Crossed products by left LCM semigroups of endomorphisms
aa r X i v : . [ m a t h . OA ] A ug CROSSED PRODUCTS BY LEFT LCM SEMIGROUPS OFENDOMORPHISMS
SAEID ZAHMATKESH
Abstract.
Let P be a left LCM semigroup, and α an action of P by endomor-phisms of a C ∗ -algebra A . We study a semigroup crossed product C ∗ -algebra inwhich the action α is implemented by partial isometries. This crossed product givesa model for the Nica-Teoplitz algebras of product systems of Hilbert bimodules(associated with semigroup dynamical systems) studied first by Fowler, for whichwe provide a structure theorem as it behaves well under short exact sequences andtensor products. Introduction
Let P be a unital semigroup whose unit element is denoted by e . Suppose that( A, P, α ) is a dynamical system consisting of a C ∗ -algebra A , and an action α : P → End( A ) of P by endomorphisms of A such that α e = id A . Note that, since the C ∗ -algebra A is not necessarily unital, we need to assume that each endomorphism α x is extendible, which means that it extends to a strictly continuous endomorphism α x of the multiplier algebra M ( A ). Recall that an endomorphism α of A is extendible ifand only if there exists an approximate identity { a λ } in A and a projection p ∈ M ( A )such that α ( a λ ) converges strictly to p in M ( A ). However, the extendibility of α doesnot necessarily imply α (1 M ( A ) ) = 1 M ( A ) .The study of C ∗ -algebras associated with semigroups and semigroup dynamicalsystems continues fascinating mathematicians (operator algebraists). In the line ofquite huge efforts in this regard, Fowler in [8], for the dynamical system ( A, P, α ),where P is the positive cone of a group G such that ( G, P ) is quasi-lattice orderedin the sense of Nica [17], defined a covariant representation called the
Nica-Toeplitzcovariant representation of the system, such that the endomorphisms α x are imple-mented by partial isometries. He then showed that there exists a universal C ∗ -algebra T cov ( X ) associated with the system generated by a universal Nica-Toeplitz covariantrepresentation of the system such that there is a bijection between the Nica-Toeplitzcovariant representations of the system and the nondegenerate representations of T cov ( X ). To be more precise, X is actually the product system of Hilbert bimodulesassociated with the system ( A, P, α ) introduced by him, and the algebra T cov ( X ) isuniversal for Toeplitz representations of X satisfying a covariance condition called Nica covariance . He called this universal algebra the
Nica-Toeplitz crossed product (or
Nica-Toeplitz algebra ) of the system (
A, P, α ) and denoted it by T cov ( A × α P ).When the group G is totally ordered and abelian (with the positive cone G + = P ),the Nica covariance condition holds automatically, and the Toeplitz algebra T ( X ) Mathematics Subject Classification.
Primary 46L55.
Key words and phrases.
LCM semigroup, crossed product, endomorphism, Nica covariance, par-tial isometry. is the partial-isometric crossed product A × piso α P of the system ( A, P, α ) introducedand studied by the authors of [16]. In other word, the semigroup crossed product A × piso α P actually gives a model for the Teoplitz algebras T ( X ) of product systems X of Hilbert bimodules associated with the systems ( A, P, α ), where P is the positivecone of a totally ordered abelian group G . Further studies on the structure of thecrossed product A × piso α P have been done progressively in [2], [3], [4], [13], and [22]since then.In the very recent years, mathematicians (operator algebraists) in [6, 11, 12], fol-lowing the idea of Fowler, have extended and studied the notion of the Nica-Toeplitzalgebra of a product system X over more general semigroups P , namely, right LCMsemigroups (see also [7]). These are the semigroups that appear as a natural general-ization of the well-known notion of quasi-latticed ordered groups introduced first byNica in [17]. Recall that the notation N T ( X ) is used for the Nica-Toeplitz algebraof X in [6, 11, 12], which are the works that brought this question to our atten-tion that whether we could define a partial-isometric crossed product correspondingto the system ( A, P, α ), where the semigroup P goes beyond the positive cones oftotally ordered abelian groups. Although, based on the work of Fowler in [8], wewere already aware that the answer to this question must be “yes” for the positivecones P of quasi-latticed ordered groups ( G, P ), [6, 11, 12] made us very enthusi-astic to seek even more than that. Hence, the initial investigations in the presentwork indicated that the semigroup P must be left LCM (see § A, P, α ), we considered the semigroup P to be left LCM(so, the opposite semigroup P o becomes right LCM). Then, following [8], we defineda covariant representation of the system satisfying a covariance condition called the (right) Nica covariance , in which the endomorphisms α x are implemented by partialisometries. We called this representation the covariant partial-isometric representa-tion of the system. More importantly, we showed that every system ( A, P, α ) admits anontrivial covariant partial-isometric representation. Next, we proved that the Nica-Toeplitz algebra
N T ( X ) of the product system X associated with the dynamicalsystem ( A, P, α ) is generated by a covariant partial-isometric representation of thesystem which is universal for covariant partial-isometric representations of the sys-tem. We called this universal algebra the partial-isometric crossed product of thesystem (
A, P, α ) and denoted it by A × piso α P (following [16]), which is unique up toisomorphism. We then studied the behavior of crossed product A × piso α P under shortexact sequences and tensor products, from which, a structure theorem followed. Inaddition, as an example, when P and P o are both left LCM semigroups we studied thedistinguished system ( B P , P, τ ), where B P is the C ∗ -subalgebra of ℓ ∞ ( P ) generatedby the characteristic functions { y : y ∈ P } , and the action τ on B P is induced by theshift on ℓ ∞ ( P ). It was shown that the algebra B P × piso τ P is universal for bicovariantpartial-isometric representations of P , which are the partial-isometric representationsof P satisfying both right and left Nica covariance conditions.Here, prior to talking about the organization of the present work, we would like tomention that, by [21], if P is the positive cone of an abelian lattice-ordered group G ,then the Nica-Toeplitz algebra T cov ( A × α P ) of the system ( A, P, α ) is a full cornerin a classical crossed product by the group G . Thus, by the present work, since A × piso α P ≃ T cov ( A × α P ), the same corner realization holds for the partial-isometric ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 3 crossed products of the systems (
A, P, α ) consisting of the positive cones P of abelianlattice-ordered groups (see also [22]).Now, the present work as an extension of the idea in [16] follows also the frameworkof [15] for partial-isometric crossed products. We begin with a preliminary sectioncontaining a summary on LCM semigroups and discrete product systems of Hilbertbimodules. In section 3 and 4, for the system ( A, P, α ) with a left LCM semigroup P , acovariant representation of the system is defined which satisfies a covariance conditioncalled the (right) Nica covariance , where the endomorphisms α x are implementedby partial isometries. This representation is called the covariant partial-isometricrepresentation of the system. We also provide an example which shows that everysystem admits a nontrivial covariant partial-isometric representation. Then, we showthat there is a C ∗ -algebra B associated with the system generated by a covariantpartial-isometric representation of the system which is universal for covariant partial-isometric representations of the system, in the sense that there is a bijection betweenthe covariant partial-isometric representations of the system and the nondegeneraterepresentations of the C ∗ -algebra B . This universal algebra B is called the partial-isometric crossed product of the system ( A, P, α ) and denoted by A × piso α P , which isunique up to isomorphism. We also show that this crossed product behaves well undershort exact sequences. In section 5, we show that under some certain conditions thecrossed product ( A ⊗ max B ) × piso ( P × S ) can be decomposed as the maximal tensorproduct of the crossed products A × piso P and B × piso S . Also, when P and the oppositesemigroup P o are both left LCM we consider the distinguished system ( B P , P, τ ),where B P is the C ∗ -subalgebra of ℓ ∞ ( P ) generated by the characteristic functions { y : y ∈ P } , and the action τ on B P is induced by the shift on ℓ ∞ ( P ). Note thateach 1 y is actually the characteristic function of the right ideal yP = { yx : x ∈ P } in P . We then show that the crossed prodcut B P × piso τ P is universal for bicovariantpartial-isometric representations of P , which are the partial-isometric representationsof P satisfying both right and left Nica covariance conditions. In section 6, for thecrossed product ( A ⊗ max B ) × piso P a composition series0 ≤ I ≤ I ≤ ( A ⊗ max B ) × piso P of ideals is obtained, for which we identify the subquotients I , I / I , and (( A ⊗ max B ) × piso P ) / I with familiar terms. 2. Preliminaries
LCM semigroups.
Let P be a unital (discrete) semigroup, which means thatthere is an element e ∈ P such that xe = ex = x for all x ∈ P . Recall that P is called right cancellative if xz = yz , then x = y for every x, y, z ∈ P . Definition . A unital semigroup P is called left LCM (least common multiple)if it is right cancellative and for every x, y ∈ P , we have either P x ∩ P y = ∅ or P x ∩ P y = P z for some z ∈ P .Let P ∗ denote the set of all invertible elements of P , which is obviously not empty as e ∈ P ∗ . In fact, P ∗ is a group with the action inherited from P . Now, if P x ∩ P y = P z ,since z = ez ∈ P z , we have sx = z = ty for some s, t ∈ P . So, z can be viewed SAEID ZAHMATKESH as a least common left multiple of x, y . However, such a least common left multiplemay not be unique. Actually one can see that if z and ˜ z are both least common leftmultiples of x, y , then there is an invertible element u of P ( u ∈ P ∗ ) such that ˜ z = uz .Note that right LCM semigroups are define similarly, for which similar facts arevalid as above. In addition, we let P o denote the opposite semigroup endowed withthe action ⋆ such that x ⋆ y = yx for all x, y ∈ P o . Clearly, P is a left LCM semigroupif and only if P o is a right LCM semigroup.2.2. Discrete product systems of Hilbert bimodules. A Hilbert bimodule overa C ∗ -algebra A is a right Hilbert A -module X together with a homomorphism φ : A → L ( X ) which defines a left action of A on X by a · x = φ ( a ) x for all a ∈ A and x ∈ X . A Toeplitz representation of X in a C ∗ -algebra B is a pair ( ψ, π ) consistingof a linear map ψ : X → B and a homomorphism π : A → B such that ψ ( x · a ) = ψ ( x ) π ( a ) , ψ ( x ) ∗ ψ ( y ) = π ( h x, y i A ) , and ψ ( a · x ) = π ( a ) ψ ( x )for all a ∈ A and x, y ∈ X . Then, there is a homomorphism (Pimsner homomorphism) ρ : K ( X ) → B such that ρ (Θ x,y ) = ψ ( x ) ψ ( y ) ∗ for all x, y ∈ X. (2.1)The Toeplitz algebra of X is the C ∗ -algebra T ( X ) which is universal for Toeplitzrepresentations of X (see [19, 9]).Recall that every right Hilbert A -module X is essential, which means that we have X = span { x · a : x ∈ X, a ∈ A } . Moreover, a Hilbert bimodule X over a C ∗ -algebra A is called essential if X = span { a · x : a ∈ A, x ∈ X } = span { φ ( a ) x : a ∈ A, x ∈ X } , which means that X is also essential as a left A -module.Now, let A be a C ∗ -algebra and S a unital (countable) discrete semigroup. Werecall from [8] that the disjoint union X = F s ∈ S X s of Hilbert bimodules X s over A is called a (discrete) product system over S if there is a multiplication( x, y ) ∈ X s × X t xy ∈ X st (2.2)on X , with which, X is a semigroup, and the map (2.2) extends to an isomorphismof the Hilbert bimodules X s ⊗ A X t and X xt for all s, t ∈ S with s = e . The bimodule X e is A A A , and the multiplications X e × X s X s and X s × X e X s are just givenby the module actions of A on X s . Note that also we write φ s : A → L ( X s ) for thehomomorphism which defines the left action of A on X s .Note that, for every s, t ∈ S with s = e , there is a homomorphism ι sts : L ( X s ) →L ( X st ) characterized by ι sts ( T )( xy ) = ( T x ) y for all x ∈ X s , y ∈ X t and T ∈ L ( X s ). In fact, ι sts ( T ) = T ⊗ id X t .A Toeplitz representation of the product system X in a C ∗ -algebra B is a map ψ : X → B such that(1) ψ s ( x ) ψ t ( y ) = ψ st ( xy ) for all s, t ∈ S , x ∈ X s , and y ∈ X t ; and(2) the pair ( ψ s , ψ e ) is a Toeplitz representation of X s in B for all s ∈ S , ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 5 where ψ s denotes the restriction of ψ to X s . For every s ∈ S , let ψ ( s ) : K ( X s ) → B be the Pimsner homomorphism corresponding to the pair ( ψ s , ψ e ) defined by ψ ( s ) (Θ x,y ) = ψ s ( x ) ψ s ( y ) ∗ for all x, y ∈ X s (see (2.1)).By [8, Proposition 2.8], for every product system X over S , there is a C ∗ -algebra T ( X ), called the Toeplitz algebra of X , which is generated by a universal Toeplitz rep-resentation i X : X → T ( X ) of X . The pair ( T ( X ) , i X ) is unique up to isomorphism,and i X is isometric.Next, we recall that for any quasi-lattice ordered group ( G, S ), the notions ofcompactly aligned product system over S and Nica covariant Toeplitz representationof it were introduced by Fowler in [8]. Then, authors in [6] extended these notions toproduct systems over right LCM semigroups. Suppose that S is a (unital) right LCMsemigroup. A product system X over S of (essential) Hilbert bimodules is called compactly aligned if for all r, t ∈ S such that rS ∩ tS = sS for some s ∈ S we have ι sr ( R ) ι st ( T ) ∈ K ( X s )for all R ∈ K ( X r ) and T ∈ K ( X t ). Let X be a compactly aligned product systemover a right LCM semigroup S of (essential) Hilbert bimodules, and ψ : X → B aToeplitz representation of X in a C ∗ -algebra B . Then, ψ is called Nica covariant if ψ ( r ) ( R ) ψ ( t ) ( T ) = ( ψ ( s ) (cid:0) ι sr ( R ) ι st ( T ) (cid:1) if rS ∩ tS = sS ,0 if rS ∩ tS = ∅ (2.3)for all r, t ∈ S , R ∈ K ( X r ) and T ∈ K ( X t ).For a compactly aligned product system X over a right LCM semigroup S of(essential) Hilbert bimodules, the Nica-Toeplitz algebra is the C ∗ -algebra generatedby a Nica covariant Toeplitz representation i X : X → N T ( X ) which is universalfor Nica covariant Toeplitz representations of X , in the sense that: for every Nicacovariant Toeplitz representation of ψ of X , there is a representation ψ ∗ of N T ( X )such that ψ ∗ ◦ i X = ψ (see [8, 6, 11]).3. Nica partial-isometric representations
Let P be a left LCM semigroup. A Nica partial-isometric representation of P on a Hilbert space H is a map V : P → B ( H ) such that each V x := V ( x ) is apartial isometry, and the map V is a unital semigroup homomorphism of P into themultiplicative semigroup B ( H ) which satisfies V ∗ x V x V ∗ y V y = ( V ∗ z V z if P x ∩ P y = P z ,0 if
P x ∩ P y = ∅ .(3.1)The equation (3.1) is called the Nica covariance condition . Of course, since the leastcommon left multiple z may not be unique, we must check that whether the Nicacovariance condition is well-defined. So, assume that P z = P x ∩ P y = P ˜ z . If followsthat V ∗ x V x V ∗ y V y = V ∗ z V z and V ∗ x V x V ∗ y V y = V ∗ ˜ z V ˜ z . SAEID ZAHMATKESH
Now, since ˜ z = uz for some invertible element u of P , we have V ∗ ˜ z V ˜ z = V ∗ uz V uz = ( V u V z ) ∗ V u V z = V ∗ z V ∗ u V u V z . But it is not difficult to see that V u is actually a unitary, and therefore, V ∗ ˜ z V ˜ z = V ∗ z V z . This implies that equation (3.1) is indeed well-defined.
