Spectral triples on irreversible C^*-dynamical systems
aa r X i v : . [ m a t h . OA ] F e b SPECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS VALERIANO AIELLO, DANIELE GUIDO, AND TOMMASO ISOLA
Abstract.
Given a spectral triple on a C ∗ -algebra A together with a unital injective endo-morphism α , the problem of defining a suitable crossed product C ∗ -algebra endowed with aspectral triple is addressed. The proposed construction is mainly based on the works of Cuntzand [14], and on our previous papers [1, 2]. The embedding of α ( A ) in A can be considered asthe dual form of a covering projection between noncommutative spaces. A main assumption isthe expansiveness of the endomorphism, which takes the form of the local isometricity of thecovering projection and is expressed via the compatibility of the Lip-norms on A and α ( A ). Introduction
The aim of this paper is to tackle the following question: is it possibile to extend the constructionof a spectral triple on a crossed product C ∗ -algebra based on a spectral triple on the base algebraas described in [14, 22, 3, 9, 15] to the case of crossed products with a single endomorphism?Even though we do not have yet a general answer to this problem, we are able to propose ageneral procedure, some steps of which can be completely described, while for others we cangive several examples, which explain what we expect to be the general case.Before describing our plan, we draw attention to a feature of our construction, namely wemore or less explicitly assume that our endomorphism is in a sense expansive, see below. Suchproperty has important consequences: the compact resolvent property for the Dirac operatorforces the spectral triple on the crossed product to be semifinite, and the bounded commutatorproperty requires a reduction of the crossed product C ∗ -algebra, namely a new definition ofcrossed product by an endomorphism.Indeed, even though there are now various notions of crossed product of a C ∗ -algebra withan endomorphism, see e.g. [21, 7, 17], we essentially follow a path outlined by Cuntz [6] andthen further developed by Stacey [26], but we are forced to adapt it to the case of expansiveendomorphisms.According to Cuntz, given a C ∗ -algebra A together with a unital injective endomorphism α ,one constructs a direct system of C ∗ -algebras A n with endomorphisms α n , whence the directlimit C ∗ -algebra A ∞ is obtained. The key point is that the endomorphism α of A becomes anautomorphism α ∞ on A ∞ , so that one may define the crossed product A ⋊ α N as the crossedproduct A ∞ ⋊ α ∞ Z .The first and second step of our construction have been studied in [1, 2], where one assumesthat a spectral triple T on A is given. Let us observe that unital injective endomorphisms of a C ∗ -algebra A can be seen as noncommutative self-coverings of the underlying noncommutativespace; the first step is then to endow any of the C ∗ -algebras A n described above with a spectraltriple T n which makes the self-covering locally isometric or, equivalently, such that the Lip-norms induced by the Dirac operators are compatible with the connecting maps (this property Date : February 11, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Crossed product, spectral triple, noncommutative coverings, Lip-semiboundedness. can and will be weakened in some cases, cf. Section 4.3). This means that the sequence ofcovering spaces consists of dilated copies of the original space. Even if we do not give a generalprocedure for this step, this is not a difficult task in all the examples considered in [1, 2].The second step consists in constructing a spectral triple T ∞ on the direct limit A ∞ which isin some sense naturally associated with the original spectral triple on A , and we do this bydefining T ∞ as a suitable limit of the triples T n on A n . This step is far from being obvious,firstly because there is no general procedure to define a limit of a sequence of spectral triples,secondly because the situations we consider are quite different, ranging from regular coveringsassociated with an action of an abelian group to (possibly ramified) coverings with trivial groupof deck transformations. Examples illustrating this step are contained in [1, 2] and brieflydescribed below. In all cases, the coverings becoming wider and wider, the spectra of theDirac operators turn more and more closely packed, so that the limit has no longer compactresolvent. However, a corresponding rescaling of the traces gives rise to a (semicontinuoussemifinite) trace on a suitable C ∗ -algebra B of geometric operators, which contains A ∞ andthe resolvents of the limiting Dirac operator, finally producing a semifinite spectral triple on A ∞ . This means in particular that the semifiniteness property is true already at the level of A ∞ , therefore determines the analogous semifiniteness property for the spectral triple on thecrossed product.The third and final step, which is the main object of this paper, consists in defining a new kindof crossed product of a C ∗ -algebra w.r.t. an endomorphism, which can be seen as a variant ofthe crossed product considered by Cuntz in [6] and Stacey in [26], and which turns out to betailored to accomodate a spectral triple in the case of expansive endomorphisms.The notion of this new crossed product with an endomorphism is given in Defintion 2.4. Onthe one hand it is a universal object, therefore defines a unique object up to isomorphisms, onthe other hand, as shown in Theorem 2.13, it coincides with a reduction by a projection of the C ∗ -algebra crossed product defined in [26], Proposition 1.13. While the latter is nothing elsethan the crossed product of A ∞ with Z w.r.t. α ∞ , our notion can be considered as the crossedproduct of A ∞ with N w.r.t. α ∞ . The advantage of such a choice is to allow the weakening of therequest of metric equicontinuity (Lip-boundedness in our paper) of [14], which, for an action α of Z and a Lipschitz element a reads sup n ∈ Z L ( α − n ( a )) < ∞ and makes sense for automorphisms, to acondition on α that we call Lip-semiboundedness, namely sup n ∈ N L ( α − n ( a )) < ∞ . More precisely,in Section 3.2, we first generalize the construction of a spectral triple on a crossed productdescribed in [14] to the case of a semifinite spectral triple, mantaining the Lip-boundednessassumption, and then modify it by replacing the crossed product of Cuntz-Stacey with ourcrossed product, and noting that in this case the request of the endomorphism being Lip-semibounded is sufficient to guarantee the bounded commutator property of the spectral triple,cf. Theorem 3.9. Moreover, such theorem shows that the metric dimension of the crossedproduct spectral triple equals the metric dimension of the base triple increased by 1.The last section of this paper is devoted to show that the self-coverings considered in [1, 2]satisfy the Lip-semiboundedness condition, hence give rise to a semifinite spectral triple on thecrossed product.The first example deals with the self-covering of a p -torus, which is a regular covering. Givena purely expanding integer valued matrix B , the covering projection goes from R p /B Z p to R p / Z p and the canonical Dirac operator on the covering makes the covering projection locallyisometric. A natural embedding of the C ∗ -algebra A n in B ( H ) ⊗ M r n ( C ) gives rise to the PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 3 embedding of the direct limit C ∗ -algebra A ∞ in B ( H ) ⊗ UHF( r ∞ ), which is the algebra B mentioned above, where r = | det( B ) | . Moreover, the Dirac operators D n converge in the normresolvent sense to a Dirac operator affiliated with B ( H ) ⊗ UHF( r ∞ ). This structure producesa semifinite spectral triple on A ∞ , as shown in [1]. Theorem 4.2 shows that the conditionof Lip-semiboundedness is satisfied, hence we get a semifinite spectral triple on our crossedproduct with N .The second example treats the case of regular noncommutative coverings of the rational rotationalgebra with abelian group of deck trasnformations as defined in [1]. The procedure and theresults are essentially the same of the previous example, but the condition r ≡ q ± k [ D n , α n ( a )] k 6 = k [ D , a ] k for a Lipschitz in A ,however k [ D n + p , α p ( a )] k is bounded in p (indeed converges) for any Lipschitz element in A n .Again we show that the condition of Lip-semiboundedness is satisfied, cf. Theorem 4.5.The fourth and last example describes the crossed product associated with a ramified coveringof the fractal called Sierpinski gasket. Such covering is not given by an action of a group of decktransformations. Here the spectral triple on A is the one described in [13], and the spectraltriples on A n make the covering maps locally isometric. The C ∗ -algebra B containing both A ∞ and the resolvents of D ∞ is an algebra of geometric operators acting on the ℓ space on theedges of the infinite Sierpinski gasket with one boundary point [30]. The proof