥ = 0. Corollary 4.7. Let ( G, P ) be a lattice ordered abelian group. Let ( A, α, P ) be aunital C*-dynamical system, with product dilation ( B, β, G ) . Then the C*-cover p A ( B ⋊ β G ) p A is the C*-envelope of A × nc α P if and only if for every finite subset F ⊆ P \ { } , K F = \ p ∈ F ker α p is an essential ideal in A .Proof. With I ⊳ B as in Theorem 4.2, the product dilation yields the C*-envelope ifand only if I = 0. But by construction, this occurs if and only if each I F = 0, whichoccurs if and only if each K ⊥ F = 0 for any finite subset F ⊆ G , or equivalently anyfinite F ⊆ P . (cid:3) The Case P = Z n + . In [6, Theorem 4.3.7], Davidson, Fuller, and Kakariadis identify the C*-envelopeof a semicrossed product A × nc α Z n + by Z n + as a full corner of a crossed productby Z n , when ( A, α, Z + n ) is a C*-dynamical system. In this section, we show thatthe C*-dynamical system ( B/I, ˙ β, Z n ) from Theorem 4.2 (in the case ( G, P ) =( Z n , Z n + )) is Z n -equivariantly ∗ -isomorphic to the C*-dynamical system constructedin [6, Section 4.3]. It follows that the latter system is a minimal automorphic Nica-covariant dilation of ( A, α, Z n + ) without nontrivial Z n -invariant boundary ideals,and we recover [6, Theorem 4.3.7] from Corollary 3.13.We now recall the construction in [6, Section 4.3]. Since our notation clasheswith the notation in that paper, we must introduce new notation. We write thestandard generators in Z + n as , . . . , n . Given x = ( x , . . . , x k ) ∈ Z n + ,supp( x ) := { k ∈ { , . . . , n } | x k > } . If x, y ∈ Z n + , we write x ⊥ y if x ∧ y = 0, or equivalently supp( x ) ∩ supp( y ) = ∅ .Moreover, let x ⊥ := { y ∈ Z n + | y ⊥ x } . Let ( A, α, Z + n ) be a C*-dynamical system.For x ∈ Z + n , define ideals Q x := \ i ∈ supp( x ) ker α i ⊥ ⊳ A, *-ENVELOPES OF SEMICROSSED PRODUCTS 23 and Q x := \ y ∈ x ⊥ α − y ( Q x ) ⊆ Q x . Form the C*-algebra C := M x ∈ Z n + AQ x . Let q x : A → A/Q x be the quotient map. Since Q = 0, η := q is a ∗ -monomorphism A ∼ = A/ → C . For convenience, we notationally identify C = X x ∈ Z n + AQ x ⊗ e x , where e x are formal generators, as in [6, Section 4.3]. Then ( C, γ, Z n + ) is an injectiveC*-dynamical system, where the action γ is determined on generators by γ i ( q x ( a ) ⊗ e x ) = ( q x ( α i ( a )) ⊗ e x + q x + i ( a ) ⊗ e x + i , i ⊥ x,q x + i ( a ) ⊗ e x + i , i ∈ supp( x ) . Since γ i ( q ( a ) ⊗ e ) = q ( α i ( a ) ⊗ e + q i ( a ) ⊗ e i has 0’th entry α i ( a ), the system( C, γ, Z n + ) dilates ( A, α, Z n + ) in the same sense as Definition 3.1.Let ( ˜ C, ˜ γ, Z n ) be the minimal automorphic extension of ( C, γ, Z n + ), from [6, The-orem 4.2.12]. This C*-dynamical system satisfies C ⊆ ˜ C, ˜ γ | C = γ and C = [ x ∈ Z n + ˜ γ − x ( C ) . Then, ( ˜ C, ˜ γ, Z n ) is a minimal Nica-covariant automorphic dilation of ( A, α, Z n + ).Nica-covariance of this