Example . Suppose that P is a left LCM semigroup and H is a Hilbert space.Define a map S : P → B ( ℓ ( P ) ⊗ H ) by( S y f )( x ) = ( f ( r ) if x = ry for some r ∈ P ,0 otherwise.for every f ∈ ℓ ( P ) ⊗ H . Note that x = ry for some r ∈ P is equivalent to sayingthat x ∈ P y . Moreover, if sy = x = ry for some r, s ∈ P , then s = r by the rightcancellativity of P , and hence f ( r ) = f ( s ). This implies that each S y is well-defined.One can see that each S y is a linear operator. We claim that each S y is actually anisometry and in particular, S e = 1. We have k S y f k = X x ∈ P k ( S y f )( x ) k = X r ∈ P k ( S y f )( ry ) k = X r ∈ P k f ( r ) k = k f k , which implies that each S y is an isometry. In particular,( S e f )( x ) = ( S e f )( xe ) = f ( x ) , which shows that S e = 1. In addition, some simple calculation shows that S x S y = S yx = S x⋆y for all x, y ∈ P. (3.2)Now we want to show that the adjoint of each S y is given by( W y f )( x ) = f ( xy )for all f ∈ ℓ ( P ) ⊗ H . For every f, g ∈ ℓ ( P ) ⊗ H , we have h S y f | g i = X x ∈ P h ( S y f )( x ) | g ( x ) i = X r ∈ P h ( S y f )( ry ) | g ( ry ) i = X r ∈ P h f ( r ) | g ( ry ) i = X r ∈ P h f ( r ) | ( W y g )( r ) i = h f | W y g i . So, S ∗ y = W y for every y ∈ P . Also, for every x, y ∈ P , by applying (3.2), we get W x W y = S ∗ x S ∗ y = [ S y S x ] ∗ = S ∗ xy = W xy . Therefore, since each W x is clearly a partial-isometry, if follows that the map W : P → B ( ℓ ( P ) ⊗ H ) define by ( W y f )( x ) = f ( xy ) is a partial-isometric representationof P on ℓ ( P ) ⊗ H . We claim that the representation W satisfies the Nica covariancecondition (3.1). Firstly, W ∗ x W x W ∗ y W y = S x S ∗ x S y S ∗ y . (3.3) ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 7
Then, for every f ∈ ℓ ( P ) ⊗ H , we have( S ∗ x S y f )( r ) = ( S ∗ x ( S y f ))( r ) = ( S y f )( rx ) for all r ∈ P. (3.4)Now, if P x ∩ P y = ∅ , since rx ∈ P x , it follows that rx P y . Therefore,( S y f )( rx ) = 0 , which implies that equation (3.3) must be zero. Suppose that we have the other case, P x ∩ P y = P z . First note that if { ε s : s ∈ P } is the usual orthonormal basis of ℓ ( P ),then each S y S ∗ y is a projection onto the closed subspace ℓ ( P y ) ⊗ H of ℓ ( P ) ⊗ H spanned by { ε sy ⊗ h : s ∈ P, h ∈ H } , which is indeed equal to the ker(1 − S y S ∗ y ). Therefore, S x S ∗ x S y S ∗ y is a projection onto the closed subspacespan { ε sy ⊗ h : s ∈ P, h ∈ H, and sy ∈ P x } = span { ε r ⊗ h : r ∈ P y ∩ P x, h ∈ H } = span { ε r ⊗ h : r ∈ P z, h ∈ H } = span { ε tz ⊗ h : t ∈ P, h ∈ H } = ℓ ( P z ) ⊗ H. See the following diagram: ℓ ( P ) ⊗ H S y S ∗ y −→ ℓ ( P y ) ⊗ H S x S ∗ x −→ ℓ ( P y ∩ P x ) ⊗ H = ℓ ( P z ) ⊗ H. (3.5)Therefore, we must have S x S ∗ x S y S ∗ y = S z S ∗ z , from which, for equation (3.3), we get W ∗ x W x W ∗ y W y = S z S ∗ z = W ∗ z W z . Consequently, W is indeed a Nica partial-isometric representation.4. Partial-isometric crossed products
Covariant partial-isometric representations.
Let P be a left LCM semi-group, and ( A, P, α ) a dynamical system consisting of a C ∗ -algebra A , and an action α : P → End( A ) of P by extendible endomorphisms of A such that α e = id A . Definition . A covariant partial-isometric representation of ( A, P, α ) on a Hilbertspace H is a pair ( π, V ) consisting of a nondegenerate representation π : A → B ( H )and a Nica partial-isometric representation V : P → B ( H ) of P such that π ( α x ( a )) = V x π ( a ) V ∗ x and V ∗ x V x π ( a ) = π ( a ) V ∗ x V x (4.1)for all a ∈ A and x ∈ P . Lemma 4.2.
Every covariant partial-isometric pair ( π, V ) extends to a covariantpartial-isometric representation ( π, V ) of the system ( M ( A ) , P, α ) , and (4.1) is equiv-alent to π ( α x ( a )) V x = V x π ( a ) and V x V ∗ x = π ( α x (1))(4.2) for all a ∈ A and x ∈ P . SAEID ZAHMATKESH
Proof.
The proof is standard. So, we skip it here. (cid:3)
Following example shows that every dynamical system (
A, P, α ) admits a nontrivial(nonzero) covariant partial-isometric representation.
Example . Suppose that (
A, P, α ) is a dynamical system, and π : A → B ( H ) isa nondegenerate representation of A on a Hilbert space H . Define a map π : A → B ( ℓ ( P ) ⊗ H ) by ( π ( a ) f )( x ) = π ( α x ( a )) f ( x )for all a ∈ A and f ∈ ℓ ( P ) ⊗ H ≃ ℓ ( P, H ). One can see that π is a representation of A on the Hilbert space ℓ ( P ) ⊗ H . Let q : ℓ ( P ) ⊗ H → ℓ ( P ) ⊗ H be a map definedby ( qf )( x ) = π ( α x (1)) f ( x )for all f ∈ ℓ ( P ) ⊗ H . It is not difficult to see that q ∈ B ( ℓ ( P ) ⊗ H ), which isactually a projection onto a closed subspace H of ℓ ( P ) ⊗ H . We claim that if { a i } is any approximate unit in A , then π ( a i ) converges strictly to q in M ( K ( ℓ ( P ) ⊗ H )) = B ( ℓ ( P ) ⊗ H ). To do so, since the net { π ( a i ) } is a norm bounded subset of B ( ℓ ( P ) ⊗ H ), and π ( a i ) ∗ = π ( a i ) for each i as well as q ∗ = q , by [20, PropositionC.7], we only need to show that π ( a i ) → q strongly in B ( ℓ ( P ) ⊗ H ). If { ε x : x ∈ P } is the usual orthonormal basis of ℓ ( P ), then it is enough to see that π ( a i )( ε x ⊗ π ( a ) h ) → q ( ε x ⊗ π ( a ) h )for each spanning element ( ε x ⊗ π ( a ) h ) of ℓ ( P ) ⊗ H (recall that π is nondegenerate).We have π ( a i )( ε x ⊗ π ( a ) h ) = ε x ⊗ π ( α x ( a i )) π ( a ) h = ε x ⊗ π ( α x ( a i ) a ) h, which is convergent to ε x ⊗ π ( α x (1) a ) h = ε x ⊗ π ( α x (1)) π ( a ) h = q ( ε x ⊗ π ( a ) h )in ℓ ( P ) ⊗ H . This is due to the extendibility of each α x . Thus, π ( a i ) → q strictly in B ( ℓ ( P ) ⊗ H ).Next, let W : P → B ( ℓ ( P ) ⊗ H ) be the Nica partial-isometric representationintroduced in Example 3.1. We aim at constructing a covariant partial-isometricrepresentation ( ρ, V ) of ( A, P, α ) on the Hilbert space (closed subspace) H by usingthe pair ( π, W ). Note that, in general, π is not nondegenerate on ℓ ( P ) ⊗ H , unless α x (1) = 1 for every x ∈ P . So, for our purpose, we first show that W x π ( a ) = π ( α x ( a )) W x and W ∗ x W x π ( a ) = π ( a ) W ∗ x W x (4.3)for all a ∈ A and x ∈ P . For every f ∈ ℓ ( P ) ⊗ H , we have( W x π ( a ) f )( r ) = ( W x ( π ( a ) f ))( r )= ( π ( a ) f )( rx )= π ( α rx ( a )) f ( rx )= π ( α r ( α x ( a )))( W x f )( r )= ( π ( α x ( a )) W x f )( r )for all r ∈ P . So, W x π ( a ) = π ( α x ( a )) W x is valid, from which, we get π ( a ) W ∗ x = W ∗ x π ( α x ( a )). One can apply these two equations to see that W ∗ x W x π ( a ) = π ( a ) W ∗ x W x ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 9 is also valid. Also, since W x W ∗ x = S ∗ x S x = 1 (see Example 3.1), each W x is a coisom-etry, and hence, by applying the equation W x π ( a ) = π ( α x ( a )) W x , we have W x π ( a ) W ∗ x = π ( α x ( a )) W x W ∗ x = π ( α x ( a )) . (4.4)Now we claim that the pair ( ρ, V ) = ( qπq, qW q ) is a covariant partial-isometricrepresentation of of ( A, P, α ) on H . More precisely, consider that maps ρ : A → qB ( ℓ ( P ) ⊗ H ) q ≃ B ( H )and V : P → qB ( ℓ ( P ) ⊗ H ) q ≃ B ( H )defined by ρ ( a ) = qπ ( a ) q = π ( a ) and V x = qW x q, for all a ∈ A and x ∈ P , respectively. Since for any approximate unit { a i } in A , ρ ( a i ) = π ( a i ) → q strongly in B ( H ), where q = 1 B ( H ) , it follows that the representa-tion ρ is nondegenerate. Moreover, by applying (4.4), we have V x ρ ( a ) V ∗ x = qW x qπ ( a ) qW ∗ x q = qW x π ( a ) W ∗ x q = qπ ( α x ( a )) q = ρ ( α x ( a )) . Also, by applying the first equation of (4.3) (equivalently, π ( a ) W ∗ x = W ∗ x π ( α x ( a )))along with the fact that ρ ( a ) = qπ ( a ) q = qπ ( a ) = π ( a ) q = π ( a ), we get V ∗ x V x ρ ( a ) = qW ∗ x qW x qπ ( a ) q = qW ∗ x qW x π ( a ) q = qW ∗ x qπ ( α x ( a )) W x q = qW ∗ x π ( α x ( a )) qW x q = qπ ( a ) W ∗ x qW x q = qπ ( a ) qW ∗ x qW x q = ρ ( a ) V ∗ x V x . Thus, it is only left to show that the map V is a Nica partial-isometric representation.To see that each V x is a partial-isometry, note that for any approximate unit { a i } in A , qW x π ( a i ) W ∗ x qW x q converges strongly to qW x qW ∗ x qW x q = V x V ∗ x V x in B ( ℓ ( P ) ⊗ H ). On the other hand, by applying the covariance equations of thepair ( π, W ), we have q [ W x π ( a i ) W ∗ x ] qW x q = qπ ( α x ( a i )) qW x q = qπ ( α x ( a i )) W x q = qW x π ( a i ) q, which converges strongly to qW x q = V x . So, we must have V x V ∗ x V x = V x , which meansthat each V x is a partial-isometry. To see V x V y = V xy for every x, y ∈ P , we first need to compute V x f for any f ∈ H . So, knowing that qf = f , we have[ V x f ]( r ) = [ qW x f ]( r ) = [ q ( W x f )]( r )= π ( α r (1))( W x f )( r )= π ( α r (1)) f ( rx )= π ( α r (1))( qf )( rx )= π ( α r (1)) π ( α rx (1)) f ( rx )= π ( α r (1) α rx (1)) f ( rx )= π ( α rx (1)) f ( rx )= ( qf )( rx ) = f ( rx ) . for all r ∈ P . Thus, by applying the above computation, we have[ V x V y f ]( r ) = [ V x ( V y f )]( r )= ( V y f )( rx )= f (( rx ) y )= f ( r ( xy )) = [ V xy f ]( r ) . It follows that V x V y = V xy for all x, y ∈ P . Finally, we show that the partial-isometricrepresentation V satisfies the Nica covariance condition. Let us first mention thatthe Hilbert space H is spanned by elements { ε r ⊗ π ( α r (1)) h : r ∈ P, h ∈ H } as closed subspace of ℓ ( P ) ⊗ H . Then, for every y ∈ P and f ∈ H ,( V ∗ y f )( r ) = ( qW ∗ y f )( r ) = ( q ( S y f ))( r ) = π ( α r (1))( S y f )( r ) , which is nonzero only if r ∈ P y . Thus, if r = sy for some s ∈ P , we get( V ∗ y f )( r ) = π ( α sy (1))( S y f )( sy ) = π ( α sy (1)) f ( s ) . If follows that, if r = sy for some s ∈ P , then[ V ∗ y V y f ]( r ) = [ V ∗ y ( V y f )]( r )= π ( α sy (1))( V y f )( s )= π ( α sy (1)) f ( sy )= ( qf )( sy ) = f ( sy ) = f ( r ) , otherwise it is zero. Therefore, each V ∗ y V y is the projection of H onto the closedsubspace H y := { f ∈ H : f ( r ) = 0 if r P y } = ker(1 − V ∗ y V y )of H , which is actually spanned by the elements { ε sy ⊗ π ( α sy (1)) h : s ∈ P, h ∈ H } . Now, if
P x ∩ P y = ∅ , then for every f ∈ H ,[ V x V ∗ y f ]( r ) = [ V x ( V ∗ y f )]( r )= ( V ∗ y f )( rx ) = 0 . This is due to the fact that since rx ∈ P x , rx P y . It follows that V x V ∗ y = 0, andhence, V ∗ x V x V ∗ y V y = 0 . If P x ∩ P y = P z , then, as V ∗ y V y is a projection onto the closed subspace H y , V ∗ x V x V ∗ y V y ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 11 is the projection onto the closed subspace { f ∈ H y : f ( r ) = 0 if r P x } = { f ∈ H : f ( r ) = 0 if ( r P y ∨ r P x ) } = { f ∈ H : f ( r ) = 0 if r ( P y ∩ P x ) } = { f ∈ H : f ( r ) = 0 if r P z } = ker(1 − V ∗ z V z ) = H z . Observe the following diagram: H V y V ∗ y −→ H y V x V ∗ x −→ H z . (4.5)Thus, we must have V ∗ x V x V ∗ y V y = V ∗ z V z . Consequently, the pair ( ρ, V ) is a (nontrivial) covariant partial-isometric representa-tion of (
A, P, α ) on H .Note that if π is faithful, then it is not difficult to see that ρ is faithful as well.So, every system ( A, P, α ) has a (nontrivial) covariant pair ( ρ, V ) with ρ faithful.4.2. Crossed products and Nica-Teoplitz algebras of Hilbert bimodules.