of the conditionof Lip-semiboundedness is contained in Theorem 4.7.In all cases, by Theorem 3.9, the spectral triples are finitely summable and their metric dimen-sion is equal to the metric dimension of A w.r.t. T plus 1, namely it is the sum of the metricdimension of A and the growth of N .Finally, we mention that even though in all of our examples the functional given by the normof the commutator with the Dirac operator is a Lip-norm in the sense of Rieffel [24] on A ,such property does not hold for the spectral triple on the crossed product. In fact any distanceon the state space of a unital C ∗ -algebra inducing the weak ∗ -topology should necessarily bebounded, and this is not the case for our construction. The reason is that the expansiveness ofthe endomorphism α produces larger and larger (quantum) covering spaces and eventually anunbounded solenoid space. This property leads to an analogous unboundedness for the distanceon the state space of the crossed product C ∗ -algebra.2. Crossed products for C ∗ -algebras Preliminaries. Inductive limit . We begin by recalling the construction of the inductivelimit C ∗ -algebra, due to Takeda [28], for the particular case of interest in this paper, to fix somenotation. Let A be a unital C ∗ -algebra, α ∈ End( A ) an injective, unital ∗ -endomorphism.Consider the following inductive system(2.1) A ϕ −−−→ A ϕ −−−→ · · · where, for all n ∈ N = { , , , . . . } , A n = A , ϕ n = α , and define, for m < n , ϕ nm : A m → A n by ϕ nm := ϕ n − ◦ · · · ◦ ϕ m ≡ α n − m , and ϕ mm := id. Consider the direct product Q ∞ n =0 A n , withpointwise operations, and set A ∞ := ( ( a n ) ∈ ∞ Y n =0 A n : ∃ m ∈ N such that a n = ϕ nm ( a m ) = α n − m ( a m ) , n ≥ m ) / ∼ , VALERIANO AIELLO, DANIELE GUIDO, AND TOMMASO ISOLA where ( a n ) ∼ ( b n ) ⇐⇒ a n = b n definitely. Then, A ∞ is a ∗ -algebra. For all n ∈ N , define ϕ ∞ n : a ∈ A n ϕ ∞ n ( a ) ∈ A ∞ , where ϕ ∞ n ( a ) ≡ ( a k ), and a k := ( , k < n,ϕ kn ( a ) = α k − n ( a ) , k ≥ n. We can introduce a seminorm p on A ∞ given by p ( a ) := lim sup n →∞ k ϕ nm ( a m ) k = k a m k , if a = ϕ ∞ m ( a m ), which is independent of the representative, and is a C ∗ -norm. Upon comple-tion, we get the desired inductive limit C ∗ -algebra, which is denoted A ∞ ≡ lim −→ A n . Crossed product . Let us recall the definition of the crossed product by an automorphism, inthe case of unital C ∗ -algebras, to fix some notation.Let A be a unital C ∗ -algebra, α ∈ Aut( A ) an automorphism. Denote by C c ( A , Z , α ) the ∗ -algebra of functions f : Z → A with finite support, pointwise addition and scalar multiplication,with product ( f g )( n ) := P k ∈ Z f ( k ) α k ( g ( n − k )), and involution f ∗ ( n ) := α n ( f ( − n ) ∗ ), f, g ∈ C c ( A , Z , α ), n ∈ Z . Define a norm on C c ( A , Z , α ) by k f k := P n ∈ Z k f ( n ) k , and denoteby ℓ ( A , Z , α ) the Banach ∗ -algebra obtained by completing C c ( A , Z , α ) with respect to thisnorm. A different description of ℓ ( A , Z , α ) is obtained by introducing the functions δ n ( k ) := δ k,n . Then, ℓ ( A , Z , α ) is the set of all sums P n ∈ Z a n δ n , with a n ∈ A , for all n ∈ Z , and P n ∈ Z k a n k < + ∞ . Let now π be a representation of A on H , V a unitary operator on H , suchthat π ( α ( a )) = V π ( a ) V ∗ , a ∈ A . The triple ( H , π, V ) is called a covariant representation of( A , α ). Then, the integrated form of ( H , π, V ) is the representation π ⋊ V of C c ( A , Z , α ) on H given by π ⋊ V ( X n ∈ Z a n δ n ) := X n ∈ Z π ( a n ) V n . (2.2)It can be proved ([23] Proposition 7.6.4) that there is a bijection between the set of non-degenerate covariant representations ( H , π, V ) of ( A , α ) on a Hilbert space H , and the set ofnon-degenerate continuous representations of ℓ ( A , Z , α ) on H . Define the universal represen-tation π u of ℓ ( A , Z , α ) to be the direct sum of all non-degenerate continuous representations of ℓ ( A , Z , α ) on Hilbert spaces. The crossed product of A by the action α of Z is the C ∗ -algebra A ⋊ α Z obtained as the norm closure of π u ( ℓ ( A , Z , α )). Reduced crossed product . Since Z is an amenable group, a different description ([23],7.7.7) of the crossed product (called the reduced crossed product, in the case of non amenablegroups) can be given. Let π be a representation of A on H , set e H := ℓ ( Z , H ) ≡ { ξ : Z → H | P n ∈ Z k ξ ( n ) k < + ∞} , and, for n ∈ Z , a ∈ A , ξ ∈ e H ,( U ξ )( n ) := ξ ( n − , ( e π ( a ) ξ )( n ) := π ( α − n ( a ))( ξ ( n )) . Observe that, e π ( α ( a )) = U e π ( a ) U ∗ , a ∈ A . Therefore, ( e H , e π, U ) is a covariant representationof ( A , Z , α ), and the representation e π ⋊ U is called a regular representation of ℓ ( A , Z , α ). Inparticular, if a = P n ∈ Z a n δ n ∈ C c ( A , Z , α ), then ( e π ⋊ U ( a ) ξ )( n ) = P k ∈ Z π ( α − n ( a k ))( ξ ( n − k )), n ∈ Z . Define the universal regular representation λ u of ℓ ( A , Z , α ) to be the direct sum of allregular representations of ℓ ( A , Z , α ) on Hilbert spaces. The (reduced) crossed product of A by the action α of Z is the C ∗ -algebra obtained as the norm closure of λ u ( ℓ ( A , Z , α )). Observethat ([23], 7.7.4), if π u is the universal representation of A , then A ⋊ α Z coincides with the PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 5 norm closure of f π u ⋊ U ( ℓ ( A , Z , α )). Therefore, we get A ⋊ α Z = h f π u ( A ) , U i , where h f π u ( A ) , U i stands for the C ∗ -algebra generated by f π u ( A ) and U . Lift of a spectral triple to a crossed product . In [3], Bellissard, Marcolli and Reihanishow how to lift a spectral triple from a unital C*-algebra A , endowed with an automorphism α , to the crossed product A ⋊ α Z . Their setting is generalised in ([14], Theorem 2.8) to thecase of the action of a discrete group. In the particular case of an automorphism, one obtains Definition 2.1.
Let A be a unital C*-algebra, α ∈ Aut( A ) a unital automorphism, ( L , H , D )a spectral triple on A such that α ( L ) ⊂ L . The automorphism is said to be Lip-bounded ifsup n ∈ Z k [ D, α − n ( a )] k < ∞ , ∀ a ∈ L . The previous notion was introduced in [14] where it is called the metric equicontinuity of theaction.
Theorem 2.2.
Let A be a unital C*-algebra, ( L , H , D ) an odd spectral triple on A , and α ∈ Aut( A ) a unital Lip-bounded automorphism. Set L ⋊ := ∗ alg( f π u ( L ) , U ) , H ⋊ := H ⊗ ℓ ( Z ) ⊗ C ,D ⋊ := D ⊗ I ⊗ ε + I ⊗ D Z ⊗ ε , Γ ⋊ := I ⊗ I ⊗ ε , where ∗ alg( f π u ( L ) , U ) is the ∗ -algebra generated by f π u ( L ) and U , ( D Z ξ )( n ) := nξ ( n ) , ∀ ξ ∈ ℓ ( Z ) ,and ε := (cid:18) (cid:19) , ε := (cid:18) − ii (cid:19) , ε := (cid:18) − (cid:19) (2.3) are the Pauli matrices.Then ( L ⋊ , H ⋊ , D ⋊ , Γ ⋊ ) is an even spectral triple on A ⋊ α Z . A new definition of crossed product by an endomorphism.
There are many dif-ferent definitions of the crossed product with an endomorphism, see e.g. [21], [7], and the verygeneral one given in [17]. We will work with a modification of the one introduced in [6, 26].Indeed, Cuntz ([6], pag. 101) considers the inductive sequence (2.1), and its inductive limitC*-algebra A ∞ , which is endowed with an automorphism α ∞ , uniquely defined by the diagram(2.4)(2.4) A α / / α (cid:15) (cid:15) A α / / α (cid:15) (cid:15) A α / / α (cid:15) (cid:15) · · · / / A ∞ α ∞ (cid:15) (cid:15) A α / / id ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A α / / id ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ A α / / id ? ? ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ · · · / / A ∞ where the diagonal maps define the inverse α − ∞ . Then Cuntz defined A ⋊ α N := q ( A ∞ ⋊ α ∞ Z ) q ,where q ∈ A ∞ is the image of 1 ∈ A , and turns out to be q = 1 in our case, since α is unital.Subsequently, Stacey [26] characterised A ⋊ α N as the solution of a universal problem.In this paper, our interest is in lifting suitable spectral triples from ( A , α ), where α ∈ End( A ),to A ⋊ α N . Since we already know how to lift a spectral triple from ( A , α ) to ( A ∞ , α ∞ ), at leastin some examples [1, 2], and the lift from ( A ∞ , α ∞ ) to A ∞ ⋊ α ∞ Z is well known [3], we found onlynatural to use Cuntz’ definition of the crossed product A ⋊ α N . Unfortunately, the spectraltriples ( L ∞ , H ∞ , D ∞ ) on ( A ∞ , α ∞ ) we constructed in [1, 2] satisfy, besides α ∞ ( L ∞ ) ⊂ L ∞ , VALERIANO AIELLO, DANIELE GUIDO, AND TOMMASO ISOLA only sup n ∈ N k [ D ∞ , α − n ∞ ( a )] k < ∞ , ∀ a ∈ L ∞ . This fact forces us to introduce a modification inCuntz’ procedure, namely to consider A ⋊ α N := p ( A ∞ ⋊ α ∞ Z ) p , where p ∈ B ( ℓ ( Z , H u )) is theprojection on the non-negative “frequencies”(2.5) ( pξ )( n ) = ( ξ ( n ) , n ≥ , , n < . Actually, we prefer to define our version of the crossed product by an endomorphism, in thesame spirit of Stacey, as the solution to a universal problem, see Definition 2.4, and then provein Theorem 2.13 that it coincides with p ( A ∞ ⋊ α ∞ Z ) p . Definition 2.3.