dilation is found in [6, Lemma 4.3.8]. The content of [6,Theorem 4.3.7] is that the natural map A × nc α Z n + → ˜ C ⋊ ˜ γ Z n is completely isometric,and via this map C ∗ e ( A × nc Z n + ) = p ( ˜ C ⋊ Z n ) p is a full corner by the projection p = 1 A ⊗ e = η (1 A ). Proposition 5.1. Let ( A, α, Z n + ) be a unital C*-dynamical system. Let ( B, β, Z n ) be the product dilation (Definition 3.4), and ( ˜ C, ˜ γ, Z n ) be the automorphic dilationdefined above. Let I ⊳ B be the unique maximum β -invariant boundary ideal as inTheorem 4.2. Then there is a Z n -equivariant ∗ -isomorphism B/I ∼ = ˜ C that fixes A .Proof. Define π : X x ∈ Z n β x ι ( A ) → ˜ C to be the unique ∗ -linear map satisfying π ( β x ι ( a )) = ˜ γ x η ( a ) . Note that π is well defined, because if F ⊆ G is finite, and a x ∈ A satisfy b := X x ∈ F β x ι ( a x ) = 0 , then each a x = 0, so such a representation is well defined. Indeed, if x ∈ F is minimal, then b x = a x = 0. By replacing F with F \ { x } and recursing,we eventually find each a x = 0. Since both ( B, β, Z n ) and ( ˜ C, ˜ γ, Z n ) are minimal automorphic dilations, π is a ∗ -linear map defined on a dense subalgebra with denserange. By construction π is Z n -equivariant. Nica-covariance of both dilations implythat, if x, y ∈ Z n and a, b ∈ G , β x ι ( a ) β y ι ( b ) = β x ∧ y ι ( α x − x ∧ y ( a ) α y − x ∧ y ( b )) , and identically ˜ γ x η ( a )˜ γ y η ( b ) = ˜ γ x ∧ y η ( α x − x ∧ y ( a ) α y − x ∧ y ( b )) . Extending linearly, it follows that π is a ∗ -homomorphism.We claim π is bounded. Given an element b = P x ∈ F β g ι ( a x ) ∈ P g β g ι ( A ) asabove, using [6, Lemma 4.3.6], we find k π ( b ) k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X x ∈ F X ≤ y ≤ x q y ( α x − y ( a x )) ⊗ e y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ sup y ∈ Z n + (cid:13)(cid:13)(cid:13) q y (cid:16) X x ∈ Fx ≥ y α x − y ( a x ) (cid:17)(cid:13)(cid:13)(cid:13) ≤ sup y ∈ Z n (cid:13)(cid:13)(cid:13) X x ∈ Fx ≥ y α x − y ( a x ) (cid:13)(cid:13)(cid:13) , or upon swapping y with − y , k π ( b ) k ≤ sup y ∈ Z n (cid:13)(cid:13)(cid:13) X x ∈ Fx + y ≥ α x + y ( a x ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X x ∈ F β x ι ( a x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k b k . So, π is contractive. Therefore, π extends uniquely to an equivariant surjective ∗ -homomorphism B → ˜ C , which we denote by the same symbol. Note also that πι = η , so π fixes the respective copies of A .The result follows if we can prove ker π = I . Since π is equivariant and isometricon ι ( A ) ∼ = η ( A ) ∼ = A , ker π is a β -invariant boundary ideal and so ker π ⊆ I , bymaximality of I .To show I ⊆ ker π , by inductivity of ideals it suffices to prove B F ∩ I H ⊆ ker π, for any grid F ⊆ Z n (Lemma 4.5) and any finite subset H ⊆ Z n . So, it suffices toassume we have an element b = X x ∈ F β x ι ( a x ) ∈ I H , where F ⊆ G is finite, and prove π ( b ) = 0. In fact, since π is Z n -equivariant, and β g ( I H ) = I H − g , we are free to apply β g for any g ≥ ( − ∧ F ) ∨ ( ∨ H ) and so assume *-ENVELOPES OF SEMICROSSED PRODUCTS 25 F ⊆ Z n + and H ⊆ − Z n + . Now, compute π ( b ) = X x ∈ F γ x η ( a x )= X y ∈ Z n + q y (cid:16) X x ∈ Fx ≥ y α x − y ( a x ) (cid:17) ⊗ e y = X y ∈ Z n + q y ( b − y ) ⊗ e y . So, we must show b − y ∈ Q y for all y ∈ Z n + . Suppose that z ∈ Z n + with z ⊥ y . Then b z − y = X x ∈ Fx + z ≥ y α x + z − y ( a x )= X x ∈ Fx ≥ y α x + z − y ( a x ) = α z ( b − y ) , since z ⊥ y implies that x + z ≥ y if and only if x ≥ y . Because b ∈ I H , we have α z ( b − y ) = b z − y ∈ J H − z + y = (cid:18) \ h ∈ Hh z − y ker α ( h − z + y ) ∨ (cid:19) ⊥ . But, we also have \ i ∈ supp( y ) ker α i ⊆ \ h ∈ Hh z − y ker α ( h − z + y ) ∨ . Indeed, if h ∈ H with h − z + y 0, then since h, − z ≤ ∅ 6 = supp(( h − z + y ) ∨ ⊆ supp( y ) , so ker α ( h − z + y ) ∨ ⊇ ker α i for at least one i ∈ supp( y ). Upon taking annihilators,which reverses containment, α z ( b − y ) ∈ \ i ∈ supp( y ) ker α i ⊥ = Q y . Therefore b − y ∈ Q y for all y ≥ 0. So, π ( b ) = 0, proving I = ker π . (cid:3) Proposition 5.1 implies there is a ∗ -isomorphism C ∗ e ( A × nc Z n + ) = p A (cid:18) BI ⋊ Z n (cid:19) p A ∼ = p (cid:16) ˜ C ⋊ Z n (cid:17) p which fixes the respective completely isometric copies of A × nc P .6. Applications and Examples Given a lattice ordered abelian group ( G, P ), we call H ⊆ G a sub-latticeordered group of G if H is a subgroup closed under ∨ and ∧ . (In fact, theidentity g + h = g ∨ h + g ∧ h shows that it is enough to assume closure underat least one of ∨ or ∧ .) For any sub-lattice ordered group, ( H, H ∩ P ) is itself alattice ordered abelian group. Suppose ( A, α, P ) is a C*-dynamical system, and set Q := H ∩ P . By [6, Theorem 4.2.9], the Fock representation is completely isometricon any Nica-covariant semicrossed product. It follows that the natural map A × nc α | Q Q → A × nc α P is completely isometric. Moreover, if G = S λ ∈ Λ G λ is an internal direct limit ofsub-lattice ordered groups G λ ⊆ G , then it follows that A × nc α P ∼ = lim −→ λ ∈ Λ A × nc α | Pλ P λ , is a direct limit. Here, P λ := G λ ∩ P . Upon identification, we think of A × nc α P = [ λ ∈ Λ A × nc α | Pλ P λ as an internal direct limit. The next result is that the respective product dilations(Definition 3.4) over P λ embed just as nicely. Proposition 6.1. Let ( G, P ) be a lattice ordered abelian group. Let ( A, α, P ) be aC*-dynamical system, with product dilation ( B, β, G ) . (1) Suppose H ⊆ G is a sub-lattice ordered group. Setting Q = H ∩ P , let ( C, γ, H ) be the product dilation for ( A, α | Q , Q ) . Then C embeds into B via an equivariant ∗ -monomorphism fixing A . (2) If G = S λ ∈ Λ G λ , for sub-lattice ordered groups G λ , let ( B λ , β λ , G