Let P be a left LCM semigroup, and ( A, P, α ) a dynamical system consisting of a C ∗ -algebra A , and an action α : P → End( A ) of P by extendible endomorphisms of A such that α e = id A Definition . A partial-isometric crossed product of ( A, P, α ) is a triple (
B, i A , i P )consisting of a C ∗ -algebra B , a nondegenerate injective homomorphism i A : A → B ,and a Nica partial-isometric representation i P : P → M ( B ) such that:(i) the pair ( i A , i P ) is a covariant partial-isometric representation of ( A, P, α ) in B ;(ii) for every covariant partial-isometric representation ( π, V ) of ( A, P, α ) on aHilbert space H , there exists a nondegenerate representation π × V : B → B ( H ) such that ( π × V ) ◦ i A = π and ( π × V ) ◦ i P = V ; and(iii) the C ∗ -algebra B is generated by { i A ( a ) i P ( x ) : a ∈ A, x ∈ P } .We call the algebra B the partial-isometric crossed product of the system ( A, P, α )and denote it by A × piso α P . Remark . Note that in the definition above, for part (iii), we actually have B = span { i P ( x ) ∗ i A ( a ) i P ( y ) : x, y ∈ P, a ∈ A } . (4.6)To see this, we only need to show that the right hand side of (4 .
6) is closed undermultiplication. To do so, we need to apply the Nica covariance condition to calculateeach product [ i P ( x ) ∗ i A ( a ) i P ( y )][ i P ( s ) ∗ i A ( b ) i P ( t )] . (4.7)We have [ i P ( x ) ∗ i A ( a ) i P ( y )][ i P ( s ) ∗ i A ( b ) i P ( t )]= i P ( x ) ∗ i A ( a ) i P ( y )[ i P ( y ) ∗ i P ( y ) i P ( s ) ∗ i P ( s )] i P ( s ) ∗ i A ( b ) i P ( t ) , which is zero if P y ∩ P s = ∅ . But if P y ∩ P s = P z for some z ∈ P , then ry = z = qs for some r, q ∈ P , and therefore by the covariance of the pair ( i A , i P ), we get[ i P ( x ) ∗ i A ( a ) i P ( y )][ i P ( s ) ∗ i A ( b ) i P ( t )]= i P ( x ) ∗ i A ( a ) i P ( y ) i P ( z ) ∗ i P ( z ) i P ( s ) ∗ i A ( b ) i P ( t )= i P ( x ) ∗ i A ( a ) i P ( y ) i P ( ry ) ∗ i P ( qs ) i P ( s ) ∗ i A ( b ) i P ( t )= i P ( x ) ∗ i A ( a )[ i P ( y ) i P ( y ) ∗ ] i P ( r ) ∗ i P ( q )[ i P ( s ) i P ( s ) ∗ ] i A ( b ) i P ( t )= i P ( x ) ∗ i A ( a ) i A ( α y (1)) i P ( r ) ∗ i P ( q ) i A ( α s (1)) i A ( b ) i P ( t )= i P ( x ) ∗ i A ( aα y (1)) i P ( r ) ∗ i P ( q ) i A ( α s (1) b ) i P ( t )= i P ( x ) ∗ i A ( aα y (1)) i P ( r ) ∗ i P ( q ) i A ( α s (1) b ) i P ( t )= i P ( x ) ∗ i P ( r ) ∗ i A ( α r ( c )) i A ( α q ( d )) i P ( q ) i P ( t )= i P ( rx ) ∗ i A ( α r ( c ) α q ( d )) i P ( qt ) , which is in the right hand side of (4 . c = aα y (1) and d = α s (1) b . Thus, (4 . A, P, α ) always exists, and it is unique up to isomorphism. Firstly, since P is a leftLCM semigroup, the opposite semigroup P o is a right LCM semigroup. Therefore,one can easily see that ( A, P o , α ) is a dynamical system in the sense of [11, Definition3.1]. Thus, following [8, §
3] or [11, § s ∈ P , let X s := { s } × α s (1) A, where α s (1) A = α s ( A ) A = span { α s ( a ) b : a, b ∈ A } as each endomorphism α s isextendible. Then, each X s is given the structure of a Hilbert bimodule over A via( s, x ) · a := ( s, xa ) , h ( s, x ) , ( s, y ) i A := x ∗ y, and a · ( s, x ) := ( s, α s ( a ) x ) . Let X = F s ∈ P X s , which is equipped with a multiplication X s × X t → X s⋆t ; (( s, x ) , ( t, y )) ( s, x )( t, y )defined by ( s, x )( t, y ) := ( ts, α t ( x ) y ) = ( s ⋆ t, α t ( x ) y )for every x ∈ α s (1) A and y ∈ α t (1) A . By [8, Lemma 3.2], X is a product system overthe opposite semigroup P o , and the left action of A on each fiber X s is by compactoperators. Let ( N T ( X ) , i X ) be the Nica-Toeplitz algebra corresponding to X definedin [11, §
3] (see also [6, § i X : X → N T ( X ). We show that this algebra is the partial-isometriccrossed product of the system ( A, P, α ). But we first need to recall that, for anyapproximate unit { a i } in A , similar to [8, Lemma 3.3], one can see that i X ( s, α s ( a i ))converges strictly in the multiplier algebra M ( N T ( X )) for every s ∈ P . Now, wehave: Proposition 4.6.
Suppose that P is a left LCM semigroup, and ( A, P, α ) a dynamicalsystem. Let { a i } be any approximate unit in A . Define the maps i A : A → N T ( X ) and i P : P → M ( N T ( X )) ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 13 by i A ( a ) := i X ( e, a ) and i P ( s ) := lim i i X ( s, α s ( a i )) ∗ for all a ∈ A and s ∈ P . Then the triple ( N T ( X ) , i A , i P ) is a partial-isometriccrossed product for ( A, P, α ) , which is unique up to isomorphism.Proof. For any approximate unit { a i } in A , i A ( a i ) = i X ( e, a i ) = i X ( e, α e ( a i ))converges strictly to 1 in the multiplier algebra M ( N T ( X )). One can see this againfrom [8, Lemma 3.3] similarly when s = e . It follows that i A is a nondegeneratehomomorphism. By a similar discussion to the first part of the proof of [8, Proposition3.4], we can see that the map i P is a partial-isometric representation such that togetherwith the (nondegenerate) homomorphism i A satisfy the covariance equations i A ( α s ( a )) = i P ( s ) i A ( a ) i P ( s ) ∗ and i A ( a ) i P ( s ) ∗ i P ( s ) = i P ( s ) ∗ i P ( s ) i A ( a )for all a ∈ A and s ∈ P . So, we only need to show that the representation i P satisfiesthe Nica covariance condition. By the same calculation as (3.7) in the proof of [8,Proposition 3.4], we have i A ( ab ∗ ) i P ( s ) ∗ i P ( s ) = i X ( s, α s ( a )) i X ( s, α s ( b )) ∗ (4.8)for all a, b ∈ A and s ∈ P , and since i X ( s, α s ( a )) i X ( s, α s ( b )) ∗ = i ( s ) X (Θ ( s,α s ( a )) , ( s,α s ( b )) ) , it follows that i A ( ab ∗ ) i P ( s ) ∗ i P ( s ) = i ( s ) X (Θ ( s,α s ( a )) , ( s,α s ( b )) ) . (4.9)Therefore, i A ( ab ∗ ) i P ( s ) ∗ i P ( s ) i A ( cd ∗ ) i P ( t ) ∗ i P ( t ) = i ( s ) X (Θ ( s,α s ( a )) , ( s,α s ( b )) ) i ( t ) X (Θ ( t,α t ( c )) , ( t,α t ( d )) ) , and since i A ( ab ∗ ) i P ( s ) ∗ i P ( s ) i A ( cd ∗ ) i P ( t ) ∗ i P ( t ) = i A ( ab ∗ ) i A ( cd ∗ ) i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t )= i A ( ab ∗ ( cd ∗ )) i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) , it follows that i A ( ab ∗ ( cd ∗ )) i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) = i ( s ) X (Θ ( s,α s ( a )) , ( s,α s ( b )) ) i ( t ) X (Θ ( t,α t ( c )) , ( t,α t ( d )) ) . (4.10)Now, if P s ∩ P t = P r for some r ∈ P , which is equivalent to saying that s ⋆ P o ∩ t ⋆ P o = r ⋆ P o , since i X is Nica-covariant, we have i A ( ab ∗ ( cd ∗ )) i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) = i ( r ) X (cid:0) ι rs (Θ ( s,α s ( a )) , ( s,α s ( b )) ) ι rt (Θ ( t,α t ( c )) , ( t,α t ( d )) ) (cid:1) . (4.11)Next, we want to calculate the product ι rs (Θ ( s,α s ( a )) , ( s,α s ( b )) ) ι rt (Θ ( t,α t ( c )) , ( t,α t ( d )) )of compact operators in K ( X r ) to show that it is equal toΘ ( r,α r ( ab ∗ )) , ( r,α r ( dc ∗ )) . Since s ⋆ p = r = t ⋆ q for some p, q ∈ P , and X t ⊗ A X q ≃ X t⋆q , it is enough to see thison the spanning elements ( t, α t (1) f )( q, α q (1) g ) of X t⋆q = X r , where f, g ∈ A . First,( t, α t (1) f )( q, α q (1) g ) by ι rt (Θ ( t,α t ( c )) , ( t,α t ( d )) ) is mapped to ι t⋆qt (cid:0) Θ ( t,α t ( c )) , ( t,α t ( d )) (cid:1)(cid:0) ( t, α t (1) f )( q, α q (1) g ) (cid:1) = (cid:0) Θ ( t,α t ( c )) , ( t,α t ( d )) ( t, α t (1) f ) (cid:1) ( q, α q (1) g )= (cid:0) ( t, α t ( c )) · (cid:10) ( t, α t ( d )) , ( t, α t (1) f ) (cid:11) A (cid:1) ( q, α q (1) g )= (cid:0) ( t, α t ( c )) · ( α t ( d ∗ ) α t (1) f ) (cid:1) ( q, α q (1) g )= (cid:0) ( t, α t ( c )) · ( α t ( d ∗ ) f ) (cid:1) ( q, α q (1) g )= (cid:0) ( t, α t ( c ) α t ( d ∗ ) f ) (cid:1) ( q, α q (1) g )= ( t, α t ( cd ∗ ) f )( q, α q (1) g )= ( qt, α q ( α t ( cd ∗ ) f ) α q (1) g )= ( qt, α q ( α t ( cd ∗ ) f ) g )= ( qt, α qt ( cd ∗ ) α q ( f ) g )= ( t ⋆ q, α t⋆q ( cd ∗ ) α q ( f ) g )= ( r, α r ( cd ∗ ) α q ( f ) g )= ( s ⋆ p, α s⋆p ( cd ∗ ) α q ( f ) g )= ( ps, α ps ( cd ∗ ) α q ( f ) g ) = ( s, α s ( cd ∗ ))( p, α p (1) α q ( f ) g ) . We then let ι rs (Θ ( s,α s ( a )) , ( s,α s ( b )) ) act on ( s, α s ( cd ∗ ))( p, α p (1) α q ( f ) g ), and hence, ι s⋆ps (cid:0) Θ ( s,α s ( a )) , ( s,α s ( b )) (cid:1)(cid:0) ( s, α s ( cd ∗ ))( p, α p (1) α q ( f ) g ) (cid:1) = (cid:0) Θ ( s,α s ( a )) , ( s,α s ( b )) ( s, α s ( cd ∗ )) (cid:1) ( p, α p (1) α q ( f ) g )= (cid:0) ( s, α s ( a )) · (cid:10) ( s, α s ( b )) , ( s, α s ( cd ∗ )) (cid:11) A (cid:1) ( p, α p (1) α q ( f ) g )= (cid:0) ( s, α s ( a )) · [ α s ( b ∗ ) α s ( cd ∗ )] (cid:1) ( p, α p (1) α q ( f ) g )= (cid:0) ( s, α s ( a )) · [ α s ( b ∗ cd ∗ )] (cid:1) ( p, α p (1) α q ( f ) g )= ( s, α s ( a ) α s ( b ∗ cd ∗ ))( p, α p (1) α q ( f ) g )= ( s, α s ( ab ∗ cd ∗ ))( p, α p (1) α q ( f ) g )= ( ps, α p ( α s ( ab ∗ cd ∗ )) α p (1) α q ( f ) g )= ( ps, α p ( α s ( ab ∗ cd ∗ )) α q ( f ) g )= ( ps, α ps ( ab ∗ cd ∗ ) α q ( f ) g )= ( s ⋆ p, α s⋆p ( ab ∗ cd ∗ ) α q ( f ) g ) = ( r, α r ( ab ∗ cd ∗ ) α q ( f ) g ) . Thus, it follows that ι rs (Θ ( s,α s ( a )) , ( s,α s ( b )) ) ι rt (Θ ( t,α t ( c )) , ( t,α t ( d )) ) (cid:0) ( t, α t (1) f )( q, α q (1) g ) (cid:1) = ( r, α r ( ab ∗ cd ∗ ) α q ( f ) g ) . (4.12)On the other hand, since( t, α t (1) f )( q, α q (1) g ) = ( qt, α q ( α t (1) f ) α q (1) g )= ( qt, α q ( α t (1) f ) g )= ( qt, α q ( α t (1)) α q ( f ) g )= ( qt, α qt (1) α q ( f ) g )= ( r, α r (1) α q ( f ) g ) , ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 15 we have Θ ( r,α r ( ab ∗ )) , ( r,α r ( dc ∗ )) (cid:0) ( t, α t (1) f )( q, α q (1) g ) (cid:1) = Θ ( r,α r ( ab ∗ )) , ( r,α r ( dc ∗ )) ( r, α r (1) α q ( f ) g )= ( r, α r ( ab ∗ )) · (cid:10) ( r, α r ( dc ∗ )) , ( r, α r (1) α q ( f ) g ) (cid:11) A = ( r, α r ( ab ∗ )) · [ α r ( dc ∗ ) ∗ α r (1) α q ( f ) g ]= ( r, α r ( ab ∗ )) · [ α r ( cd ∗ ) α r (1) α q ( f ) g ]= ( r, α r ( ab ∗ )) · [ α r ( cd ∗ ) α q ( f ) g ]= ( r, α r ( ab ∗ ) α r ( cd ∗ ) α q ( f ) g )= ( r, α r ( ab ∗ cd ∗ ) α q ( f ) g ) . (4.13)So, we conclude by (4.12) and (4.13) that ι rs (Θ ( s,α s ( a )) , ( s,α s ( b )) ) ι rt (Θ ( t,α t ( c )) , ( t,α t ( d )) ) = Θ ( r,α r ( ab ∗ )) , ( r,α r ( dc ∗ )) . (4.14)Consequently, if P s ∩ P t = P r , then by applying (4.14), (4.11), and (4.9), we get i A ( ab ∗ ( cd ∗ )) i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) = i ( r ) X (cid:0) ι rs (Θ ( s,α s ( a )) , ( s,α s ( b )) ) ι rt (Θ ( t,α t ( c )) , ( t,α t ( d )) ) (cid:1) = i ( r ) X (cid:0) Θ ( r,α r ( ab ∗ )) , ( r,α r ( dc ∗ )) (cid:1) = i A ( ab ∗ ( dc ∗ ) ∗ ) i P ( r ) ∗ i P ( r )= i A ( ab ∗ ( cd ∗ )) i P ( r ) ∗ i P ( r ) . We therefore have i A ( ab ∗ cd ∗ ) i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) = i A ( ab ∗ cd ∗ ) i P ( r ) ∗ i P ( r )(4.15)for all a, b, c, d ∈ A . Since A contains approximate unit, it follows by (4.15) that i A ( a ) i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) = i A ( a ) i P ( r ) ∗ i P ( r )(4.16)for all a ∈ A . So, if { a i } is any approximate unit in A , then i A ( a i ) i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) = i A ( a i ) i P ( r ) ∗ i P ( r ) . (4.17)Now, in (4.17), the left hand side converges strictly to i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t )in M ( N T ( X )), while the right hand side converges strictly to i P ( r ) ∗ i P ( r ). Hence,we must have i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) = i P ( r ) ∗ i P ( r )when P s ∩ P t = P r .If
P s ∩ P t = ∅ , then again, since i X is Nica-covariant, the right hand side of (4.10)is zero, and therefore, i A ( ab ∗ ( cd ∗ )) i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) = 0(4.18)for all a, b, c, d ∈ A . Then, similar to the above, as A contains approximate unit, wecan show that we must have i P ( s ) ∗ i P ( s ) i P ( t ) ∗ i P ( t ) = 0 . Consequently, the pair ( i A , i P ) is a covariant partial-isometric representation of ( A, P, α )in the algebra