Let A be a unital C ∗ -algebra, α ∈ End( A ) a ∗ -endomorphism. Let π : A → B ( H ) be a representation, W ∈ B ( H ) an isometry. We say that ( H , π, W ) is a covariantrepresentation of ( A , α ) on H , if π ( α ( a )) W = W π ( a ) , a ∈ A ,W k W ∗ k ∈ π ( A ) ′ , k ∈ N . Definition 2.4.
Let A be a unital C ∗ -algebra, α ∈ End( A ) an injective, unital ∗ -endomorphism.The crossed product of A with N by α is a unital C ∗ -algebra B , together with a unital ∗ -monomorphism i A : A → B , and an isometry t ∈ B , such that(1) B is the C ∗ -algebra generated by i A ( A ) and t ,(2) i A ( α ( a )) t = t i A ( a ), a ∈ A ,(3) t k ( t ∗ ) k commutes with i A ( A ), k ∈ N ,(4) for every covariant representation ( H , π, W ) of ( A , α ), there exists a non-degeneraterepresentation b π of B on H , such that b π ◦ i A = π , and b π ( t ) = W .We denote by A ⋊ α N the above algebra B . We have defined our crossed product as a universalobject, which guarantees its uniqueness. For its existence, we will prove in Proposition 2.13that it is a reduction by a projection of the C ∗ -algebra crossed product defined by Cuntz in [6].2.3. Existence of the universal object.
Let us now consider the commutative diagram(2.4). It follows from ([31], Theorem L.2.1) that the vertical maps determine a ∗ -homomorphism α ∞ : A ∞ → A ∞ , and the diagonal maps define the inverse of α ∞ . Proposition 2.5.
Let A be a unital C ∗ -algebra, α a unital, injective ∗ -endomorphism of A .Then, there exists a covariant representation ( H , π, W ) of ( A , α ) .Proof. Let ψ be a faithful representation of A ∞ ⋊ α ∞ Z on a Hilbert space H , and T ∈ B ( K )an isometry on a Hilbert space K . If π u is the universal representation of A ∞ , let f π u : A ∞ → A ∞ ⋊ α ∞ Z , U ∈ U ( A ∞ ⋊ α ∞ Z ) be such that A ∞ ⋊ α ∞ Z = h f π u ( A ∞ ) , U i , and set H := H ⊗ K , π := ψ ◦ f π u ◦ ϕ ∞ ⊗ A → B ( H ), which is a representation of A on H , and W := ψ ( U ) ⊗ T ∈ B ( H ),which is an isometry on H . Moreover, for all a ∈ A , k ∈ N , by using that ϕ ∞ ◦ α = α ∞ ◦ ϕ ∞ and f π u ( α ∞ ( x )) = U f π u ( x ) U ∗ , we get π ( α ( a )) W = ( ψ ◦ f π u ◦ ϕ ∞ ( α ( a )) · ψ ( U )) ⊗ T = ψ ( f π u ◦ α ∞ ◦ ϕ ∞ ( a ) · U ) ⊗ T = ψ ( U · f π u ◦ ϕ ∞ ( a )) ⊗ T = ( ψ ( U ) ⊗ T )( ψ ◦ f π u ◦ ϕ ∞ ( a ) ⊗ W π ( a ) ,W k W ∗ k = ψ ( U k U ∗ k ) ⊗ T k T ∗ k = 1 ⊗ T k T ∗ k ∈ π ( A ) ′ . (cid:3) PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 7 We now prove that any covariant representation of ( A , α ) lifts to a covariant representation of( A ∞ , α ∞ ). Proposition 2.6.
Let A be a unital C ∗ -algebra, α a unital, injective ∗ -endomorphism of A , anddenote by A ∞ the C ∗ -algebra inductive limit of the inductive system (2.1) , and denote by α ∞ the automorphism of A ∞ induced by α . Let ( H , π, W ) be a covariant representation of ( A, α ) ,and denote by H ∞ ≡ lim −→ H n the Hilbert space inductive limit of the inductive system (2.6) H S −−−→ H S −−−→ · · · where, for all n ∈ N , H n := H , S n := W . Then, there exist W ∞ ∈ U ( H ∞ ) , and a covariantrepresentation ( H ∞ , π ∞ , W ∞ ) of ( A ∞ , α ∞ ) , such that π ∞ ◦ ϕ ∞ n ( a ) S ∞ n = S ∞ n π ( a ) , n ∈ N ∪ { } , a ∈ A ,W ∞ S ∞ = S ∞ W, where S ∞ n : ξ ∈ H n ( ξ k ) ∈ H ∞ , ξ k := ( , k < n,W k − n ξ, k ≥ n. Proof.
Denote by W ∞ the unitary operator on the inductive limit H ∞ ≡ lim −→ H n defined by thefollowing diagram H W / / W (cid:15) (cid:15) H W / / W (cid:15) (cid:15) H W / / W (cid:15) (cid:15) · · · / / H ∞ W ∞ (cid:15) (cid:15) H W / / id ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ H W / / id ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ H W / / id ? ? ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ · · · / / H ∞ so that W ∞ S ∞ n = S ∞ ,n − , for all n ∈ N , n ≥
1, and W ∞ S ∞ = S ∞ W .Introduce a map ψ : A → B ( H ∞ ) by ψ ( a ) S ∞ m ξ := S ∞ m π ( α m ( a )) ξ, a ∈ A , m ∈ N , ξ ∈ H m ≡ H , which is well defined, because, if S ∞ m ξ = S ∞ ,m − η = S ∞ m W η , then ξ = W η , and S ∞ m π ( α m ( a )) ξ = S ∞ m π ( α m ( a )) W η = S ∞ m W π ( α m − ( a )) η = S ∞ ,m − π ( α m − ( a )) η. Let us prove that ψ is a representation of A . Indeed, for a, b ∈ A , we get, for all m ∈ N , ξ ∈ H m , ψ ( ab ) S ∞ m ξ = S ∞ m π ( α m ( ab )) ξ = S ∞ m π ( α m ( a )) π ( α m ( b )) ξ = ψ ( a ) S ∞ m π ( α m ( b )) ξ = ψ ( a ) ψ ( b ) S ∞ m ξ. Moreover, for a ∈ A ∞ , ξ, η ∈ H , m, n ∈ Z , we get, if n < m ,( S ∞ m ξ, ψ ( a ) ∗ S ∞ n η ) = ( ψ ( a ) S ∞ m ξ, S ∞ n η ) = ( S ∞ m π ( α m ( a )) ξ, S ∞ n η )= ( S ∞ m π ( α m ( a )) ξ, S ∞ m S mn η ) = ( π ( α m ( a )) ξ, S mn η )= ( ξ, π ( α m ( a ∗ )) W m − n η ) = ( ξ, W m − n π ( α n ( a ∗ )) η )= ( S ∞ m ξ, S ∞ n π ( α n ( a ∗ )) η ) = ( S ∞ m ξ, ψ ( a ∗ ) S ∞ n η ) . VALERIANO AIELLO, DANIELE GUIDO, AND TOMMASO ISOLA
Setting, for all n ∈ N , ψ n := Ad( W ∗∞ ) n ◦ ψ , we get, for m ≥ n + 1, ψ n +1 ( α ( a )) S ∞ m = ( W ∗∞ ) n +1 ψ ( α ( a )) W n +1 ∞ S ∞ m = ( W ∗∞ ) n +1 ψ ( α ( a )) S ∞ ,m − n − = ( W ∗∞ ) n +1 S ∞ ,m − n − π ( α m − n ( a )) = ( W ∗∞ ) n S ∞ ,m − n π ( α m − n ( a ))= ( W ∗∞ ) n ψ ( a ) S ∞ ,m − n = ( W ∗∞ ) n ψ ( a ) W n ∞ S ∞ m = ψ n ( a ) S ∞ m , so that ψ n +1 ( α ( a )) = ψ n ( a ). Therefore, the following diagram commutes A ϕ −−−→ A ϕ −−−→ A ϕ −−−→ · · · −−−→ A ∞ ψ y ψ y ψ y π ∞ y B ( H ∞ ) id −−−→ B ( H ∞ ) id −−−→ B ( H ∞ ) id −−−→ · · · −−−→ B ( H ∞ )so that there is a unique ∗ -homomorphism π ∞ : A ∞ → B ( H ∞ ) such that π ∞ ◦ ϕ ∞ n = ψ n , forall n ∈ N . Therefore, for all n ∈ N , a ∈ A , we have π ∞ ◦ ϕ ∞ n ( a ) S ∞ n = ψ n ( a ) S ∞ n = W ∗ n ∞ ψ ( a ) W n ∞ S ∞ n = W ∗ n ∞ ψ ( a ) S ∞ = W ∗ n ∞ S ∞ π ( a ) = S ∞ n π ( a ) . (2.7)Finally, for all n ∈ N , n ≥ a ∈ A n = A , we have π ∞ ◦ α ∞ ◦ ϕ ∞ n ( a ) = π ∞ ◦ ϕ ∞ n ◦ α ( a ) = ψ n ◦ α ( a ) = ψ n − ( a )= Ad( W ∞ ) ◦ ψ n ( a ) = Ad( W ∞ ) ◦ π ∞ ◦ ϕ ∞ n ( a ) , so that π ∞ ◦ α ∞ = Ad( W ∞ ) ◦ π ∞ , that is ( H ∞ , π ∞ , W ∞ ) is a covariant representation of( A ∞ , α ∞ ). (cid:3) We recall that in the construction of A ∞ ⋊ α ∞ Z we denoted by π u the universal representationof A ∞ on H u , so that A ∞ ⋊ α ∞ Z = h f π u ( A ∞ ) , U i .Define the projection p ∈ B ( ℓ ( Z , H u )) as in (2.5), so that p f π u ( a ) = f π u ( a ) p , a ∈ A ∞ , and set t := pU p ≡ U p , so that t ∗ t = p , and t f π u ( a ) = f π u ( α ∞ ( a )) t , a ∈ A ∞ . Set i A ( a ) := f π u ◦ ϕ ∞ ( a ) p , whichis a representation of A on pℓ ( Z , H u ), and denote by C ∗ ( A , α, N ) the C ∗ -algebra generated by i A ( A ) and t on pℓ ( Z , H u ). Proposition 2.7.