λ ) be theproduct dilation for ( A, α λ , P λ ) = ( A, α | P λ , G λ ∩ P ) . Then up to identifica-tion, we have B ∼ = [ λ ∈ Λ B λ ∼ = lim −→ λ ∈ Λ B λ . Proof. As in the proof of Proposition 5.1, there is a well defined ∗ -homomorphism π : P g ∈ G β g η ( A ) → B with ( β H ) g η ( a ) β g ι ( a ). Here ι : A → B and η : A → C are the usual inclusions. Then, (1) follows if we can prove π is isometric. Let b = X g ∈ F β − g ι ( a g ) , where F ⊆ H is finite and a g ∈ A . Then k b k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X g ∈ F γ − g ι H ( a g ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = sup h ∈ H (cid:13)(cid:13)(cid:13) X g ∈ Fg ≤ h α h − g ( a g ) (cid:13)(cid:13)(cid:13) ≤ sup k ∈ G (cid:13)(cid:13)(cid:13) X g ∈ Fg ≤ k α k − g ( a g ) (cid:13)(cid:13)(cid:13) = k π ( b ) k . Conversely, given k ∈ G , since H is ∨ -closed we have { g ∈ F | g ≤ k } = { g ∈ F | g ≤ h } , *-ENVELOPES OF SEMICROSSED PRODUCTS 27 where h = ∨{ g ∈ F | g ≤ k } ∈ H . Then, k π ( b ) k k = (cid:13)(cid:13)(cid:13) X g ∈ Fg ≤ k α k − g ( a g ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) α k − h (cid:16) X g ∈ Fg ≤ h α h − g ( a g ) (cid:17)(cid:13)(cid:13)(cid:13) = k α k − h ( b h ) k ≤ k b h k ≤ k b k . So, k π ( b ) k = k b k and π extends to a ∗ -monomorphism.For claim (2), it follows from (1) that each B λ embeds in B . By minimality, B = X g ∈ G β g ι ( A ) = [ λ ∈ Λ X g ∈ G λ β g ι ( A ) = [ λ ∈ Λ B λ , as claimed. (cid:3) Since the embedding in Proposition 6.1 is equivariant and fixes the copy of A ,and since all groups involved are abelian and so exact, we also get a ∗ -embedding B λ ⋊ β λ G λ ⊆ B ⋊ β G. And, this embedding restricts to the natural embedding A × nc P λ ⊆ A × nc P .Moreover B ⋊ β G ∼ = [ λ ∈ Λ B λ ⋊ β λ G λ is again a direct product. It’s then tempting to ask when this result still holds afterpassing to quotients by Silov ideals. That is, when is C ∗ e ( A × nc α P ) ∼ = lim −→ λ ∈ Λ C ∗ e ( A × nc α λ P λ )?This does occur for surjective systems over totally ordered groups. Proposition 6.2. Let ( G, P ) be a totally ordered abelian group, and suppose that ( A, α, P ) is a unital surjective C*-dynamical system. (1) Let H ⊆ G be a subgroup, and Q = H ∩ P . Let ( B, β, G ) (resp. ( C, γ, H ) )be the product dilation for ( A, α, P ) (resp. ( A, α | Q , Q ) ). Let I (resp. J ) bethe unique maximum G -invariant (resp. H -invariant) A -boundary ideal in B (resp. C ). After identifying C ⊆ B , J = I ∩ C. (2) Suppose G = S λ ∈ Λ G λ is a directed limit of subgroups. If B (resp. B λ ) isthe product dilation for ( A, α, P ) (resp. ( A, α λ , P λ ) = ( A, α λ , G λ ∩ P ) ), and I ⊳ B and I λ ⊳ B λ are the respective maximum G or G λ -invariant boundaryideals, then I λ = I ∩ B λ and I = [ λ ∈ Λ I λ . Proof. To prove (1), we use Proposition 6.1 to identify C ⊆ B . Since I is a G -invariant boundary ideal, I ∩ C is an H -invariant boundary ideal in C . So, I ∩ B H ⊆ J . Conversely, suppose x ∈ J . By Lemma 4.5, and inductivity of ideals, it sufficesto assume x has the form x = X g ∈ F β − g ι ( a g ) ∈ I ( H ) S := { y ∈ C | y h ∈ J S − h ⊳ A for all h ∈ H } for some grid F ⊆ H ⊆ G , and some finite subset S ⊆ H . We will prove x ∈ I S = { y ∈ B | y g ∈ J S − g for all g ∈ G } ⊆ I. Let g ∈ G , and b ∈ K S − g = \ s ∈ Ss g ker α ( s − g ) ∨ = \ s ∈ Ss>g ker α s − g . The second equality is where we use the assumption that G is totally ordered. Asin the proof of Proposition 6.1.(1), we find x g = α g − h ( x h ) , where h = _ { k ∈ F ∪ S | k ≤ g } ∈ H. Since the action α is by surjections, we can write b = α g − h ( c ) for some c ∈ A . Thenbecause b ∈ K S − g , it follows that c ∈ \ s ∈ Ss>h ker α s − h = K S − h . Because x ∈ I ( H ) S , we conclude x h c = 0, so x g b = α g − h ( x h c ) = 0. Thus x ∈ I S ⊆ I ,as needed.Claim (2) follows because from (1) and the identification B = S λ B λ (Proposi-tion 6.1.(2)), because in this case inductivity of ideals implies I = [ λ ∈ Λ I ∩ B λ . But by (1), I ∩ B λ = I λ . (cid:3) Corollary 6.3. Suppose ( G, P ) is a totally ordered group with G = S λ ∈ Λ G λ , forsubgroups G λ . If ( A, α, P ) is a surjective unital C*-dynamical system, then C ∗ e ( A × nc α P ) ∼ = lim −→ λ ∈ Λ C ∗ e ( A × nc α | Pλ P λ ) , where P λ = G λ ∩ P . Corollary 6.3 applies to the totally ordered group ( Q , Q + ), where we can decom-pose Q = [ n ≥ Z n !as a direct limit of an increasing sequence of totally ordered subgroups. Moregenerally, it applies to any subgroup of R which is built as a union of an increasingsequence of cyclic subgroups, such as the dyadic rationals. It is not clear that onecan obtain Corollary 6.3 in vacuo without the explicit description of the Silov idealfrom Theorem 4.2.The following examples show that the hypotheses of surjectivity or total orderingof G cannot be dropped from Proposition 6.2. *-ENVELOPES OF SEMICROSSED PRODUCTS 29 Example 6.4. Define an action ϕ of R + on [ − , 1] by the continuous maps ϕ x ( t ) = ( t, x = 0 ,e − x | t | , x > . Then ϕ is a semigroup action, which is jointly continuous away from x = 0 ∈ R + .This induces an action α of R + on A = C ([ − , ∗ -homomorphisms α t ( f ) = f ◦ ϕ t . For any x > ϕ x is not injective, and so α x is not surjective. Indeed, for any f ∈ C ([ − , α x ( f ) is an even function.Restrict α to get C*-dynamical systems ( A, α, Z + ) and ( A, α, Z + / B, β, Z / 2) for ( A, α, Z + / A, α, Z + ) as the C*-subalgebra B = X n ∈ Z β n ι ( A ) . We will show that the maximum boundary ideal I for A in ( B , β, Z ) is not asubset of the maximum boundary ideal I ⊳ B in ( B, β, Z / I = n x ∈ B (cid:12)(cid:12)(cid:12) x n/ ∈ (ker α / ) ⊥ for all n ∈ Z , and lim n →∞ x n = 0 o , and I = n x ∈ B (cid:12)(cid:12)(cid:12) x n ∈ (ker α ) ⊥ for all n ∈ Z , and lim n →∞ x n = 0 o . Suppose that we had I ⊆ I . Then it would follow that α / (cid:16) (ker α ) ⊥ (cid:17) ⊆ (ker α / ) ⊥ . To prove this, suppose a ∈ (ker α ) ⊥ . Then a − β − ι ( α ( a )) ∈ I , so by assumption a − β − ι ( α ( a )) ∈ I . Then[ a − β − ι ( α ( a ))] / = α / ( a ) ∈ (ker α / ) ⊥ . However, in our case, for x > α x = (cid:8) f ∈ A | f | [0 ,e − x ] = 0 (cid:9) . So, (ker α x ) ⊥ = C ((0 , e − x )) = (cid:8) f ∈ A | supp( f ) ⊆ [0 , e − x ] (cid:9) . We certainly cannot have α / (cid:0) C ((0 , e − )) (cid:1) ⊆ C (0 , e − / ) , because α / is always an even function and α / = 0. For instance, f ( x ) =max { x (1 − ex ) , } satisfies f ∈ C (0 , e − ) and α / ( f ) C (0 , e − / ) , because α / ( f )( − e − / / 2) = f ( e − / > 0. So, we cannot have I ⊆ I and theconclusion in Proposition 6.2.(1) fails for the sub-lattice ordered group Z ⊆ Z / α is not surjective. Example 6.5. Proposition 6.2.(1) fails in the case H = Z ⊕ ⊆ Z ⊕ Z = G , evenfor surjective actions. Take any C*-dynamical system ( A, α, Z ). Using the samenotation as Proposition 6.2, let C and B be the respective product dilations for( A, α, Z + ⊕ 0) and ( A, α, Z ). Let J and I be the respective maximum invariantboundary ideals in C and B . As in Proposition 6.1.(1), identify C ⊆ B . Then,suppose for a contradiction that J ⊆ I .As H ∼ = Z , Proposition 3.16 gives J = ( x ∈ B ⊆ Y Z A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ( n, ∈ (ker α ) ⊥ for all n ∈ Z , and lim n →∞ x ( n, = 0 ) . Therefore, if a ∈ (ker α ) ⊥ , we have x = ι ( a ) − β − ια ( a ) ∈ J. Given ε > 0, there is a finite subset F ⊆ Z and an element y ∈ I F with k x − y k < ε .Since { I F | F ⊆ G finite } is directed, we are free to enlarge F so that (1 , ∈ F .Set k = max { m | ( n, m ) ∈ F } . Then for j ≥ k , we have y (0 ,j ) ∈ (cid:18) \ ( n,m ) ∈ F ( n,m − j ) ker α ( n,m − j ) ∨ (cid:19) ⊥ = (cid:18) \ ( n,m ) ∈ Fn> ker α n (cid:19) ⊥ ⊆ (ker α ) ⊥ , so dist( α j ( a ) , (ker α ) ⊥ ) ≤ k x − y k < ε. This proves that for any commuting unital endomorphisms α , α ∈ End( A ), andany a ∈ (ker α ) ⊥ , that(6) lim j →∞ dist( α j ( a ) , (ker α ) ⊥ ) = lim j →∞ k α j ( a ) + (ker α ) ⊥ k = 0 . However, the identity (6) fails in general. Let X = [0 , × [0 , 1] and A = C ( X ).The two injective continuous maps ϕ , ϕ : X → X defined by ϕ ( s, t ) = (cid:16) s , t (cid:17) , ϕ ( s, t ) = (cid:18) s , t (cid:19) commute and define surjective ∗ -endomorphisms α i ∈ End( A ), where α i ( f ) = f ◦ ϕ i ,for i = 1 , 2. Then (ker α ) ⊥ = C ([0 , / × [0 , , and(ker α ) ⊥ = C ([0 , / × [0 , / . Pick any f ∈ (ker α ) ⊥ with f ( s, t ) = 1 whenever s ∈ [0 , / α j ( f )(3 / , 0) = 1 for any j ≥ 1. So, k α j ( f ) + (ker α ) ⊥ k = (cid:13)(cid:13)(cid:13) α j ( f ) | [1 / , × [0 , (cid:13)(cid:13)(cid:13) ≥ *-ENVELOPES OF SEMICROSSED PRODUCTS 31 for all j , and (6) does not hold. 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