N T ( X ). So, the condition (i) in Definition 4.4 is satisfied.Suppose that ( π, V ) is a covariant partial-isometric representation of ( A, P, α ) on aHilbert space H . Then, the pair ( π, V ∗ ) is a representation of the system ( A, P o , α ) inthe sense of [11, Definition 3.2], which is Nica-covariant. Note that the homomorphism V ∗ : P o → B ( H ) is defined by s V ∗ s . Therefore, by [11, Proposition 3.11], the map ψ : X → B ( H ) defined by ψ ( s, x ) := V ∗ s π ( x )is a nondegenerate Nica-covariant Toeplitz representation of X on H . So, there is ahomomorphism ψ ∗ : N T ( X ) → B ( H ) such that ψ ∗ ◦ i X = ψ (see [6, 11, 8]), which isnondegenerate. Let π × V = ψ ∗ . Then( π × V )( i A ( a )) = ψ ∗ ( i X ( e, a )) = ψ ( e, a ) = V ∗ e π ( a ) = π ( a )for all a ∈ A . Also, since π × V is nondegenerate, we have( π × V )( i P ( s )) = ( π × V )(lim i i X ( s, α s ( a i )) ∗ )= lim i ( π × V )( i X ( s, α s ( a i )) ∗ )= lim i ψ ∗ ( i X ( s, α s ( a i ))) ∗ = lim i ψ ( s, α s ( a i )) ∗ = lim i [ V ∗ s π ( α s ( a i ))] ∗ = lim i [ π ( a i ) V ∗ s ] ∗ (by the covariance of ( π, V ))= lim i V s π ( a i ) = V s for all s ∈ P . Thus, the condition (ii) in Definition 4.4 is satisfied, too.The condition (iii) is also satisfied. This is due to the facts that the elements ofthe form i X ( s, α s (1) a ) generate N T ( X ), and i X ( s, α s (1) a ) = lim i i X ( s, α s ( a i ) a )= lim i [ i X ( s, α s ( a i )) i X ( e, a )]= lim i [ i X ( s, α s ( a i ))] i X ( e, a ) = i P ( s ) ∗ i A ( a ) . To see that the homomorphism i A is injective, we recall from Example 4.3 thatthe system ( A, P, α ) admits covariant partial-isometric representations ( π, V ) with π faithful. Therefore, it follows from the equation ( π × V ) ◦ i A = π that i A must beinjective.For uniqueness, suppose that ( C, j A , j P ) is another triple which satisfies conditions(i)-(iii) in Definition 4.4. Then by applying the universal properties (condition (ii)) ofthe algebras C and N T ( X ), respectively, once can see that there is an isomorphismof C onto N T ( X ) which maps the pair ( j A , j P ) into the pair ( i A , i P ). (cid:3) Suppose that (
A, P, α ) is a dynamical system, an I is an ideal of A such that α s ( I ) ⊂ I for all s ∈ P . To define a crossed product I × piso α P which we want it tosit naturally in A × piso α P as an ideal, we need some extra condition. So, we need torecall a definition from [1]. Let α be an extendible endomorphism of a C ∗ -algebra A ,and I an ideal of A . Suppose that ψ : A → M ( I ) is the canonical nondegeneratehomomorphism defined by ψ ( a ) i = ai for all a ∈ A and i ∈ I . Then, we say I is extendible α -invariant if it is α -invariant, which means that α ( I ) ⊂ I , and theendomorphism α | I is extendible, such that α ( u λ ) → ψ ( α (1 M ( A ) ))strictly in M ( I ), where { u λ } is an approximate unit in I .In addition, if ( A, P, α ) is a dynamical system and I is an ideal of A , then there isa dynamical system ( A/I, P, ˜ α ) with extendible endomorphisms given by ˜ α s ( a + I ) = α s ( a ) + I for every a ∈ A and s ∈ P (see again [1]).The following Theorem is actually a generalization of [4, Theorem 3.1]: ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 17
Theorem 4.7.
Let ( A × piso α P, i A , V ) be the partial-isometric crossed product of adynamical system ( A, P, α ) , and I an extendible α x -invariant ideal of A for every x ∈ P . Then, there is a short exact sequence −→ I × piso α P µ −→ A × piso α P ϕ −→ A/I × piso˜ α P −→ of C ∗ -algebras, where µ is an isomorphism of I × piso α P onto the ideal E := span { V ∗ s i A ( i ) V t : i ∈ I, s, t ∈ P } of A × piso α P . If q : A → A/I is the quotient map, and the triples ( I × piso α P, i I , W ) and( A/I × piso˜ α P, i
A/I , U ) are the crossed products of the systems (
I, P, α ) and (
A/I, P, ˜ α ),respectively, then µ ◦ i I = i A | I , µ ◦ W = V and ϕ ◦ i A = i A/I ◦ q, ϕ ◦ V = U. Proof.
We first show that E is an ideal of A × piso α P . To do so, it suffices to see on thespanning elements of E that V ∗ r E , i A ( a ) E , and V r E are all contained in E for every a ∈ A and r ∈ P . This first one is obvious, and the second one follows easily byapplying the covariance equation i A ( a ) V ∗ s = V ∗ s i A ( α s ( a )). For the third one, we have V r V ∗ s i A ( i ) V t = V r [ V ∗ r V r V ∗ s V s ] V ∗ s i A ( i ) V t , which is zero if P r ∩ P s = ∅ . But if P r ∩ P s = P z for some z ∈ P , then there are x, y ∈ P such that xr = z = ys , and therefore it follows that V r V ∗ s i A ( i ) V t = V r [ V ∗ z V z ] V ∗ s i A ( i ) V t = V r V ∗ xr V ys V ∗ s i A ( i ) V t = V r [ V x V r ] ∗ V y V s V ∗ s i A ( i ) V t = [ V r V ∗ r ] V ∗ x V y [ V s V ∗ s ] i A ( i ) V t = i A ( α r (1)) V ∗ x V y i A ( α s (1)) i A ( i ) V t (by Lemma (4.2))= V ∗ x i A ( α x ( α r (1))) V y i A ( α s (1) i ) V t (by Lemma (4.2))= V ∗ x i A ( α xr (1)) i A ( α y ( α s (1) i )) V y V t = V ∗ x i A ( α z (1)) i A ( α y ( α s (1)) α y ( i )) V yt = V ∗ x i A ( α z (1)) i A ( α ys (1) α y ( i )) V yt = V ∗ x i A ( α z (1)) i A ( α z (1) α y ( i )) V yt = V ∗ x i A ( α z (1) α z (1) α y ( i )) V yt = V ∗ x i A ( α z (1) α y ( i )) V yt , which belongs to E . Thus, E is an ideal of A × piso α P . Let φ : A × piso α P → M ( E ) be thecanonical nondegenerate homomorphism defined by φ ( ξ ) η = ξη for all ξ ∈ A × piso α P and η ∈ E . Suppose that now the maps k I : I → M ( E ) and S : P → M ( E )are defined by the compositions I i A | I −→ A × piso α P φ −→ M ( E ) and P V −→ M ( A × piso α P ) φ −→ M ( E ) , respectively. We claim that the triple ( E , k I , S ) is a partial-isometric crossed productof the system ( I, P, α ). First, exactly by the same discussion as in the proof of [4,Theorem 3.1] using the extendibility of the ideal I , it follows that the homomorphism k I is nondegenerate. Also, it follows easily by the definition of the map S that it is indeed a Nica partial-isometric representation. Then, by some routine calculations,one can see that the pair ( k I , S ) satisfies the covariance equations k I ( α t ( i )) = S t k I ( i ) S ∗ t and S ∗ t S t k I ( i ) = k I ( i ) S ∗ t S t for all i ∈ I and t ∈ P .Next, suppose that the pair ( π, T ) is a covariant partial-isometric representation of( I, P, α ) on a Hilbert space H . Let ψ : A → M ( I ) be the canonical nondegeneratehomomorphism which was mentioned about earlier. Let the map ρ : A → B ( H ) bedefined by the composition A ψ −→ M ( I ) π −→ B ( H ) , which is a nondegenerate representation of A on H . We claim that the pair ( ρ, T ) isa covariant partial-isometric representation of ( A, P, α ) on H . To prove our claim, weonly need to show that the pair ( ρ, T ) satisfies the covariance equations (4.1). Sincethe ideal I is extendible, we have α s | I ◦ ψ = ψ ◦ α s for all s ∈ P . It therefore followsthat ρ ( α s ( a )) = ( π ◦ ψ )( α s ( a ))= π ( ψ ◦ α s ( a ))= π ( α s | I ◦ ψ ( a ))= ( π ◦ α s | I )( ψ ( a ))= T s π ( ψ ( a )) T ∗ s = T s ρ ( a ) T ∗ s . Also, one can easily see that we have T ∗ s T s ρ ( a ) = ρ ( a ) T ∗ s T s . Thus, there is a non-degenerate representation ρ × T of A × piso α P on H , whose restriction ( ρ × T ) | E is anondegenerate representation of E on H satisfying( ρ × T ) | E ◦ k I = π and ( ρ × T ) | E ◦ S = T. Finally, the elements of the form S ∗ s k I ( i ) S t = φ ( V s ) ∗ φ ( i A ( i )) φ ( V t )= φ ( V ∗ s i A ( i ) V t ) = V ∗ s i A ( i ) V t obviously span the algebra E . Thus, ( E , k I , S ) is a partial-isometric crossed productof ( I, P, α ). So, by Proposition 4.6, there is an isomorphism µ : A × piso α P → E suchthat µ ( i I ( i ) W t ) = k I ( i ) S t = φ ( i A | I ( i )) φ ( V t ) = φ ( i A | I ( i ) V t ) = i A | I ( i ) V t , from which, it follows that µ ◦ i I = i A | I and µ ◦ W = V. To get the desired homomorphism ϕ , let the homomorphism j A : A → A/I × piso˜ α P be given by the composition A q −→ A/I i A/I −→ A/I × piso˜ α P, which is nondegenerate. Then, it is not difficult to see that the pair ( j A , U ) is acovariant partial-isometric representation of ( A, P, α ) in the algebra
A/I × piso˜ α P . Thus,there is a nondegenerate homomorphism ϕ := j A × U : A × piso α P → A/I × piso˜ α P suchthat ϕ ◦ i A = j A = i A/I ◦ q and ϕ ◦ V = U, ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 19 which implies that ϕ is onto.Finally, we show that µ ( I × piso α P ) = E is equal to ker ϕ which means that (4.19) isexact. The inclusion E ⊂ ker ϕ is immediate. To see the other inclusion, take a non-degenerate representation Π of A × piso α P on a Hilbert space H with ker Π = E . Since I ⊂ ker(Π ◦ i A ), the composition Π ◦ i A gives a (well-defined) nodegenerate represen-tation e Π of
A/I on H . Also, the composition Π ◦ V defines a Nica partial-isometricrepresentation P on H , such that together with e Π forms a covariant partial-isometricrepresentation of (
A/I, P, ˜ α ) on H . Then the corresponding (nondegenerate) repre-sentation e Π × (Π ◦ V ) lifts to Π, which means that [ e Π × (Π ◦ V )] ◦ ϕ = Π, from whichthe inclusion ker ϕ ⊂ E follows. This completes the proof. (cid:3) Example . Suppose that S is a (unital) right LCM semigroup. See in [5, 18] thatassociated to S there is a universal C ∗ -algebra C ∗ ( S ) = span { W s W ∗ t : s, t ∈ S } generated by a universal isometric representation W : S → C ∗ ( S ), which is Nica-covariant , which means that it satisfies W r W ∗ r W s W ∗ s = ( W t W ∗ t if rS ∩ sS = tS ,0 if rS ∩ sS = ∅ .(4.20)In addition, by [6, Corollary 7.11], C ∗ ( S ) is isomorphic to the Nica-Toeplitz algebra N T ( X ) of the compactly aligned product system X over S with fibers X s = C forall s ∈ S . Now, consider the trivial dynamical system ( C , P, id), where P is a leftLCM semigroup. So, the opposite semigroup P o is right LCM. Then, it follows byProposition 4.6 that there is an isomorphism i p ( x ) ∈ ( C × pisoid P ) W ∗ x ∈ C ∗ ( P o )for all x ∈ P , where W is the universal Nica-covariant isometric representation of P o which generates C ∗ ( P o ).5. Tensor products of crossed products
Let (
A, P, α ) and (
B, S, β ) be dynamical systems in which P and S are left LCMsemigroups. Then, P × S is a unital semigroup with the unit element ( e P , e S ), where e P and e S are the unit elements of P and S , respectively. In addition, since( P × S )( x, r ) ∩ ( P × S )( y, s ) = ( P x × Sr ) ∩ ( P y × Ss )= ( P x ∩ P y ) × ( Sr ∩ Ss ) , (5.1)it follows that P × S is a left LCM semigroup. More precisely, if P x ∩ P y = P z and Sr ∩ Ss = St for some z ∈ P and t ∈ S , then it follows by (5.1) that( P × S )( x, r ) ∩ ( P × S )( y, s ) = P z × St = ( P × S )( z, t ) , which means that ( z, t ) is a least common left multiple of ( x, r ) and ( y, s ) in P × S .Otherwise, ( P × S )( x, r ) ∩ ( P × S )( y, s ) = ∅ . Thus, P × S is actually a left LCMsemigroup (note that, the similar fact holds if P and S are right LCM semigroups).Next, for every x ∈ P and r ∈ S , as α x and β r are endomorphisms of the algebras A and B , respectively, it follows by [20, Lemma B. 31] that there is an endomorphism α x ⊗ β r of the maximal tensor product A ⊗ max B such that ( α x ⊗ β r )( a ⊗ b ) = α x ( a ) ⊗ β r ( b ) for all a ∈ A and b ∈ B . We therefore have an action α ⊗ β : P × S → End( A ⊗ max B )of P × S on A ⊗ max B by endomorphisms such that( α ⊗ β ) ( x,r ) = α x ⊗ β r for all ( x, r ) ∈ P × S. Moreover, it follows by the extendibility of the actions α and β that the action α ⊗ β on A ⊗ max B is actually given by extendible endomorphisms (see [15, Lemma 2.3]).Thus, we have a dynamical system ( A ⊗ max B, P × S, α ⊗ β ) for which we can talkabout the corresponding partial-isometric crossed product. We actually aim to showthat under some certain conditions we have the following isomorphism:( A ⊗ max B ) × piso α ⊗ β ( P × S ) ≃ ( A × piso α P ) ⊗ max ( B × piso β S ) . In fact, those conditions are to ensure that the Nica partial-isometric representationsof P and S are ∗ -commuting. Hence, we first need to assume that the (unital)semigroups P , P o , S , and S o are all left LCM. It thus turns out that all of them areboth left and right LCM semigroups. The other condition comes from the followingdefinition: Definition . Suppose that P and P o are both left LCM semigroups. A bicovariantpartial-isometric representation of P on a Hilbert space H is a Nica partial-isometricrepresentation V : P → B ( H ) which satisfies V r V ∗ r V s V ∗ s = ( V t V ∗ t if rP ∩ sP = tP ,0 if rP ∩ sP = ∅ .(5.2)Note the equation (5.2) is a kind of Nica covariance condition, too. So, to distinguishit from the covariance equation (3.1), we may view (3.1) as the right Nica covariancecondition and (5.2) as the left Nica covariance condition .Note that similar to (3.1), we can see that the equation (5.2) is also well-defined. Lemma 5.2.