We have that i A ( α ( a )) t = t i A ( a ) , a ∈ A .Proof. Indeed, for all a ∈ A , i A ( α ( a )) t = f π u ◦ ϕ ∞ ◦ α ( a ) t = f π u ◦ α ∞ ◦ ϕ ∞ ( a ) t = t f π u ◦ ϕ ∞ ( a ) p = t i A ( a ) . (cid:3) In order to the help the reader with the understanding of the following statements and proofs,we exhibit two tables with the C ∗ -algebras considered, and their representations on the variousHilbert spaces Aut( · ) H ∞ H u ℓ ( Z ; H u ) A ∞ α ∞ π ∞ π u f π u C ( Z ∞ ) β ρ ∞ ρ u e ρ u C ( Z ∞ ; A ∞ ) γ ≡ α ∞ ⊗ β σ ∞ σ u f σ u A ∞ ⋊ α ∞ Z - π ∞ ⋊ W ∞ - f π u ⋊ UC ( Z ∞ ; A ∞ ) ⋊ γ Z - σ ∞ ⋊ W ∞ - χ ≡ f σ u ⋊ U C ≡ χ ( C ( Z ∞ ; A ∞ ) ⋊ γ Z ) - b χ ≡ ( σ ∞ ⋊ W ∞ ) ◦ χ − - id PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 9 and End( · ) H H ∞ pℓ ( Z ; H u ) A α π ψ i A ≡ f π u ◦ ϕ ∞ ( · ) pC ∗ ( A , α, N ) - b π ≡ S ∗∞ b χ ( · ) S ∞ b χ | C ∗ ( A ,α, N ) idWe want to prove that C ∗ ( A , α, N ) is isomorphic to the crossed product of A by α . Actually,the proof of property (4) in Definition 2.4 will force us to a long detour. We start with somepreliminary results. Denote by Z ∞ := Z ∪ { + ∞} the spectrum of the C ∗ -algebra of functionson Z , vanishing at −∞ , and having finite limit for n → + ∞ , and let β be the automorphismof C ( Z ∞ ) given by β ( f )( n ) := f ( n − n ∈ Z .Let ( H , π, W ) be a covariant representation of ( A, α ), and recall from Proposition 2.6 thatthere exist W ∞ ∈ U ( H ∞ ), and a covariant representation ( H ∞ , π ∞ , W ∞ ) of ( A ∞ , α ∞ ), on H ∞ ≡ lim −→ H n , the Hilbert space inductive limit of the inductive system (2.6), such that π ∞ ◦ ϕ ∞ n ( a ) S ∞ n = S ∞ n π ( a ), for all n ∈ N , a ∈ A , and W ∞ S ∞ = S ∞ W .We now construct a representation ρ ∞ of C ( Z ∞ ) on H ∞ such that [ π ∞ ( a ) , ρ ∞ ( f )] = 0, for all a ∈ A ∞ , f ∈ C ( Z ∞ ). Proposition 2.8.
Set P := S ∞ S ∗∞ , P n := Ad ( W n ∞ )( P ) , n ∈ Z . Then (1) { P n : n ∈ Z } is a decreasing family of projections in B ( H ∞ ) , (2) there exists P + ∞ := lim n → + ∞ P n , in the strong operator topology of B ( H ∞ ) , (3) lim n →−∞ P n = 1 , in the strong operator topology of B ( H ∞ ) , (4) { P n : n ∈ Z ∞ } ⊂ π ∞ ( A ∞ ) ′ .Proof. (1) Let n ∈ Z . If n ≥
0, then P n = W n ∞ S ∞ S ∗∞ W n ∗∞ = S ∞ W n W n ∗ S ∗∞ ≥ S ∞ W n +1 ( W ∗ ) n +1 S ∗∞ = P n +1 . If n = − k ≤
0, then P n = W ∗ k ∞ S ∞ S ∗∞ W k ∞ = S ∞ k S ∗∞ k = S ∞ ,k +1 W W ∗ S ∗∞ ,k +1 ≤ S ∞ ,k +1 S ∗∞ ,k +1 = P n − . (2) follows from (1).(3) We have to prove that lim k → + ∞ S ∞ k S ∗∞ k = 1, in the strong operator topology, and it sufficesto prove it on the dense subset of H ∞ spanned by { S ∞ n ξ : n ∈ N , ξ ∈ H } . Let us fix n ∈ N , ξ ∈ H , and compute, for k > n , S ∞ k S ∗∞ k S ∞ n ξ = S ∞ k S ∗∞ k S ∞ k S kn ξ = S ∞ k S kn ξ = S ∞ n ξ , and thethesis follows.(4) Let us first prove that π ∞ ( x ) P = P π ∞ ( x ) for x ∈ A ∞ . It suffices to show the equality for x ∈ { ϕ ∞ n ( a ) : n ∈ N , a ∈ A } . We have, from equation (2.7), π ∞ ◦ ϕ ∞ n ( a ) P = π ∞ ◦ ϕ ∞ n ( a ) S ∞ S ∗∞ = π ∞ ◦ ϕ ∞ n ( a ) S ∞ n W n S ∗∞ = S ∞ n π ( a ) W n W ∗ n S ∗∞ n = S ∞ n W n W ∗ n π ( a ) S ∗∞ n = S ∞ W ∗ n S ∗∞ n π ∞ ◦ ϕ ∞ n ( a ) = P π ∞ ◦ ϕ ∞ n ( a ) . Then, for any x ∈ A ∞ , k ∈ Z , π ∞ ( x ) P k = π ∞ ( x ) W k ∞ P W ∗ k ∞ = W k ∞ π ∞ ( α − k ∞ ( x )) P W ∗ k ∞ = W k ∞ P π ∞ ( α − k ∞ ( x )) W ∗ k ∞ = W k ∞ P W ∗ k ∞ π ∞ ( x ) = P k π ∞ ( x ) . Finally, P + ∞ ∈ π ∞ ( A ∞ ) ′ , because of (2). (cid:3) Proposition 2.9.
There exists a representation ρ ∞ of C ( Z ∞ ) on H ∞ , such that, for any f ∈ C ( Z ∞ ) , ρ ∞ ( f ) ∈ π ∞ ( A ∞ ) ′ ,ρ ∞ ( β ( f )) = W ∞ ρ ∞ ( f ) W ∗∞ . Proof.