Suppose that the (unital) semigroups P , P o , S , and S o are all leftLCM. Let V and W be bicovariant partial-isometric representations of P and S ona Hilbert space H , respectively, such that each V p ∗ -commutes with each W s for all p ∈ P and s ∈ S . Then, there exits a bicovariant partial-isometric representation U of P × S on H such that U ( p,s ) = V p W s . Moreover, every bicovariant partial-isometricrepresentation of P × S arises this way.Proof. Define a map U : P × S → B ( H ) by U ( p,s ) = V p W s for all ( p, s ) ∈ P × S . Since each V p ∗ -commutes with each W s , it follows that each U ( p,s ) is a partial isometry, as U ( p,s ) U ∗ ( p,s ) U ( p,s ) = V p W s [ V p W s ] ∗ V p W s = V p W s [ W s V p ] ∗ V p W s = V p W s V ∗ p W ∗ s V p W s = V p V ∗ p W s V p W ∗ s W s = V p V ∗ p V p W s W ∗ s W s = V p W s = U ( p,s ) . ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 21
Also, a simple computation shows that U ( p,s ) U ( q,t ) = U ( p,s )( q,t ) for every ( p, s ) and ( q, t ) in P × S . Thus, the map U is a (unital) semigroup homo-morphism (with partial-isometric values). Next, we want to show that it satisfies theNica covariance conditions (3.1) and (5.2), and hence, it is bicovariant. To see (3.1),we first have U ∗ ( p,s ) U ( p,s ) U ∗ ( q,t ) U ( q,t ) = [ V p W s ] ∗ V p W s [ V q W t ] ∗ V q W t = [ W s V p ] ∗ V p W s [ W t V q ] ∗ V q W t = V ∗ p W ∗ s V p W s V ∗ q W ∗ t V q W t = V ∗ p V p W ∗ s V ∗ q W s V q W ∗ t W t = V ∗ p V p V ∗ q W ∗ s V q W s W ∗ t W t = [ V ∗ p V p V ∗ q V q ][ W ∗ s W s W ∗ t W t ] . (5.3)If ( P × S )( p, s ) ∩ ( P × S )( q, t ) = ( P × S )( z, r ) = P z × Sr for some ( z, r ) ∈ P × S , then it follows by (5.1) (for the left hand side in above) that( P p ∩ P q ) × ( Ss ∩ St ) = P z × Sr.
Thus, we must have
P p ∩ P q = P z and Ss ∩ St = Sr, and hence, for (5.3), we get U ∗ ( p,s ) U ( p,s ) U ∗ ( q,t ) U ( q,t ) = [ V ∗ p V p V ∗ q V q ][ W ∗ s W s W ∗ t W t ]= V ∗ z V z W ∗ r W r = V ∗ z W ∗ r V z W r = [ W r V z ] ∗ U ( z,r ) = [ V z W r ] ∗ U ( z,r ) = U ∗ ( z,r ) U ( z,r ) . If ( P × S )( p, s ) ∩ ( P × S )( q, t ) = ∅ , then again, by (5.1), we get( P p ∩ P q ) × ( Ss ∩ St ) = ∅ . It follows that
P p ∩ P q = ∅ ∨ Ss ∩ St = ∅ , which implies that, V ∗ p V p V ∗ q V q = 0 ∨ W ∗ s W s W ∗ t W t = 0 . Thus, for (5.3), we have U ∗ ( p,s ) U ( p,s ) U ∗ ( q,t ) U ( q,t ) = [ V ∗ p V p V ∗ q V q ][ W ∗ s W s W ∗ t W t ] = 0 . A similar discussion shows that the representation U satisfies the Nica covariancecondition (5.2), too, namely, we have U ( p,r ) U ∗ ( p,r ) U ( q,s ) U ∗ ( q,s ) = ( U ( x,t ) U ∗ ( x,t ) if ( p, r )( P × S ) ∩ ( q, s )( P × S ) = ( x, t )( P × S ),0 if ( p, r )( P × S ) ∩ ( q, s )( P × S ) = ∅ .Therefore, U is a bicovariant partial-isometric representation of P × S on H satisfying U ( p,s ) = V p W s . Conversely, suppose that U is any bicovariant partial-isometric representation of P × S on a Hilbert space H . Define the maps V : P → B ( H ) and W : S → B ( H )by V p := U ( p,e S ) and W s := U ( e P ,s ) for all p ∈ P and s ∈ S , respectively. It is easy to see that each V p is a partial isometryas well as each W s , and the maps V and W are (unital) semigroup homomorphisms.Next, we show that the presentation V is bicovariant, and we skip the proof for thepresentation W as it follows similarly. To see that the presentation V satisfies theNica covariance condition (3.1), firstly, V ∗ p V p V ∗ q V q = U ∗ ( p,e S ) U ( p,e S ) U ∗ ( q,e S ) U ( q,e S ) . (5.4)Now, if P p ∩ P q = P z for some z ∈ P , then it follows by (5.1) that( P × S )( p, e S ) ∩ ( P × S )( q, e S ) = ( P p ∩ P q ) × ( S ∩ S )= P z × S = P z × Se S = ( P × S )( z, e S ) . Therefore, since U is bicovariant, for (5.4), we have V ∗ p V p V ∗ q V q = U ∗ ( p,e S ) U ( p,e S ) U ∗ ( q,e S ) U ( q,e S ) = U ∗ ( z,e S ) U ( z,e S ) = V ∗ z V z . If P p ∩ P q = ∅ , then it follows again by (5.1) that( P × S )( p, e S ) ∩ ( P × S )( q, e S ) = ( P p ∩ P q ) × ( S ∩ S ) = ∅ × S = ∅ . Therefore, for (5.4), we get V ∗ p V p V ∗ q V q = U ∗ ( p,e S ) U ( p,e S ) U ∗ ( q,e S ) U ( q,e S ) = 0 , as U is bicovariant. A similar discussion shows that the representation V satisfies theNica covariance condition (5.2), too. We skip it here. Finally, as we obviously have V p W s = W s V p = U ( p,s ) , (5.5)it is only left to show that V ∗ p W s = W s V ∗ p for all p ∈ P and s ∈ S . To do so, wefirst need to recall that the product vw of two partial isometries v and w is a partialisometry if and only if v ∗ v commutes with ww ∗ (see [10, Lemma 2]). This fact canbe applied to the partial isometries V p and W s due to (5.5). Now, we have V ∗ p W s = V ∗ p [ V p V ∗ p W s W ∗ s ] W s = U ∗ ( p,e S ) [ U ( p,e S ) U ∗ ( p,e S ) U ( e P ,s ) U ∗ ( e P ,s ) ] U ( e P ,s ) . (5.6)Since, ( p, e S )( P × S ) ∩ ( e P , s )( P × S ) = ( pP × S ) ∩ ( P × sS )= ( pP ∩ P ) × ( S ∩ sS )= pP × sS = ( p, s )( P × S ) , ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 23 it follows that V p V ∗ p W s W ∗ s = U ( p,e S ) U ∗ ( p,e S ) U ( e P ,s ) U ∗ ( e P ,s ) = U ( p,s ) U ∗ ( p,s ) = U ( p,e S ) U ( e P ,s ) [ U ( e P ,s ) U ( p,e S ) ] ∗ = U ( p,e S ) U ( e P ,s ) U ∗ ( p,e S ) U ∗ ( e P ,s ) = V p W s V ∗ p W ∗ s , (5.7)as U is bicovariant. Therefore, by (5.6) and (5.7), we get V ∗ p W s = V ∗ p [ V p V ∗ p W s W ∗ s ] W s = V ∗ p [ V p W s V ∗ p W ∗ s ] W s = [ V ∗ p V p W s W ∗ s ] W s V ∗ p [ V p V ∗ p W ∗ s W s ]= [ W s W ∗ s V ∗ p V p ] W s V ∗ p [ W ∗ s W s V p V ∗ p ]= W s W ∗ s V ∗ p [ V p W s V ∗ p W ∗ s ] W s V p V ∗ p ]= W s W ∗ s V ∗ p [ V p V ∗ p W s W ∗ s ] W s V p V ∗ p ] (by (5.7))= W s W ∗ s V ∗ p W s V p V ∗ p = W s [( V p W s ) ∗ ( V p W s )] V ∗ p = W s [ U ∗ ( p,s ) U ( p,s ) ] V ∗ p , and hence, V ∗ p W s = W s [ U ∗ ( p,s ) U ( p,s ) ] V ∗ p . (5.8)Moreover, since similarly( P × S )( e P , s ) ∩ ( P × S )( p, e S ) = ( P × S )( p, s ) , we have U ∗ ( e P ,s ) U ( e P ,s ) U ∗ ( p,e S ) U ( p,e S ) = U ∗ ( p,s ) U ( p,s ) , as U is bicovariant. By applying this to (5.8), we finally get V ∗ p W s = W s [ U ∗ ( e P ,s ) U ( e P ,s ) U ∗ ( p,e S ) U ( p,e S ) ] V ∗ p = [ W s W ∗ s W s ][ V ∗ p V p V ∗ p ]= W s V ∗ p . This completes the proof. (cid:3)
Definition . Let (
A, P, α ) be a dynamical system, in which P and P o are both leftLCM semigroups. The action α is called left-Nica covariant if it satisfies α x (1) α y (1) = ( α z (1) if xP ∩ yP = zP ,0 if xP ∩ yP = ∅ .(5.9)We should mention that the above definition is well-defined. This is due to thefact that if tP = xP ∩ yP = zP , then there is an invertible element u of P suchthat t = zu . Since u is invertible, α u becomes an automorphism of A , and hence α u (1) = 1. So, it follows that α t (1) = α zu (1) = α z ( α u (1)) = α z (1) . Remark . Let (
A, P, α ) be a dynamical system, in which P and P o are both leftLCM semigroups, and the action α is left Nica-covariant. If ( π, V ) is covariant partial-isometric of the system, then the representation V satisfies the left Nica-covariancecondition (5.2). One can easily see this by applying the equation V x V ∗ x = π ( α x (1))(see Lemma 4.2). Thus, the representation V is actually bicovariant. Let us also recall that for C ∗ -algebras A and B , there are nondegenerate homo-morphisms k A : A → M ( A ⊗ max B ) and k B : B → M ( A ⊗ max B )such that k A ( a ) k B ( b ) = k B ( b ) k A ( a ) = a ⊗ b for all a ∈ A and b ∈ B (see [20, Theorem B. 27]). Moreover, One can see that theextensions k A and k B of the nondegenerate homomorphisms k A and k B , respectively,have also commuting ranges. Therefore, there is a homomorphism k A ⊗ max k B : M ( A ) ⊗ max M ( B ) → M ( A ⊗ max B ) , which is the identity map on A ⊗ max B (see [15, Remark 2.2]). Theorem 5.5.
Suppose that the (unital) semigroups P , P o , S , and S o are all leftLCM. Let ( A, P, α ) and ( B, S, β ) be dynamical systems in which the actions α and β are both left Nica-covariant. Then, we have the following isomorphism: ( A ⊗ max B ) × piso α ⊗ β ( P × S ) ≃ ( A × piso α P ) ⊗ max ( B × piso β S ) . (5.10) Proof.
Let the triples ( A × piso α P, i A , i P ) and ( B × piso α S, i B , i S ) be the partial-isometriccrossed products of the dynamical systems ( A, P, α ) and (
B, S, β ), respectively. Sup-pose that ( k A × α P , k B × α S ) is the canonical pair of the algebras A × piso α P and B × piso α S into the multiplier algebra M (( A × piso α P ) ⊗ max ( B × piso β S )). Define the map j A ⊗ max B : A ⊗ max B → ( A × piso α P ) ⊗ max ( B × piso β S )by j A ⊗ max B := i A ⊗ max i B (see [20, Lemma B. 31]), and therefore, we have j A ⊗ max B ( a ⊗ b ) = ( i A ⊗ max i B )( a ⊗ b ) = i A ( a ) ⊗ i B ( b ) = k A × α P ( i A ( a )) k B × α S ( i B ( b ))for all a, b ∈ A . Also, define a map j P × S : P × S → M (( A × piso α P ) ⊗ max ( B × piso β S ))by j P × S ( x, t ) = k A × α P ⊗ max k B × α S ( i P ( x ) ⊗ i S ( t )) = k A × α P ( i P ( x )) k B × α S ( i S ( t ))for all ( x, t ) ∈ P × S . We claim that the triple (cid:0) ( A × piso α P ) ⊗ max ( B × piso β S ) , j A ⊗ max B , j P × S (cid:1) is a partial-isometric crossed product of the system ( A ⊗ max B, P × S, α ⊗ β ). To proveour claim, first note that, since the homomorphisms i A and i B are nondegenerate, sois the homomorphism j A ⊗ max B . Next, we show that the map j P × S is a bicovariantpartial-isometric representation of P × S . To do so, first note that, since the actions α and β are left Nica-covariant, the representations i P and i S are bicovariant (seeRemark 5.4). It follows that the maps j P : P → M (( A × piso α P ) ⊗ max ( B × piso β S ))and j S : S → M (( A × piso α P ) ⊗ max ( B × piso β S )) ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 25 given by compositions P i P −→ M ( A × piso α P ) k A × αP −→ M (( A × piso α P ) ⊗ max ( B × piso β S ))and S i S −→ M ( B × piso β S ) k B × βS −→ M (( A × piso α P ) ⊗ max ( B × piso β S )) , respectively, are bicovariant partial-isometric representations of P and S in the mul-tiplier algebra M (( A × piso α P ) ⊗ max ( B × piso β S )). Moreover, since the homomorphisms k A × α P and k B × α S have commuting ranges, each j P ( x ) ∗ -commutes with each j S ( t ).Therefore, as j P × S ( x, t ) = k A × α P ( i P ( x )) k B × α S ( i S ( t )) = j P ( x ) j S ( t ) , it follows by Lemma (5.2) that the map j P × S must be a bicovariant partial-isometricrepresentation of P × S . Moreover, by using the covariance equations of the pairs( i A , i P ) and ( i B , i S ), and the commutativity of the ranges of the homomorphisms k A × α P and k B × α S , one can see that the pair ( j A ⊗ max B , j P × S ) satisfies the covarianceequations j A ⊗ max B (( α ⊗ β ) ( x,t ) ( a ⊗ b )) = j P × S ( x, t ) j A ⊗ max B ( a ⊗ b ) j P × S ( x, t ) ∗ and j P × S ( x, t ) ∗ j P × S ( x, t ) j A ⊗ max B ( a ⊗ b ) = j A ⊗ max B ( a ⊗ b ) j P × S ( x, t ) ∗ j P × S ( x, t ) . Next, suppose that the pair ( π, U ) is covariant partial-isometric representation of( A ⊗ max B, P × S, α ⊗ β ) on a Hilbert space H . We want to get a nondegeneraterepresentation π × U of ( A × piso α P ) ⊗ max ( B × piso β S ) such that( π × U ) ◦ j A ⊗ max B = π and ( π × U ) ◦ j P × S = U. Let ( k A , k B ) be the canonical pair of the C ∗ -algebras A and B into the multiplieralgebra M ( A ⊗ max B ). The compositions A k A −→ M ( A ⊗ max B ) π −→ B ( H )and B k B −→ M ( A ⊗ max B ) π −→ B ( H )give us the nondegenerate representations π A and π B of A and B on H with commut-ing ranges, respectively. This is due to the fact that the ranges of i A and i B commute(see also [20, Corollary B. 22]). Also, define the maps V : P → B ( H ) and W : S → B ( H )by V x := U ( x,e S ) and W t := U ( e P ,t ) for all x ∈ P and t ∈ S , respectively. Since the representation U already satisfiesthe right Nica covariance condition (3.1), if we show that it satisfies the left Nicacovariance condition (5.2), too, then it is bicovariant. Therefore, it follows by Lemma5.2 that the maps V and W are bicovariant partial-isometric representations suchthat each V x ∗ -commutes with each W t for all x ∈ P and t ∈ S . By Remark 5.4, we only need to verify that the action α ⊗ β in the system ( A ⊗ max B, P × S, α ⊗ β ) isleft Nica-covariant. Firstly, we have( α ⊗ β ) ( x,t ) ◦ ( k A ⊗ max k B ) = α x ⊗ β t ◦ ( k A ⊗ max k B )= ( k A ◦ α x ) ⊗ max ( k B ◦ β t )(5.11)for all ( x, t ) ∈ P × S (see [15, Lemma 2.3]). It follows that( α ⊗ β ) ( x,r ) (1 M ( A ⊗ max B ) )( α ⊗ β ) ( y,s ) (1 M ( A ⊗ max B ) )= (cid:2) α x ⊗ β r (cid:0) k A (1 M ( A ) ) k B (1 M ( B ) ) (cid:1)(cid:3)(cid:2) α y ⊗ β s (cid:0) k A (1 M ( A ) ) k B (1 M ( B ) ) (cid:1)(cid:3) = k A (cid:0) α x (1 M ( A ) ) (cid:1) k B (cid:0) β r (1 M ( B ) ) (cid:1) k A (cid:0) α y (1 M ( A ) ) (cid:1) k B (cid:0) β s (1 M ( B ) ) (cid:1) = k A (cid:0) α x (1 M ( A ) ) (cid:1) k A (cid:0) α y (1 M ( A ) ) (cid:1) k B (cid:0) β r (1 M ( B ) ) (cid:1) k B (cid:0) β s (1 M ( B ) ) (cid:1) = k A (cid:0) α x (1 M ( A ) ) α y (1 M ( A ) ) (cid:1) k B (cid:0) β r (1 M ( B ) ) β s (1 M ( B ) ) (cid:1) , and hence, ( α ⊗ β ) ( x,r ) (1 M ( A ⊗ max B ) )( α ⊗ β ) ( y,s ) (1 M ( A ⊗ max B ) )= k A (cid:0) α x (1 M ( A ) ) α y (1 M ( A ) ) (cid:1) k B (cid:0) β r (1 M ( B ) ) β s (1 M ( B ) ) (cid:1) (5.12)for all ( x, r ) , ( y, s ) ∈ P × S . Now, if( x, r )( P × S ) ∩ ( y, s )( P × S ) = ( z, t )( P × S )for some ( z, t ) ∈ P × S , then, since xP ∩ yP = zP and rS ∩ sS = tS, and the actions α and β are left Nica-covariant, for (5.12), we get( α ⊗ β ) ( x,r ) (1 M ( A ⊗ max B ) )( α ⊗ β ) ( y,s ) (1 M ( A ⊗ max B ) )= k A (cid:0) α z (1 M ( A ) ) (cid:1) k B (cid:0) β t (1 M ( B ) ) (cid:1) = ( k A ◦ α z ) ⊗ max ( k B ◦ β t )(1 M ( A ) ⊗ M ( B ) )= α z ⊗ β t (cid:0) k A ⊗ max k B (1 M ( A ) ⊗ M ( B ) ) (cid:1) [by (5.11)]= ( α ⊗ β ) ( z,t ) (1 M ( A ⊗ max B ) ) . If ( x, r )( P × S ) ∩ ( y, s )( P × S ) = ∅ , then xP ∩ yP = ∅ ∨ rS ∩ sS = ∅ , which implies that α x (1 M ( A ) ) α y (1 M ( A ) ) = 0 ∨ β r (1 M ( B ) ) β s (1 M ( B ) ) = 0as α and β are left Nica-covariant. Thus, for (5.12), we get( α ⊗ β ) ( x,r ) (1 M ( A ⊗ max B ) )( α ⊗ β ) ( y,s ) (1 M ( A ⊗ max B ) ) = 0 . So, the action α ⊗ β is left Nica-covariant.Now, consider the pairs ( π A , V ) and ( π B , W ). They are indeed the covariant partial-isometric representations of the systems ( A, P, α ) and (
B, S, β ) on H , respectively. ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 27
We only show this for ( π A , V ) as the proof for ( π B , W ) follows similarly. We have toshow that the pair ( π A , V ) satisfies the covariance equations (4.1). We have V x π A ( a ) V ∗ x = U ( x,e S ) π ( k A ( a )) U ∗ ( x,e S ) = U ( x,e S ) π ( k A ( a )) π (1 M ( A ⊗ max B ) ) U ∗ ( x,e S ) = U ( x,e S ) π ( k A ( a )) π ( k B (1 M ( B ) )) U ∗ ( x,e S ) = U ( x,e S ) π ( k A ( a ) k B (1 M ( B ) )) U ∗ ( x,e S ) = π (cid:0) ( α ⊗ β ) ( x,e S ) ( k A ( a ) k B (1 M ( B ) )) (cid:1) [by the covariance of ( π, U )]= π (cid:0) α x ⊗ β e S ( k A ( a ) k B (1 M ( B ) )) (cid:1) = π (cid:0) k A ( α x ( a )) k B ( β e S (1 M ( B ) )) (cid:1) [by (5.11)]= π (cid:0) k A ( α x ( a )) k B (id M ( B ) (1 M ( B ) )) (cid:1) = π (cid:0) k A ( α x ( a )) k B (1 M ( B ) ) (cid:1) = π (cid:0) k A ( α x ( a ))1 M ( A ⊗ max B ) (cid:1) = π (cid:0) k A ( α x ( a )) (cid:1) = π A ( α x ( a ))for all a ∈ A and x ∈ P . Also, by a similar calculation using the covariance of thepair ( π, U ), it follows that π A ( a ) V ∗ x V x = V ∗ x V x π A ( a ) . Consequently, there are nondegenerate representations π A × V and π B × W of thealgebras A × piso α P and B × piso β S on H , respectively, such that( π A × V ) ◦ i A = π A , π A × V ◦ i P = V and ( π B × W ) ◦ i B = π B , π B × W ◦ i S = W. Next, we aim to show that the representations π A × V and π B × W have commutingranges, from which, it follows that there is a representation ( π A × V ) ⊗ max ( π B × W )of ( A × piso α P ) ⊗ max ( B × piso β S ), which is the desired (nondegenerate) representation π × U . So, it suffices to see that the pairs ( π A , π B ), ( V, W ), ( V ∗ , W ), ( π A , W ), and( π B , V ) have commuting ranges. We already saw that this is indeed true for the firstthree pairs. So, we compute to show that this is also true for the pair ( π A , W ) andskip the similar computation for the pair ( π B , V ). We have W t π A ( a ) = U ( e P ,t ) π ( k A ( a ))= U ( e P ,t ) π ( k A ( a )1 M ( A ⊗ max B ) )= U ( e P ,t ) π ( k A ( a ) k B (1 M ( B ) ))= π (cid:0) ( α ⊗ β ) ( e P ,t ) ( k A ( a ) k B (1 M ( B ) )) (cid:1) U ( e P ,t ) [by the covariance of ( π, U )]= π (cid:0) α e P ⊗ β t ( k A ( a ) k B (1 M ( B ) )) (cid:1) U ( e P ,t ) = π (cid:0) k A ( α e P ( a )) k B ( β t (1 M ( B ) )) (cid:1) U ( e P ,t ) [by (5.11)]= π (cid:0) k A (id A ( a )) k B ( β t (1 M ( B ) )) (cid:1) U ( e P ,t ) = ( π ◦ k A )( a )( π ◦ k B ) (cid:0) β t (1 M ( B ) ) (cid:1) U ( e P ,t ) = π A ( a ) π B (cid:0) β t (1 M ( B ) ) (cid:1) W t = π A ( a ) W t W ∗ t W t [by the covariance of ( π B , W )]= π A ( a ) W t . Thus, there is a representation ( π A × V ) ⊗ max ( π B × W ) of ( A × piso α P ) ⊗ max ( B × piso β S )on H such that( π A × V ) ⊗ max ( π B × W )( ξ ⊗ η ) = ( π A × V )( ξ )( π B × W )( η ) for all ξ ∈ ( A × piso α P ) and η ∈ ( B × piso β S ). Let π × U = ( π A × V ) ⊗ max ( π B × W ) , which is nondegenerate as both representations π A × V and π B × W are. Then, wehave π × U ( j A ⊗ max B ( a ⊗ b )) = π × U ( i A ( a ) ⊗ i B ( b ))= ( π A × V )( i A ( a ))( π B × W )( i B ( b ))= π A ( a ) π B ( b )= π ( k A ( a )) π ( k B ( b ))= π ( k A ( a ) k B ( b )) = π ( a ⊗ b ) . To see ( π × U ) ◦ j P × S = U , we apply the equation π × U ◦ ( k A × α P ⊗ max k B × α S ) = ( π A × V ) ⊗ max ( π B × W ) ◦ ( k A × α P ⊗ max k B × α S )= ( π A × V ) ⊗ max ( π B × W ) , which is valid by [15, Lemma 2.4]. Therefore, we have π × U ( j P × S ( x, t )) = π × U (cid:0) k A × α P ( i P ( x )) k B × α S ( i S ( t )) (cid:1) = π × U (cid:0) k A × α P ⊗ max k B × α S ( i P ( x ) ⊗ i S ( t )) (cid:1) = π × U ◦ ( k A × α P ⊗ max k B × α S )( i P ( x ) ⊗ i S ( t ))= ( π A × V ) ⊗ max ( π B × W )( i P ( x ) ⊗ i S ( t ))= ( π A × V )( i P ( x ))( π B × W )( i S ( t ))= V x W t = U ( x,e S ) U ( e P ,t ) = U ( x,t ) . Finally, as the algebras A × piso α P and B × piso α S are spanned by the elements i P ( x ) ∗ i A ( a ) i P ( y )and i S ( r ) ∗ i B ( b ) i S ( t ), respectively, the algebra ( A × piso α P ) ⊗ max ( B × piso α S ) is spannedby the elements[ i P ( x ) ∗ i A ( a ) i P ( y )] ⊗ [ i S ( r ) ∗ i B ( b ) i S ( t )]= k A × α P (cid:0) i P ( x ) ∗ i A ( a ) i P ( y ) (cid:1) k B × α S (cid:0) i S ( r ) ∗ i B ( b ) i S ( t ) (cid:1) = k A × α P ( i P ( x ) ∗ ) k A × α P ( i A ( a )) k A × α P ( i P ( y )) k B × α S ( i S ( r ) ∗ ) k B × α S ( i B ( b )) k B × α S ( i S ( t ))= k A × α P ( i P ( x ) ∗ ) k A × α P ( i A ( a )) k B × α S ( i S ( r ) ∗ ) k A × α P ( i P ( y )) k B × α S ( i B ( b )) k B × α S ( i S ( t ))= k A × α P ( i P ( x ) ∗ ) k B × α S ( i S ( r ) ∗ ) k A × α P ( i A ( a )) k B × α S ( i B ( b )) k A × α P ( i P ( y )) k B × α S ( i S ( t ))= k A × α P ( i P ( x )) ∗ k B × α S ( i S ( r )) ∗ j A ⊗ max B ( a ⊗ b ) j P × S ( y, t )= (cid:2) k B × α S ( i S ( r )) k A × α P ( i P ( x )) (cid:3) ∗ j A ⊗ max B ( a ⊗ b ) j P × S ( y, t )= (cid:2) k A × α P ( i P ( x )) k B × α S ( i S ( r )) (cid:3) ∗ j A ⊗ max B ( a ⊗ b ) j P × S ( y, t )= j P × S ( x, r ) ∗ j A ⊗ max B ( a ⊗ b ) j P × S ( y, t ) . So, the triple (cid:0) ( A × piso α P ) ⊗ max ( B × piso β S ) , j A ⊗ max B , j P × S (cid:1) is a partial-isometric crossed product of the system ( A ⊗ max B, P × S, α ⊗ β ). It thusfollows that there is an isomorphismΓ : (cid:0) ( A ⊗ max B ) × piso α ⊗ β ( P × S ) , i A ⊗ max B , i P × S (cid:1) → ( A × piso α P ) ⊗ max ( B × piso β S )such thatΓ (cid:0) i P × S ( x, r ) ∗ i A ⊗ max B ( a ⊗ b ) i P × S ( y, t ) (cid:1) = j P × S ( x, r ) ∗ j A ⊗ max B ( a ⊗ b ) j P × S ( y, t )= [ i P ( x ) ∗ i A ( a ) i P ( y )] ⊗ [ i S ( r ) ∗ i B ( b ) i S ( t )] . This completes the proof. (cid:3)
ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 29
Let P be a unital semigroup such that itself and the opposite semigroup P o areboth left LCM. For every y ∈ P , define a map 1 y : P → C by1 y ( x ) = ( x ∈ yP ,0 otherwise,which is the characteristic function of yP . Each 1 y is obviously a function in ℓ ∞ ( P ).Then, since P is right LCM, one can see that we have1 x y = ( z if xP ∩ yP = zP ,0 xP ∩ yP = ∅ .Note that, if ˜ zP = xP ∩ yP = zP , then there is an invertible element u of P suchthat ˜ z = zu . It therefore follows that s ∈ zP if and only if s ∈ ˜ zP for all s ∈ P , whichimplies that we must have 1 z = 1 ˜ z . So, the above equation is well-defined. Also, weclearly have 1 ∗ y = 1 y for all y ∈ P . Therefore, if B P is the C ∗ -subalgebra of ℓ ∞ ( P )generated by the characteristic functions { y : y ∈ P } , then we have B P = span { y : y ∈ P } . Note that the algebra B P is abelian and unital, whose unit element is 1 e which is aconstant function on P with the constant value 1. One can see that, in fact, 1 u = 1 e forevery u ∈ P ∗ . In addition, the shift on ℓ ∞ ( P ) induces an action on B P by injectiveendomorphisms. More precisely, for every x ∈ P , the map α x : ℓ ∞ ( P ) → ℓ ∞ ( P )defined by α x ( f )( t ) = ( f ( r ) if t = xr for some r ∈ P ( ≡ t ∈ xP ),0 otherwise,for every f ∈ ℓ ∞ ( P ) is an injective endomorphism of ℓ ∞ ( P ). Also, the map α : P → End( ℓ ∞ ( P )); x α x is a semigroup homomorphism such that α e = id, which gives us an action of P on ℓ ∞ ( P ) by injective endomorphisms. Since α x (1 y ) = 1 xy for all x, y ∈ P , α x ( B P ) ⊂ B P ,and therefore the restriction of the action α to B P gives an action τ : P → End( B P )by injective endomorphisms such that τ x (1 y ) = 1 xy for all x, y ∈ P . Note that τ x (1 e ) = 1 x = 1 e for all x ∈ P \ P ∗ . Consequently, we obtain a dynamical system( B P , P, τ ), for which, we want to describe the corresponding partial-isometric crossedproduct ( B P × piso τ P, i B P , i P ). More precisely, we want to show that the algebra B P × piso τ P is universal for bicovariant partial-isometric representations of P . Once,we have done this, it would be proper to denote B P × piso τ P by C ∗ bicov ( P ). So, thisactually generalizes [8, Proposition 9.6] from the positive cones of quasi lattice-orderedgroups (in the sense of Nica [17]) to LCM semigroups.To start, for our purpose, we borrow some notations from quasi lattice-orderedgroups. So, for every x, y ∈ P , if xP ∩ yP = zP for some z ∈ P , which means that z is a least common right multiple of x and y , then we denote such an element z by x ∨ lt y , which may not be unique. If xP ∩ yP = ∅ , then we denote x ∨ lt y = ∞ . Notethat we are using the notation ∨ lt to indicate that we are treating P as a right LCM semigroup. But if we are treating P as a left LCM semigroup, then we use the notation ∨ rt to distinguish it from ∨ lt . Moreover, if F = { x , x , ..., x n } is any finite subset of P , then σF is written for x ∨ lt x ∨ lt ... ∨ lt x n . Therefore, if T x ∈ F xP = T ni =1 x i P = ∅ , σF denotes an element in { y : n \ i =1 x i P = yP } , and if T ni =1 x i P = ∅ , then σF = ∞ . Lemma 5.6.