Set E n := P n − P n +1 , n ∈ Z , E + ∞ := P + ∞ . Then, { E n : n ∈ Z ∞ } is a spectral family on H ∞ , and E n +1 = W ∞ E n W ∗∞ , n ∈ N , E + ∞ = W ∞ E + ∞ W ∗∞ . Define, for f ∈ C ( Z ∞ ), ρ ∞ ( f ) := P n ∈ Z ∞ f ( n ) E n , where the series converges in the strong operator topology of B ( H ∞ ). Then, ρ ∞ is a representation of C ( Z ∞ ) on H ∞ , such that ρ ∞ ( f ) ∈ π ∞ ( A ∞ ) ′ , for any f ∈ C ( Z ∞ ),and ρ ∞ ( β ( f )) = W ∞ ρ ∞ ( f ) W ∗∞ , f ∈ C ( Z ∞ ). (cid:3) It follows from [25], Proposition 1.22.3, that A ∞ ⊗ C ( Z ∞ ) ∼ = C ( Z ∞ , A ∞ ), that is two-sidedsequences of elements in A ∞ , vanishing at −∞ , and having norm-limit for n → + ∞ . Proposition 2.10. (1)
There is a unique automorphism γ ∈ Aut( C ( Z ∞ ; A ∞ )) such that γ ( a ⊗ f ) = α ∞ ( a ) ⊗ β ( f ) , a ∈ A ∞ , f ∈ C ( Z ∞ ) . (2) There is a unique representation σ ∞ of C ( Z ∞ ; A ∞ ) on H ∞ , such that σ ∞ ( a ⊗ f ) = π ∞ ( a ) ρ ∞ ( f ) , a ∈ A ∞ , f ∈ C ( Z ∞ ) . Moreover, σ ∞ ( γ ( g )) = W ∞ σ ∞ ( g ) W ∗∞ , g ∈ C ( Z ∞ ; A ∞ ) . (3) There is a unique representation σ ∞ ⋊ W ∞ of C ( Z ∞ ; A ∞ ) ⋊ γ Z on H ∞ such that σ ∞ ⋊ W ∞ ( gδ n ) = σ ∞ ( g ) W n ∞ , g ∈ C ( Z ∞ ; A ∞ ) , n ∈ Z .Proof. (1) This follows from [29], Proposition IV.4.22.(2) This follows from [29], Proposition IV.4.7.(3) This follows from [23], Proposition 7.6.4. (cid:3) In Proposition 2.11, we construct a more convenient representation of C ( Z ∞ ; A ∞ ) ⋊ γ Z on ℓ ( Z , H u ). Let ρ u be the representation of C ( Z ∞ ) on H u given by ρ u ( f ) ξ = f (0) ξ , f ∈ C ( Z ∞ ), ξ ∈ H u . It follows from [29], Proposition IV.4.7, that there is a unique representation σ u of C ( Z ∞ ; A ∞ ) on H u , such that σ u ( a ⊗ f ) = π u ( a ) ρ u ( f ), a ∈ A ∞ , f ∈ C ( Z ∞ ).Introduce the representations e ρ u of C ( Z ∞ ) and f σ u of C ( Z ∞ ; A ∞ ) on ℓ ( Z , H u ) given by, for a ∈ A ∞ , f ∈ C ( Z ∞ ), ξ ∈ ℓ ( Z , H u ), n ∈ Z ,( e ρ u ( f ) ξ )( n ) := ρ u ( β − n ( f )) ξ ( n ) = f ( n ) ξ ( n ) , ( f σ u ( a ⊗ f ) ξ )( n ) := σ u ( γ − n ( a ⊗ f )) ξ ( n ) . Proposition 2.11. (1) U e ρ u ( f ) U ∗ = e ρ u ( β ( f )) , f ∈ C ( Z ∞ ) . (2) The representation f σ u of C ( Z ∞ ; A ∞ ) on ℓ ( Z , H u ) is faithful, and f σ u ( C ( Z ∞ ; A ∞ )) = h f π u ( A ∞ ) , e ρ u ( C ( Z ∞ )) i ,U f σ u ( a ⊗ f ) U ∗ = f σ u ( γ ( a ⊗ f )) , a ∈ A ∞ , f ∈ C ( Z ∞ ) . (3) The regular representation χ := f σ u ⋊ U of C ( Z ∞ , A ∞ ) ⋊ γ Z , induced from σ u on ℓ ( Z , H u ) ,is faithful.Proof. (1) is a computation.(2) It is easy to see that ( f σ u ( g ) ξ )( k ) = π u ( α − k ∞ ( g ( k ))) ξ ( k ), k ∈ Z , ξ ∈ ℓ ( Z , H u ), g ∈ C ( Z ∞ , A ∞ ),from which it follows that f σ u is faithful. Moreover, for a ∈ A ∞ , f ∈ C ( Z ∞ ), one has f σ u ( γ ( a ⊗ f )) = f σ u ( α ∞ ( a ) ⊗ β ( f )) = f π u ( α ∞ ( a )) e ρ u ( β ( f ))= U f π u ( a ) U ∗ U e ρ u ( f ) U ∗ = U f σ u ( a ⊗ f ) U ∗ . PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 11 (3) This follows from [32], Theorem 7.13. (cid:3) It follows from the previous Proposition that C := h f π u ( A ∞ ) , e ρ u ( C ( Z ∞ )) , U i ⊂ B ( ℓ ( Z , H u ))is isomorphic, via χ − , to ( A ∞ ⊗ C ( Z ∞ )) ⋊ γ Z .Let us set b χ := σ ∞ ⋊ W ∞ ◦ χ − , which is a representation of C on H ∞ . Proposition 2.12. (1) b χ ( x ) = π ∞ ⋊ W ∞ ( x ) , for all x ∈ A ∞ ⋊ α ∞ Z ≡ h f π u ( A ∞ ) , U i , (2) b χ ( p ) = P = S ∞ S ∗∞ .Proof. It follows from Proposition 2.11 that, for P n ∈ Z ( a n ⊗ f n ) δ n ∈ C c ( C ( Z ∞ ; A ∞ ) , Z , γ ), wehave χ ( P n ∈ Z ( a n ⊗ f n ) δ n ) = P n ∈ Z f π u ( a n ) e ρ u ( f n ) u n , so that b χ (cid:18) X n ∈ Z f π u ( a n ) e ρ u ( f n ) U n (cid:19) = σ ∞ ⋊ W ∞ (cid:18) X n ∈ Z ( a n ⊗ f n ) δ n (cid:19) = X k ∈ Z σ ∞ ( a n ⊗ f n ) W n ∞ = X k ∈ Z π ∞ ( a n ) ρ ∞ ( f n ) W n ∞ . (1) Indeed, with { e n : n ∈ N } an approximate unit of C ( Z ∞ ), we get, for all a ∈ A ∞ , k ∈ Z , b χ ( f π u ( a ) U k ) = lim n →∞ b χ ( f π u ( a ) e ρ u ( e n ) U k ) = lim n →∞ π ∞ ( a ) ρ ∞ ( e n ) W k ∞ = π ∞ ( a ) W k ∞ = π ∞ ⋊ W ∞ ( f π u ( a ) U k ) , and the thesis follows.(2) If f ( n ) = ( , n < , , n ≥ , then b χ ( p ) = b χ ( e ρ u ( f )) = ρ ∞ ( f ) = P . (cid:3) Let us still denote by b χ the restriction of b χ to the subalgebra C ∗ ( A , α, N ) ≡ h p f π u ( A ∞ ) p, pU p i of C ≡ h f π u ( A ∞ ) , e ρ u ( C ( Z ∞ )) , U i . Theorem 2.13. C ∗ ( A , α, N ) satisfies all the properties in Definition 2.4, namely is the crossedproduct of A with N by α .Proof. Property (1) in Definition 2.4 follows by definition, while property (2) has been provedin Proposition 2.7. Let us prove property (3), namely, for any a ∈ A , k ∈ N , t k ( t ∗ ) k i A ( a ) = i A ( a ) t k ( t ∗ ) k . Indeed, since t k = U k p = pU k p , we get t k ( t ∗ ) k i A ( a ) = U k p ( U ∗ ) k f π u ◦ ϕ ∞ ( a ) p = U k p f π u ◦ α − k ∞ ◦ ϕ ∞ ( a )( U ∗ ) k p = U k f π u ◦ α − k ∞ ◦ ϕ ∞ ( a ) p ( U ∗ ) k p = f π u ◦ ϕ ∞ ( a ) U k p ( U ∗ ) k p = f π u ◦ ϕ ∞ ( a ) pU k p ( U ∗ ) k p = i A ( a ) t k ( t ∗ ) k . It remains to prove property (4). Let ( H , π, W ) be a covariant representation of ( A, α ), andrecall from Proposition 2.6 that there exist W ∞ ∈ U ( H ∞ ), and a covariant representation( H ∞ , π ∞ , W ∞ ) of ( A ∞ , α ∞ ), on H ∞ ≡ lim −→ H n , the Hilbert space inductive limit of the inductivesystem (2.6), such that π ∞ ◦ ϕ ∞ n ( a ) S ∞ n = S ∞ n π ( a ), for all n ∈ N , a ∈ A , and W ∞ S ∞ = S ∞ W .Let b χ be the representation of C ∗ ( A , α, N ) on H ∞ constructed in Proposition 2.12. Let us nowprove that P ∈ b χ ( C ∗ ( A , α, N )) ′ , that is b χ ( C ∗ ( A , α, N )) S ∞ H ⊂ S ∞ H . Moreover, becauseof Proposition 2.7 it is enough to prove that b χ ( t ) S ∞ H ⊂ S ∞ H , b χ ( t ∗ ) S ∞ H ⊂ S ∞ H , and b χ ( i A ( a )) S ∞ H ⊂ S ∞ H , for all a ∈ A . Indeed, for all a ∈ A , ξ ∈ H , we have b χ ( t ) S ∞ ξ = b χ ( pU p ) S ∞ ξ = P W ∞ P S ∞ ξ ∈ S ∞ H , b χ ( t ∗ ) S ∞ ξ = b χ ( pU ∗ p ) S ∞ ξ = P W ∗∞ P S ∞ ξ ∈ S ∞ H , b χ ( i A ( a )) S ∞ ξ = b χ ◦ f π u ◦ ϕ ∞ ( a ) P S ∞ ξ = π ∞ ◦ ϕ ∞ ( a ) S ∞ ξ = S ∞ π ( a ) ξ ∈ S ∞ H . Recall from the proof of Proposition 2.6 that there is a representation ψ of A on H ∞ such that ψ ( a ) S ∞ = S ∞ π ( a ), a ∈ A , and π ∞ ◦ ϕ ∞ = ψ . Finally, define b π ( x ) := S ∗∞ b χ ( x ) S ∞ , x ∈ C ∗ ( A , α, N ) , which is a representation of C ∗ ( A , α, N ) on H , because P ∈ b χ ( C ∗ ( A , α, N )) ′ . Then, b π ( t ) = S ∗∞ b χ ( t ) S ∞ = S ∗∞ P W ∞ P S ∞ = S ∗∞ W ∞ S ∞ = S ∗∞ S ∞ W = W, and, for all a ∈ A , b π ( i A ( a )) = S ∗∞ b χ ( f π u ◦ ϕ ∞ ( a ) p ) S ∞ = S ∗∞ π ∞ ◦ ϕ ∞ ( a ) S ∞ S ∗∞ S ∞ = S ∗∞ ψ ( a ) S ∞ = S ∗∞ S ∞ π ( a ) = π ( a ) . (cid:3) Some results on semifinite spectral triples
In this section we discuss some generalizations of results well-known for type I spectral triples.Some of these results have already been proved in [16] and some are new.First of all we recall the following definitions:
Definition 3.1.