Let P be a unital semigroup such that itself and the opposite semigroup P o are both left LCM. Let V be any bicovariant partial-isometric representation of P on a Hilbert space H . Then: (i) there is a (unital) representation π V of B P on H such that π V (1 x ) = V x V ∗ x for all x ∈ P ; (ii) the pair ( π V , V ) is a covariant partial-isometric representation of ( B P , P, τ ) on H .Proof. We prove (i) by making some adjustment to the proof of [14, Proposition 1.3(2)] for the particular family { L x := V x V ∗ x : x ∈ P } of projections, which satifies L e = 1 and L x L y = L x ∨ lt y . Note that L ∞ is defined to be zero (projection). Now, define a map π : span { x : x ∈ P } → B ( H )by π (cid:0) n X i =1 λ x i x i (cid:1) = n X i =1 λ x i L x i = n X i =1 λ x i V x i V ∗ x i , where λ x i ∈ C for each i . It is obvious that π is linear. Next, we show that (cid:13)(cid:13) X x ∈ F λ x L x (cid:13)(cid:13) ≤ (cid:13)(cid:13) X x ∈ F λ x x (cid:13)(cid:13) (5.13)for any finite subset F of P . So, it follows that the map π is a well-defined boundedlinear map, and therefore, it extends to a bounded linear map of B p in B ( H ) suchthat 1 x V x V ∗ x for all x ∈ P . To see (5.13), we exactly follow [14, Lemma 1.4] toobtain an expression for the norm of the forms P x ∈ F λ x L x by using an appropriateset of mutually orthogonal projections. So, if F is any finite subset of P , then forevery nonempty proper subset A of F , take Q LA = Π x ∈ F \ A ( L σA − L σA ∨ lt x ). Moreover,let Q L ∅ = Π x ∈ F (1 − L x ) and Q LF = Π x ∈ F L x = L σF . Then, exactly by following theproof of [14, Lemma 1.4], we can show that { Q LA : A ⊂ F } is a decomposition of theidentity into mutually orthogonal projections, such that X x ∈ F λ x L x = X A ⊂ F (cid:18) X x ∈ A λ x (cid:19) Q LA (5.14) ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 31 and (cid:13)(cid:13) X x ∈ F λ x L x (cid:13)(cid:13) = max (cid:26)(cid:12)(cid:12) X x ∈ A λ x (cid:12)(cid:12) : A ⊂ F and Q LA = 0 (cid:27) . (5.15)Also, we have a fact similar to [14, Remark 1.5]. Hence, suppose that, similarly, { Q A : A ⊂ F } is the decomposition of the identity corresponding to the family ofprojections { x : x ∈ F } . Consider Q A = Π x ∈ F \ A (1 σA − σA ∨ lt x )for any nonempty proper subset A ⊂ F . If σA ∈ x P for some x ∈ F \ A , then( σA ) P = T y ∈ A yP ⊂ x P which implies that ( σA ) P ∩ x P = ( σA ) P . So, we have σA ∨ lt x = σA , and therefore,1 σA − σA ∨ lt x = 1 σA − σA = 0 . Thus, we get Q A = 0. Note that when we say σA ∈ x P (for some x ∈ F \ A ), itmeans that at least one element in { z : \ y ∈ A yP = zP } (5.16)belongs to x P , from which, it follows that all elements in (5.16) must belong to x P . This is due to the fact that if z, ˜ z are in (5.16), then ˜ z = zu for some invertibleelement u of P . Now, conversely, suppose that0 = Q A = Π x ∈ F \ A (1 σA − σA ∨ lt x ) . This implies that we must have Q A ( r ) = 0 for all r ∈ P , in particular when r = σA ,and hence,0 = Q A ( r ) = Π x ∈ F \ A (cid:0) r ( r ) − r ∨ lt x ( r ) (cid:1) = Π x ∈ F \ A (cid:0) − r ∨ lt x ( r ) (cid:1) . Therefore, there is at least one element x ∈ F \ A such that 1 r ∨ lt x ( r ) = 1, whichimplies that we must have r ∈ ( r ∨ lt x ) P = rP ∩ x P . It follows that σA = r ∈ x P (and hence, r ∨ lt x = σA ∨ lt x = σA ). Consequently, we have Q A = 0 if and only if A = { x ∈ F : σA ∈ xP } . Eventually, we conclude that if Q LA = 0, then Q A = 0. This is due to the fact that, if Q A = 0, then there is x ∈ F \ A such that σA ∈ x P . Therefore, we get Q LA = 0 asthe factor L σA − L σA ∨ lt x in Q LA becomes zero. Thus, it follows that (cid:26)(cid:12)(cid:12) X x ∈ A λ x (cid:12)(cid:12) : A ⊂ F and Q LA = 0 (cid:27) ⊂ (cid:26)(cid:12)(cid:12) X x ∈ A λ x (cid:12)(cid:12) : A ⊂ F and Q A = 0 (cid:27) , which implies that the inequality (5.13) is valid for any finite subset F of P . So,we have a bounded linear map π V : B p → B ( H ) (the extension of π ) such that π V (1 x ) = V x V ∗ x for all x ∈ P . Furthermore, since π V (1 x ) π V (1 y ) = V x V ∗ x V y V ∗ y = V x ∨ lt y V ∗ x ∨ lt y = π V (1 x ∨ lt y ) = π V (1 x y ) , and obviously, π V (1 x ) ∗ = π V (1 ∗ x ) = π V (1 x ), it follows that the map π V is actually a ∗ -homomorphism, which is clearly unital. This completes the proof of (i). To see (ii), it is enough to show that the pair ( π V , V ) satisfies the covarianceequations (4.1) on the spanning elements of B P . For all x, y ∈ P , we have π V ( τ x (1 y )) = 1 xy = V xy V ∗ xy = V x V y [ V x V y ] ∗ = V x V y V ∗ y V ∗ x = V x π V (1 y ) V ∗ x . Also, since the product of partial isometries V x and V y is a partial isometry, namely, V x V y = V xy , by [10, Lemma 2], each V ∗ x V x commutes with each V y V ∗ y . Hence, we have V ∗ x V x π V (1 y ) = V ∗ x V x V y V ∗ y = V y V ∗ y V ∗ x V x = π V (1 y ) V ∗ x V x . So, we are done with (ii), too. (cid:3)
Proposition 5.7.
Suppose that P is a unital semigroup such that itself and the op-posite semigroup P o are both left LCM. Then, the map i P : P → B P × piso τ P is a bicovariant partial-isometric representation of P whose range generates the C ∗ -algebra B P × piso τ P . Moreover, for every bicovariant partial-isometric representation V of P , there is a (unital) representation V ∗ of B P × piso τ P such that V ∗ ◦ i P = V .Proof. To see that i P is a bicovariant partial-isometric representation of P , we onlyneed to show that it satisfies the left Nica covariance condition (5.2). Since i B P (1 y ) = i B P ( τ y (1 e )) = i P ( y ) i B P (1 e ) i P ( y ) ∗ = i P ( y ) i P ( y ) ∗ for all y ∈ P , it follows that i P indeed satisfies (5.2). Then, as the elements { y : y ∈ P } generate (span) the algebra B P , the C ∗ -algebra B P × piso τ P is generated by theelements i B P (1 y ) i P ( x ) = i P ( y ) i P ( y ) ∗ i P ( x ) , which implies that i P ( P ) generates B P × piso τ P .Suppose that now V is a bicovariant partial-isometric representation of P on aHilbert space H . Then, by Lemma 5.6, there is a covariant partial-isometric repre-sentation ( π V , V ) of ( B P , P, τ ) on H , such that π V (1 x ) = V x V ∗ x for all x ∈ P . Thecorresponding (unital) representation π V × V of ( B P × piso τ P, i B P , i P ) on H is thedesired representation V ∗ which satisfies V ∗ ◦ i P = V . (cid:3) So, as we mentioned earlier, we denote the algebra B P × piso τ P by C ∗ bicov ( P ), whichis universal for bicovariant partial-isometric representations of P . Corollary 5.8.
Suppose that the (unital) semigroups P , P o , S , and S o are all leftLCM. Then, C ∗ bicov ( P × S ) ≃ C ∗ bicov ( P ) ⊗ max C ∗ bicov ( S ) . (5.17) Proof.
Corresponding to the pairs (
P, P o ) and ( S, S o ) we have the dynamical systems( B P , P, τ ) and ( B S , S, β ) along with their associated C ∗ -algebras (cid:0) C ∗ bicov ( P ) = B P × piso τ P, i B P , V (cid:1) and (cid:0) C ∗ bicov ( S ) = B S × piso β S, i B S , W (cid:1) , ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 33 respectively. Now, by Theorem 5.5, there is an isomorphismΓ : (cid:0) ( B P ⊗ max B S ) × piso τ ⊗ β ( P × S ) , i B P ⊗ max B S , T (cid:1) → ( B P × piso τ P ) ⊗ max ( B S × piso β S )such that Γ (cid:0) i B P ⊗ max B S (1 x ⊗ t ) T ( p,s ) (cid:1) = [ i B P (1 x ) V p ] ⊗ [ i B S (1 t ) W s ]for all x, p ∈ P and t, s ∈ S . On the other hand, as the (unital) semigroups P × S and ( P × S ) o are both left LCM, we have the dynamical system ( B ( P × S ) , P × S, α )along with its associated C ∗ -algebra (cid:0) C ∗ bicov ( P × S ) = B ( P × S ) × piso α ( P × S ) , i B ( P × S ) , U (cid:1) , where the action α : P × S → End( B ( P × S ) ) is induced by the shift on ℓ ∞ ( P × S ) suchthat α ( p,s ) (1 ( x,t ) ) = 1 ( p,s )( x,t ) = 1 ( px,st ) . Moreover, since there is an isomorphism ψ : (1 x ⊗ t ) ∈ ( B P ⊗ max B S ) ( x,t ) ∈ B ( P × S ) such that ψ ◦ ( τ ⊗ β ) = α ◦ ψ , there is an isomorphismΛ : ( B P ⊗ max B S ) × piso τ ⊗ β ( P × S ) → B ( P × S ) × piso α ( P × S )such that Λ ◦ i B P ⊗ max B S = i B ( P × S ) ◦ ψ and Λ ◦ T = U (see Remark 6.1) . Eventually, the composition C ∗ bicov ( P × S ) Λ − −→ ( B P ⊗ max B S ) × piso τ ⊗ β ( P × S ) Γ −→ C ∗ bicov ( P ) ⊗ max C ∗ bicov ( S )of isomorphisms gives the desired isomorphism (5.17) such that U ( p,s ) V p ⊗ W s for all ( p, s ) ∈ P × S . (cid:3) Ideals in tensor products
Suppose that P is a left LCM semigroup. Let α and β be the actions of P on C ∗ -algebras A and B by extendible endomorphisms, respectively. Then, there is anaction α ⊗ β : P → End( A ⊗ max B )of P on the maximal tensor product A ⊗ max B by extendible endomorphisms suchthat ( α ⊗ β ) x = α x ⊗ β x for all x ∈ P. Note that the extendibility of α ⊗ β follows by the extendibility of the actions α and β (see [15, Lemma 2.3]). Therefore, we obtain a dynamical system ( A ⊗ max B, P, α ⊗ β ).Let ( A ⊗ max B ) × piso α ⊗ β P be the partial-isometric crossed product of ( A ⊗ max B, P, α ⊗ β ).Our main goal in this section is to obtain a composition series0 ≤ I ≤ I ≤ ( A ⊗ max B ) × piso α ⊗ β P of ideals, and then identify the subquotients I , I / I , and (( A ⊗ max B ) × piso α ⊗ β P ) / I with familiar terms. To do so, let us first mention some point in the following remarkwhich is required. Remark . Let (
A, P, α ) and (
B, P, β ) be dynamical systems, and ψ : A → B anondegenerate homomorphism such that ψ ◦ α x = β x ◦ ψ for all x ∈ P . Suppose that( A × piso α P, i ) and ( B × piso β P, j ) are the partial-isometric crossed products of the systems(
A, P, α ) and (
B, P, β ), respectively. Then, one can see that the pair ( j B ◦ ψ, j P ) iscovariant partial-isometric representation of ( A, P, α ) in the algebra B × piso β P . Hence,there is a nondegenerate homomorphism ψ × P := [( j B ◦ ψ ) × j P ] : A × piso α P → B × piso β P such that ( ψ × P ) ◦ i A = j B ◦ ψ and ψ × P ◦ i P = j P . One can see that if ψ is an isomorphism, so is ψ × P . Lemma 6.2. [15, Lemma 3.2]
Suppose that α and β are extendible endomorphismsof C ∗ -algebras A and B , respectively. If I is an extendible α -invariant of A and J isan extendible β -invariant ideal of B , then the ideal I ⊗ max J of A ⊗ max B is extendible α ⊗ β -invariant.Remark . It follows by the above lemma that if (
A, P, α ) and (
B, P, β ) are dy-namical systems, and I is an extendible α x -invariant of A and J is an extendible β x -invariant ideal of B for every x ∈ P , then I ⊗ max J is an extendible ( α ⊗ β ) x -invariant ideal of A ⊗ max B for all x ∈ P . Therefore, by Theorem 4.7, the crossedproduct ( I ⊗ max J ) × piso α ⊗ β P sits in the algebra ( A ⊗ max B ) × piso α ⊗ β P as an ideal (this willbe the ideal I shortly later). As an application, we observe that, by [20, PropositionB. 30], the short exact sequence0 −→ J −→ B q J −→ B/J −→ −→ A ⊗ max J −→ A ⊗ max B id A ⊗ max q J −→ A ⊗ max B/J −→ , (6.1)where A ⊗ max J is an extendible ( α ⊗ β ) x -invariant ideal of A ⊗ max B for all x ∈ P .Thus, (6.1) itself by Theorem 4.7 gives rise to the following short exact sequence:0 −→ ( A ⊗ max J ) × piso α ⊗ β P µ −→ (cid:0) ( A ⊗ max B ) × piso α ⊗ β P, i (cid:1) φ −→ (cid:0) ( A ⊗ max B/J ) × piso α ⊗ ˜ β P, j (cid:1) , where ˜ β : P → End(
B/J ) is the (extendible) action induced by β . More precisely,by Remark 6.1, the isomorphism( A ⊗ max B ) / ( A ⊗ max J ) ≃ A ⊗ max B/J, which intertwines the actions ^ α ⊗ β and α ⊗ ˜ β of the algebras ( A ⊗ max B ) / ( A ⊗ max J )and A ⊗ max B/J , respectively, induces an isomorphism[( A ⊗ max B ) / ( A ⊗ max J )] × piso ] α ⊗ β P ≃ ( A ⊗ max B/J ) × piso α ⊗ ˜ β P. Therefore, the surjective homomorphism φ is actually given by (id A ⊗ max q J ) × P suchthat[(id A ⊗ max q J ) × P ] ◦ i A ⊗ max B = j A ⊗ max B/J ◦ (id A ⊗ max q J ) and [(id A ⊗ max q J ) × P ] ◦ i P = j P . ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 35
In the following proposition and theorem, for the maximal tensor product betweenthe C ∗ -algebras involved, we simply write ⊗ for convenience. Proposition 6.4.