Let ( M , τ ) be a von Neumann algebra with a normal semifinite faithful (n.s.f.)trace, T b ∈ M a self-adjoint operator. We use the notation e T (Ω) for the spectral projectionof T relative to the measurable set Ω ⊂ R , λ t ( T ) := τ ( e | T | ( t, + ∞ )), Λ t ( T ) := τ ( e | T | [0 , t )), µ t ( T ) := inf { s > λ T ( s ) ≤ t } , t >
0. The operator T is said to be τ -measurable if λ t ( T ) → t → + ∞ , and τ -compact if µ t ( T ) → t → + ∞ , or equivalently, λ t ( T ) < + ∞ , ∀ t > Definition 3.2.
Let A be a unital C ∗ -algebra. An odd semifinite spectral triple ( L , H , D ; M , τ )on A , with respect to a semifinite von Neumann algebra M ⊂ B ( H ) endowed with a n.s.f. trace τ , is given by a unital, norm-dense, ∗ -subalgebra L ⊂ A , a (separable) Hilbert space H , a faithfulrepresentation π : A → B ( H ) such that π ( A ) ⊂ M , and an unbounded self-adjoint operator D b ∈ M such that(1) (1 + D ) − is a τ -compact operator, i.e. λ t ((1 + D ) − ) < + ∞ , ∀ t > t ( D ) < + ∞ , ∀ t > π ( a )(Dom D ) ⊂ Dom D , and [ D, π ( a )] ∈ M , for all a ∈ L .The spectral triple ( L , H , D ; M , τ ) is even if, in addition,(3) there is a self-adjoint unitary operator ( i.e. a Z -grading) Γ ∈ M such that π ( a )Γ =Γ π ( a ), ∀ a ∈ A , and D Γ = − Γ D .The spectral triple ( L , H , D ; M , τ ) is finitely summable if, in addition,(4) there exists a δ > τ ((1 + D ) − δ/ ) < + ∞ . PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 13 Definition 3.3.
Given a finitely summable semifinite spectral triple ( L , H , D ; M , τ ), the num-ber d = inf { α > τ ((1 + D ) − α/ ) < + ∞} is called the metric or Hausdorff dimension of thetriple, since it is the unique exponent, if any, such that the logarithmic Dixmier trace is finitenon zero on (1 + D ) − α/ (cf. [12], Theorem 2.7). Proposition 3.4.
Let ( L , H , D ; M , τ ) be a finitely summable semifinite spectral triple. Then d = lim sup t →∞ log Λ t ( D )log t .Proof. We first observe that, by [8], Proposition 2.7, τ ((1 + D ) − α/ ) = Z + ∞ µ t ((1 + D ) − α/ ) dt = Z + ∞ µ αt ((1 + D ) − / ) dt. Therefore, d = (cid:18) lim inf t →∞ log µ t ((1 + D ) − / )log(1 /t ) (cid:19) − = lim sup s → log λ s ((1 + D ) − / )log(1 /s )= lim sup t →∞ log Λ t ((1 + D ) / )log t = lim sup t →∞ log Λ t ( D )log t , where the first equality follows by [12] Theorem 1.4, the second by [11] Proposition 1.13, thethird by definition of Λ, the last by simple estimates. (cid:3) The case of the tensor product.
Let us recall the definition of tensor product ofsemifinite spectral triples.
Definition 3.5.
Let A , A be unital C ∗ -algebras, with respective semifinite spectral triples T := ( L , H , D , Γ ; M , τ ), T := ( L , H , D , Γ ; M , τ ), and define T × T ≡ ( L , H , D, Γ; M , τ )as follows:if T and T are both even L := L ⊙ L , H := H ⊗ H , D := D ⊗ I + Γ ⊗ D , Γ := Γ ⊗ Γ , M := M ⊗ M , τ := τ ⊗ τ , if T is even, and T is odd, L := L ⊙ L , H := H ⊗ H , D := D ⊗ I + Γ ⊗ D , Γ := I ⊗ I , M := M ⊗ M , τ := τ ⊗ τ , if T is odd, and T is even, L := L ⊙ L , H := H ⊗ H , D := D ⊗ Γ + I ⊗ D , Γ := I ⊗ I , M := M ⊗ M , τ := τ ⊗ τ , if T and T are both odd, L := L ⊙ L , H := H ⊗ H ⊗ C , D := D ⊗ I ⊗ ε + I ⊗ D ⊗ ε , Γ := I ⊗ I ⊗ ε , M := M ⊗ M ⊗ M ( C ) , τ := τ ⊗ τ ⊗ T r, where ε , ε , ε are the Pauli matrices, see (2.3). Proposition 3.6.
Let A , A be unital C ∗ -algebras, with respective semifinite spectral triples T := ( L , H , D , Γ ; M , τ ) , T := ( L , H , D , Γ ; M , τ ) . Then T × T is a semifinitespectral triple on the spatial tensor product A ⊗ A . Moreover, the Hausdorff dimension d of T × T satisfies d ≤ d + d , where d , d are the Hausdorff dimensions of the factor spectraltriples. Finally, if lim t →∞ log Λ t ( D )log t exists, the equality d = d + d holds. Proof.
In case T and T are not both odd, the result is proved in [16], Theorem 2.13, andLemma 2.19. In the remaining case, one can proceed analogously. We now give an alternativeproof of the formula for the Hausdorff dimension, valid in all cases. Since D = D ⊗ I + I ⊗ D ,in all cases, if d denotes the dimension of ( L , H , D ; M , τ ), we have that d = lim sup t →∞ log Λ t ( D )log t = lim sup t →∞ log τ ( e D ( − t, t ))log t = lim sup t →∞ log τ ( χ [0 ,t ) ( D ⊗ I + I ⊗ D ))log t . If σ i denotes the spectrum of D i , i = 1 ,
2, the representations of C ( σ i ) on H i with image in M i given by functional calculus, i = 1 ,
2, together with the Radon measures ν i on σ i inducedby the traces τ i , i = 1 ,
2, give rise to a representation j of C ( σ × σ ) on H ⊗ H with imagein M ⊗ M together with the Radon measure ν := ν ⊗ ν on σ × σ induced by the trace τ := τ ⊗ τ such that j ( f ⊗ f ) = f ( D ) ⊗ f ( D ) and R f ⊗ f dν = τ ( f ( D )) τ ( f ( D )).Then, denoting by B r the disk of radius r centered in the origin of the plane, and by Q r thesquare [ − r, r ] × [ − r, r ] in the plane, χ [0 ,t ) ( D ⊗ I + I ⊗ D ) = j ( χ B t ) . Then the inclusions Q t/ √ ⊂ B t ⊂ Q t give the inequalities τ (Λ t/ √ ( D )) · τ (Λ t/ √ ( D )) ≤ ν ( Q t/ √ ) ≤ ν ( B t ) ≤ ν ( Q t ) ≤ τ (Λ t ( D )) · τ (Λ t ( D )) , from which we getlim inf t →∞ log Λ t ( D )log t + lim sup t →∞ log Λ t ( D )log t ≤ lim sup t →∞ log Λ t ( D )log t ≤ lim sup t →∞ log Λ t ( D )log t + lim sup t →∞ log Λ t ( D )log t . (cid:3) The cases of the crossed products.
Let A be a unital C ∗ -algebra, α ∈ Aut( A ) a unitalautomorphism, and ( L , H , D ; M , τ ) a semifinite spectral triple on A . Assume that α is Lip-bounded, that is α ( L ) ⊂ L , and, for any a ∈ L , sup n ∈ Z k [ D, α − n ( a )] k < ∞ . Then, following[3], we can construct a semifinite spectral triple ( L ⋊ , H ⋊ , D ⋊ ; M ⋊ , τ ⋊ ) on the crossed product C ∗ -algebra A ⋊ α Z = h f π u ( A ) , U i , which is defined as follows:(1) if ( L , H , D, Γ; M , τ ) is even, L ⋊ := ∗ alg( f π u ( L ) , U ) , H ⋊ := H ⊗ ℓ ( Z ) ,D ⋊ := D ⊗ I + Γ ⊗ D Z , Γ ⋊ := I ⊗ I, M ⋊ := M ⊗ B ( ℓ ( Z )) , τ ⋊ := τ ⊗ T r, where ∗ alg( f π u ( L ) , U ) is the ∗ -algebra generated by f π u ( L ) and U , ( D Z ξ )( n ) := nξ ( n ), ∀ ξ ∈ ℓ ( Z ), and T r is the usual trace on B ( ℓ ( Z )),(2) if ( L , H , Γ; M , τ ) is odd, L ⋊ := ∗ alg( f π u ( L ) , U ) , H ⋊ := H ⊗ ℓ ( Z ) ⊗ C ,D ⋊ := D ⊗ I ⊗ ε + I ⊗ D Z ⊗ ε , Γ ⋊ := I ⊗ I ⊗ ε , M ⋊ := M ⊗ B ( ℓ ( Z )) ⊗ M ( C ) , τ ⋊ := τ ⊗ T r ⊗ tr, where tr is the normalized trace on M ( C ). PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 15 In case α satisfies a weaker condition, we have the following result. Definition 3.7.
Let A be a unital C*-algebra, α ∈ Aut( A ) a unital automorphism, ( L , H , D )a spectral triple on A such that α ( L ) ⊂ L . The automorphism is said to be Lip-semiboundedif sup n ∈ N k [ D, α − n ( a )] k < ∞ , ∀ a ∈ L . Proposition 3.8.
Let A be a unital C ∗ -algebra, α ∈ Aut( A ) a unital automorphism, ( L , H , D ; M , τ ) a semifinite spectral triple on A , and assume α is Lip-semibounded. Then we can constructa semifinite spectral triple ( L ⋊ , H ⋊ , D ⋊ ; M ⋊ , τ ⋊ ) on the crossed product C ∗ -algebra A ⋊ α N = h i A ( A ) , t i , which is defined as follows: (1) if ( L , H , D, Γ; M , τ ) is even, L ⋊ := ∗ alg( i A ( L ) , t ) , H ⋊ := H ⊗ ℓ ( N ) ,D ⋊ := D ⊗ I + Γ ⊗ D N , Γ ⋊ := I ⊗ I, M ⋊ := M ⊗ B ( ℓ ( N )) , τ ⋊ := τ ⊗ T r, where ∗ alg( i A ( L ) , t ) is the ∗ -algebra generated by i A ( L ) and t , ( D N ξ )( n ) := nξ ( n ) , ∀ ξ ∈ ℓ ( N ) , and T r is the usual trace on B ( ℓ ( N )) , (2) if ( L , H , Γ; M , τ ) is odd, L ⋊ := ∗ alg( i A ( L ) , t ) , H ⋊ := H ⊗ ℓ ( N ) ⊗ C ,D ⋊ := D ⊗ I ⊗ ε + I ⊗ D N ⊗ ε , Γ ⋊ := I ⊗ I ⊗ ε , M ⋊ := M ⊗ B ( ℓ ( N )) ⊗ M ( C ) , τ ⋊ := τ ⊗ T r ⊗ tr, where tr is the normalized trace on M ( C ) .Moreover, in both cases, if d is the dimension of the original spectral triple, then thedimension of the new spectral triple is d + 1 .Proof. We only prove the even case, the odd case being similar. Let us first observe that, since α is an automorphism, A ∞ = A , α ∞ = α , and i A ( a ) = f π u ( a ) p , ∀ a ∈ A . Let π : A → B ( H ) bethe representation implied by the spectral triple ( L , H , D, Γ; M , τ ), and consider ( e π ( a ) ξ )( n ) := π ( α − n ( a )) ξ ( n ), ∀ a ∈ A , ξ ∈ H ⊗ ℓ ( N ), n ∈ N , which is a representation of A on H ⊗ ℓ ( N ),and the shift operator ( W ξ )( n ) := ( , n = 0 ,ξ ( n − , n ≥ . Then, it is easy to see that ( H ⊗ ℓ ( N ) , e π, W ) is a covariant representation of ( A , α, N ) on H ⊗ ℓ ( N ), in the sense of Definition 2.3. Therefore it induces a non-degenerate representation b π of A ⋊ α N = h i ( A ) , t i on H ⊗ ℓ ( N ), such that b π ◦ i A = e π , and b π ( t ) = W . Hence b π ( A ⋊ α N ) ⊂ M ⋊ ,while the facts that D ⋊ b ∈ M ⋊ , and (1 + D ⋊ ) − is τ ⋊ -compact follow from Proposition 3.6. Itremains to prove that k [ D ⋊ , b π ( a )] k < ∞ , ∀ a ∈ L ⋊ . Since the commutators [Γ ⊗ D N , b π ( a )]and [ D ⊗ I, W ] vanish, while k [Γ ⊗ D N , W ] k ≤
1, it is enough to estimate the commutators k [ D ⊗ I, b π ( a )] k = k diag { [ D, π ( α − n ( a ))] : n ∈ N }k = sup n ∈ N k [ D, π ( α − n ( a ))] k < ∞ , and theclaim follows.We now prove the statement about the dimension, which in turn implies (again) the τ -compactnessof the resolvent. By Proposition 3.4, the Hausdorff dimension of D ⋊ is given bylim sup t → + ∞ log(Λ t ( D ⋊ ))log t . We observe that Λ t ( D N ) = [ t ] and thuslim sup t →∞ log Λ t ( D N )log t = lim t →∞ log([ t ])log t = 1 . Now by applying Proposition 3.6 we are done. (cid:3)
The next result has to do with the case of crossed products with respect to endomorphisms.
Theorem 3.9.
Let A be a unital C ∗ -algebra, α ∈ End( A ) an injective, unital ∗ -endomorphism, A ∞ = lim −→ A the inductive limit described in (2.1) , and ( L ∞ , H ∞ , D ∞ ; M ∞ , τ ∞ ) a semifinitespectral triple of dimension p on A ∞ . If the morphism α ∞ ∈ Aut( A ∞ ) is Lip-semibounded,then there exists a semifinite spectral triple ( L ⋊ , H ⋊ , D ⋊ ; M ⋊ , τ ⋊ ) of dimension p + 1 on thecrossed product C ∗ -algebra A ⋊ α N .Proof. Note that A ⋊ α N = A ∞ ⋊ α ∞ N . Now the claim follows by applying the previousproposition. (cid:3) Spectral triples for crossed products generated by self-coverings
In this section we exhibit some examples of semifinite spectral triples for crossed products withrespect to an endomorphism: the self-covering of a p -torus, the self-covering of the rationalrotation algebra, the endomorphism UHF algebra given by the shift, and the self-covering ofthe Sierpinski gasket. In this paper we consider two pictures of the inductive limits. One iswhat we call the Cuntz picture. The other one deals with an increasing sequence of algebras A i with the morphisms ϕ i : A i → A i +1 being the inclusions, which entails that the morphisms α i : A i → A i are injective. The following result gives a more detailed description of the secondpicture. Proposition 4.1.