Let ( A, P, α ) and ( B, P, β ) be dynamical systems, and I an ex-tendible α x -invariant of A and J an extendible β x -invariant ideal of B for every x ∈ P . Assume that ˜ α : P → End(
A/I ) and ˜ β : P → End(
B/J ) are the actionsinduced by α and β , respectively. Then, the following diagram (6.2) 0 0 00 ( I ⊗ J ) × piso α ⊗ β P ( I ⊗ B ) × piso α ⊗ β P ( I ⊗ B/J ) × piso α ⊗ ˜ β P
00 ( A ⊗ J ) × piso α ⊗ β P ( A ⊗ B ) × piso α ⊗ β P ( A ⊗ B/J ) × piso α ⊗ ˜ β P
00 (
A/I ⊗ J ) × piso˜ α ⊗ β P ( A/I ⊗ B ) × piso˜ α ⊗ β P ( A/I ⊗ B/J ) × piso˜ α ⊗ ˜ β P
00 0 0 φ ϕ q × Pφ ϕ ϕ φ commutes, where φ := (id I ⊗ q J ) × P, φ := (id A ⊗ q J ) × P, φ := (id A/I ⊗ q J ) × P,ϕ := ( q I ⊗ id J ) × P, ϕ := ( q I ⊗ id B ) × P, and ϕ := ( q I ⊗ id B/J ) × P. Also, there is a surjective homomorphism q : A ⊗ B → ( A/I ) ⊗ ( B/J ) which inter-twines the actions α ⊗ β and ˜ α ⊗ ˜ β , and therefore, we have a homomorphism q × P of ( A ⊗ B ) × piso α ⊗ β P onto ( A/I ⊗ B/J ) × piso˜ α ⊗ ˜ β P induced by q . Moreover, we have ker( q × P ) = ( A ⊗ J ) × piso α ⊗ β P + ( I ⊗ B ) × piso α ⊗ β P. (6.3) Proof.
First of all, in the diagram, each row as well as each column is obtained bythe similar discussion to Remark 6.3, and hence, it is exact.Next, for the quotient maps q I : A → A/I and q J : B → B/J , by [20, Lemma B.31], there is a homomorphism q I ⊗ q J : A ⊗ B → ( A/I ) ⊗ ( B/J ), which we denote itby q , such that q ( a ⊗ b ) = ( q I ⊗ q J )( a ⊗ b ) = q I ( a ) ⊗ q J ( b ) = ( a + I ) ⊗ ( b + J ) for all a ∈ A and b ∈ B . It is obviously surjective. Moreover, q (( α ⊗ β ) x ( a ⊗ b )) = q (( α x ⊗ β x )( a ⊗ b ))= q ( α x ( a ) ⊗ β x ( b ))= ( α x ( a ) + I ) ⊗ ( β x ( b ) + J )= ˜ α x ( a + I ) ⊗ ˜ β x ( b + J )= ( ˜ α x ⊗ ˜ β x )(( a + I ) ⊗ ( b + J ))= ( ˜ α ⊗ ˜ β ) x ( q ( a ⊗ b ))(6.4)for all x ∈ P . Therefore, by Remark 6.1, there is a (nondegenerate) homomorphism q × P : (cid:0) ( A ⊗ B ) × piso α ⊗ β P, i (cid:1) → (cid:0) ( A/I ⊗ B/J ) × piso˜ α ⊗ ˜ β P, k (cid:1) such that ( q × P ) ◦ i A ⊗ B = k A/I ⊗ B/J ◦ q and q × P ◦ i P = k P . One can easily see that as q is surjective, so is q × P .Now, by inspection on spanning elements, it follows that the diagram commutes.Finally, to see (6.3), we only show thatker( q × P ) ⊂ ( A ⊗ J ) × piso α ⊗ β P + ( I ⊗ B ) × piso α ⊗ β P as the other inclusion can be verified easily. To do so, take a nondegenerate repre-sentation π : ( A ⊗ B ) × piso α ⊗ β P → B ( H )with ker π = ( A ⊗ J ) × piso α ⊗ β P + ( I ⊗ B ) × piso α ⊗ β P. Then, define a map ρ : ( A/I ⊗ B/J ) → B ( H ) by ρ ( q ( ξ )) = π ( i A ⊗ B ( ξ ))for all ξ ∈ ( A ⊗ B ). Since( A ⊗ J ) + ( I ⊗ B ) = ker q ⊂ ker( π ◦ i A ⊗ B ) , it follows that the map ρ is well-defined, which is actually a nondegenerate represen-tation. Also, the composition P i P −→ M (cid:0) ( A ⊗ B ) × piso α ⊗ β P (cid:1) π −→ B ( H )gives a (right) Nica partial-isometric representation W : P → B ( H ). Now, by ap-plying the covariance equations of the pair ( i A ⊗ B , i P ) and (6.4), one can see that thepair ( ρ, W ) is a covariant partial-isometric representation of ( A/I ⊗ B/J, P, ˜ α ⊗ ˜ β )on H . The corresponding representation ρ × W lifts to π , which means that( ρ × W ) ◦ ( q × P ) = π, from which, it follows thatker( q × P ) ⊂ ker π = ( A ⊗ J ) × piso α ⊗ β P + ( I ⊗ B ) × piso α ⊗ β P. Thus, the equation (6.3) holds. (cid:3)
ROSSED PRODUCTS BY LEFT LCM SEMIGROUPS 37
Theorem 6.5.
Let ( A, P, α ) and ( B, P, β ) be dynamical systems, and I an extendible α x -invariant of A and J an extendible β x -invariant ideal of B for every x ∈ P .Assume that ˜ α : P → End(
A/I ) and ˜ β : P → End(
B/J ) are the actions induced by α and β , respectively. Then, there is a composition series ≤ I ≤ I ≤ ( A ⊗ B ) × piso α ⊗ β P of ideals, such that: (i) the ideal I is (isomorphic to) ( I ⊗ J ) × piso α ⊗ β P ; (ii) I / I ≃ ( A/I ⊗ J ) × piso˜ α ⊗ β P ⊕ ( I ⊗ B/J ) × piso α ⊗ ˜ β P ; (iii) the surjection q × P induces an isomorphism of (cid:0) ( A ⊗ B ) × piso α ⊗ β P (cid:1) / I onto ( A/I ⊗ B/J ) × piso˜ α ⊗ ˜ β P .Proof. For (i), as we mentioned in Remark 6.3, I ⊗ J is an extendible ( α ⊗ β ) x -invariant ideal of A ⊗ B for all x ∈ P . Therefore, by Theorem 4.7, the crossedproduct ( I ⊗ J ) × piso α ⊗ β P sits in the algebra ( A ⊗ B ) × piso α ⊗ β P as an ideal, which wedenote it by I .To get (ii), we first define I := ( A ⊗ J ) × piso α ⊗ β P + ( I ⊗ B ) × piso α ⊗ β P, which is an ideal of ( A ⊗ B ) × piso α ⊗ β P as each summand is. Note that we have[( A ⊗ J ) × piso α ⊗ β P ] ∩ [( I ⊗ B ) × piso α ⊗ β P ] = ( I ⊗ J ) × piso α ⊗ β P Also, by the similar discussion to Remark 6.3 (see also diagram (6.2)), it follows that I / I = (cid:2) ( A ⊗ J ) × piso α ⊗ β P + ( I ⊗ B ) × piso α ⊗ β P (cid:3) / ( I ⊗ J ) × piso α ⊗ β P = (cid:2) ( A ⊗ J ) × piso α ⊗ β P (cid:3) / [( I ⊗ J ) × piso α ⊗ β P ] ⊕ (cid:2) ( I ⊗ B ) × piso α ⊗ β P (cid:3) / [( I ⊗ J ) × piso α ⊗ β P ] ≃ [( A ⊗ J ) / ( I ⊗ J )] × piso ] α ⊗ β P ⊕ [( I ⊗ B ) / ( I ⊗ J )] × piso ] α ⊗ β P ≃ ( A/I ⊗ J ) × piso˜ α ⊗ β P ⊕ ( I ⊗ B/J ) × piso α ⊗ ˜ β P. Finally, for (iii), we recall from Proposition (6.4) that we have a surjective homomor-phism q × P : ( A ⊗ B ) × piso α ⊗ β P → ( A/I ⊗ B/J ) × piso˜ α ⊗ ˜ β P with ker( q × P ) = ( A ⊗ J ) × piso α ⊗ β P + ( I ⊗ B ) × piso α ⊗ β P = I . Therefore, we have (cid:0) ( A ⊗ B ) × piso α ⊗ β P (cid:1) / I = (cid:0) ( A ⊗ B ) × piso α ⊗ β P (cid:1) / ker( q × P ) ≃ ( A/I ⊗ B/J ) × piso˜ α ⊗ ˜ β P. (cid:3) References [1] S. Adji,
Crossed products of C ∗ -algebras by semigroups of endomorphisms , Ph.D. thesis, Uni-versity of Newcastle, 1995.[2] S. Adji and A. Hosseini, The Partial-Isometric Crossed Products of c by the Forward and theBackward Shifts , Bull. Malays. Math. Sci. Soc. (2) (3) (2010), 487–498.[3] S. Adji, S. Zahmatkesh, Partial-isometric crossed products by semigroups of endomorphisms asfull corners , J. Aust. Math. Soc. (2014), 145–166.[4] S. Adji, S. Zahmatkesh, The composition series of ideals of the partial-isometric crossed productby semigroup of endomorphisms , J. Korean. Math. Soc. (2015), no. 4, 869–889.[5] N. Brownlowe, N. S. Larsen and N. Stammeier, On C ∗ -algebras associated to right LCM semi-groups , Trans. Amer. Math. Soc. (2017), no. 1, 31–68.[6] N. Brownlowe, N. S. Larsen, and N. Stammeier, C ∗ -algebras of algebraic dynamical systems andright LCM semigroups , Indiana Univ. Math. J. (2018), 2453–2486.[7] J. Fletcher, A uniqueness theorem for the Nica–Toeplitz algebra of a compactly aligned productsystem , Rocky Mountain J. Math. (2019), no. 5, 1563–1594.[8] N. J. Fowler, Discrete product systems of Hilbert bimodules , Pacific J. Math. (2002), no. 2,335–375.[9] N. J. Fowler, I. Raeburn,
The Toeplitz algebra of a Hilbert bimodule , Indiana Univ. Math. J. (1999), 155–181.[10] P. R. Halmos, L. J. Wallen, Powers of partial isometries , Indiana Univ. Math. J. (1970),657–663.[11] B. K. Kwa´sniewski, N. S. Larsen, Nica-Toeplitz algebras associated with product systems overright LCM semigroups , J. Math. Anal. Appl. (2019), no. 1, 532–570.[12] B. K. Kwa´sniewski, N. S. Larsen,
Nica-Toeplitz algebras associated withright-tensor C ∗ -precategories over right LCM semigroups , Int. J. Math.,https://doi.org/10.1142/S0129167X19500137, arXiv:1611.08525v3.[13] W. Lewkeeratiyutkul, S. Zahmatkesh, The primitive ideal space of the partial-isometric crossedproduct of a system by a single automorphism , Rocky Mountain J. Math. (2017), no. 8,2699–2722.[14] M. Laca and I. Raeburn, Semigroup crossed products and the Toeplitz algebras of nonabeliangroups , J. Funct. Anal. (1996), 415–440.[15] N. S. Larsen,
Nonunital semigroup crossed products , Math. Proc. Royal Irish Acad. (2000), 205–218.[16] J. Lindiarni and I. Raeburn,
Partial-isometric crossed products by semigroups of endomor-phisms , J. Operator Theory (2004), 61–87.[17] A. Nica, C ∗ -algebras generated by isometries and Wiener-Hopf operators , J. Operator Theory (1992), 17–52.[18] M. D. Norling, Inverse semigroup C ∗ -algebras associated with left cancellative semigroups , Proc.Edinb. Math. Soc. (2014), no. 2, 533–564.[19] M. V. Pimsner, A class of C ∗ -algebras generalizing both Cuntz-Krieger algebras and crossedproducts by Z , in Free probability theory (Waterloo, ON, 1995), Amer. Math. Soc., Providence,RI, 1997, 189–212.[20] I. Raeburn and D. P. Williams, Morita Equivalence and Continuous-Trace C ∗ -Algebras, Math-ematical Surveys and Monographs, 60 (American Mathematical Society, Providence, RI, 1998).[21] S. Zahmatkesh, The Nica-Toeplitz algebras of abelian lattice-ordered groups are full corners ingroup crossed products , submitted, arXiv:1912.09682v2.[22] S. Zahmatkesh,
The Partial-isometric crossed products by semigroups of endomorphisms areMorita-Equivalent to crossed products by groups , New Zealand J. Math. (2017), 121–139. Department of Mathematics, Faculty of Science, King Mongkut’s University ofTechnology Thonburi, Bangkok 10140, THAILAND
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