Given a family of algebras { A i } i ≥ , a morphism α : A → A , a collectionof isomorphisms β i : A i → A i +1 for all i ∈ N , one can obtain the following commuting diagram A ϕ / / α (cid:15) (cid:15) A ϕ / / α (cid:15) (cid:15) A ϕ / / α (cid:15) (cid:15) · · · / / A ∞ α ∞ (cid:15) (cid:15) A ϕ / / β > > ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ A ϕ / / β > > ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ A ϕ / / β > > ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ · · · / / A ∞ where the morphisms α i : A i → A i are defined by the formula α i := β i − ◦ α i − ◦ β − i − for i ≥ , ϕ := β ◦ α , ϕ i := α i +1 ◦ β i = β i ◦ α i for i ≥ . Moreover, the morphisms { ϕ i } i ≥ give rise toan inductive limit that we denote by A ∞ and the former morphisms { α i } i ≥ and { β i } i ≥ inducemorphisms α ∞ , β ∞ : A ∞ → A ∞ that are inverses of each other.Proof. The first part of the statement, namely the one concerning the commuting diagram,follows by direct computations. Now we take care of the second part concerning the morphisms α ∞ and β ∞ . We observe that α ∞ ( f , f , . . . ) = ( α ( f ) , α ( f ) , . . . ) β ∞ ( f , f , . . . ) = (0 , β ( f ) , β ( f ) , . . . ) PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 17 for all ( f , f , . . . ) ∈ A ∞ . On the one hand, we have that α ∞ ◦ β ∞ ( f , f , . . . ) = α ∞ (0 , β ( f ) , β ( f ) , . . . )= (0 , α ◦ β ( f ) , α ◦ β ( f ) , . . . ) . On the other hand, we have that β ∞ ◦ α ∞ ( f , f , . . . ) = β ∞ ( α ( f ) , α ( f ) , . . . )= (0 , β ◦ α ( f ) , β ◦ α ( f ) , . . . ) . Since α i +1 ◦ β i = β i ◦ α i we are done. (cid:3) Before the discussion of the examples, we introduce some notation. We will consider an invert-ible matrix B ∈ M p ( Z ) and we will set A := ( B T ) − . The following exact sequence will play arole in the definition of some of the Dirac operators0 → Z p → A Z p → c Z B := A Z p / Z p → . Moreover, we will consider a section s : c Z B → A Z p such that s ( · ) ∈ [0 , p . We set s h ( x ) := A h − s ( x ) as in [1], p. 1387-1388. Note that | c Z B | = | det( B ) | =: r .4.1. The crossed product for the self-coverings of the p -torus. We begin with the caseof tori. The p -torus T p := R p / Z p can be endowed with a Dirac operator acting on the Hilbertspace H := C [ p/ ⊗ L ( T p , dm ) D := − i p X a =1 ε a ⊗ ∂ a , where the matrices ε a = ( ε a ) ∗ ∈ M [ p/ ( C ), ε a ε b + ε b ε a = 2 δ a,b , furnish a representation of theClifford algebra for the p -torus (see [19] for more information on Dirac operators). Then, wemay consider the following spectral triple( L := C ( T p ) , H , D ) . We recall that the spectral triple considered for the torus is even precisely when p is even.With the above notation and B ∈ M p ( Z ), let π : t ∈ T p Bt ∈ T p be the self-covering, α ( f )( t ) = f ( Bt ) the associated endomorphism of A = C ( T p ). Then we consider the inductivesystem (2.1) and construct the inductive limit A ∞ = lim −→ A n . An alternative description isgiven by the following isomorphic inductive family: A n consists of continuous B n Z p -periodicfunctions on R p , and the embedding is the inclusion.Assume now that B is purely expanding, namely k B n v k goes to infinity for all vectors v =0, hence k A k <
1, where A = ( B T ) − . In [1], we produced a semifinite spectral triple on A ∞ = lim −→ C ( T n ). More precisely, we constructed a Dirac operator D ∞ acting on H ∞ := C [ p/ ⊗ L ( T p , dm ) ⊗ L ( R , τ ) D ∞ := D ⊗ I − π p X a =1 ε a ⊗ I ⊗ (cid:18) ∞ X h =1 I ⊗ h − ⊗ diag( s h ( · ) a ) (cid:19) , the algebra L ∞ := ∪ n ∈ N C ( T n ) ⊂ A ∞ embeds into the injective limitlim −→ B ( H ) ⊗ M r n ( C ) = B ( C [ p/ ⊗ L ( T p , dm )) ⊗ UHF( r ∞ ) , which in turn embeds into M ∞ := B ( C [ p/ ⊗ L ( T p , dm )) ⊗ R , where R denotes the uniqueinjective type II factor obtained as the weak closure of the UHF algebra in the GNS rep-resentation of the unital trace, and we denote by τ ∞ := T r ⊗ τ R the trace on M ∞ . Then( L ∞ , H ∞ , D ∞ ; M ∞ , τ ∞ ) is a finitely summable, semifinite, spectral triple on lim −→ A n , with Haus-dorff dimension p . Theorem 4.2.
Under the above hypotheses and with the notation of the former section, C ( T p ) ⋊ α N can be endowed with the finitely summable semifinite spectral triple ( L ⋊ , H ⋊ , D ⋊ ; M ⋊ , τ ⋊ ) ofTheorem 3.9, with Hausdorff dimension p + 1 .Proof. In order to construct a spectral triple on C ( T p ) ⋊ α N , according to Theorem 3.9, we onlyneed to check that α ∞ is Lip-semibounded, that issup {k [ D ∞ , α − n ∞ ( f )] k , n ∈ N } < ∞ , ∀ f ∈ L ∞ = ∪ n ∈ N C ( T n ) . Let f ∈ C ( T k ). As observed in [1], the seminorms L D ∞ , L D , L D , . . . are compatible andwe have that k [ D ∞ , α − n ∞ ( f )] k = k [ D , f ◦ B − n ] k Moreover, by using the relation ε a ε b + ε b ε a = 2 δ a,b we obtain the following equalities k [ D , f ] k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ε a ⊗ ∂ a ( f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ε a ⊗ ∂ a ( f ) ! p X a =1 ε a ⊗ ∂ a ( f ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ( ε a ) ⊗ ( ∂ a f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ⊗ ( ∂ a f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 19 Now we compute k [ D , f ◦ B − n ] k . Setting X = B − n for simplicity, we have that k [ D , f ◦ X ] k = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ε a ⊗ ∂ a ( f ◦ X ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ε a ⊗ ∂ a ( f ◦ X ) ! p X a =1 ε a ⊗ ∂ a ( f ◦ X ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ε a ⊗ p X i =1 X a,i ∂ i ( f ◦ X ) !! p X b =1 ε b ⊗ p X j =1 X b,j ∂ j ( f ◦ X ) !!(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ( ε a ) ⊗ p X i,j =1 X a,i X a,j ∂ i ( f ◦ X ) ∂ j ( f ◦ X )+ X ab ε a ε b ⊗ p X i,j =1 X a,i X b,j ∂ i ( f ◦ X ) ∂ j ( f ◦ X ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ⊗ p X i,j =1 X a,i X a,j ∂ i ( f ◦ X ) ∂ j ( f ◦ X ) !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k ( ∇ ( f ◦ X ) , X ∗ X ∇ ( f ◦ X )) k≤ k X ∗ X k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p X a =1 ⊗ ( ∂ a f ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = k X k k [ D, f ] k . These computations and the hypothesis on B being purely expanding (cf. Proposition 2.6 in[1]) imply thatsup {k [ D ∞ , α − n ∞ ( f )] k , n ∈ N } ≤ sup {k B − n kk [ D ∞ , f ] k , n ∈ N } < ∞ . (cid:3) The following example is associated with a regular noncommutative self-covering with finiteabelian group of deck transformations [1].
Definition 4.3.
A finite (noncommutative) covering with abelian group is an inclusion of(unital) C ∗ -algebras A ⊂ B together with an action of a finite abelian group Γ on B such that A = B Γ . We will say that B is a covering of A with deck transformations given by the groupΓ.4.2. The crossed product for the self-coverings of the rational rotation algebra.
Weare now going to give a description of the rational rotation algebra making small modificationsto the description of A θ , θ = p/q ∈ Q , seen in [4]. Consider the following matrices( U ) hk = δ h,k e πi ( k − θ , ( V ) hk = δ h +1 ,k + δ h,q δ k, ∈ M q ( C ) and W ( n ) := U n V n , for all n = ( n , n ) ∈ Z . Let p ′ , p ′′ ∈ N , p ′ , p ′′ < q , be such that pp ′ + 1 = n ′ q , pp ′′ − n ′′ q , for some n ′ , n ′′ ∈ N , and introduce P := (cid:18) p ′ p ′′ (cid:19) , and e γ n ( f )( t ) := ad( W ( P n ))[ f ( t + n )] = V − p ′′ n U − p ′ n f ( t + n ) U p ′ n V p ′′ n , for all t ∈ R , n ∈ Z . We have the following description of A θ (cf. [4]) A θ = { f ∈ C ( R , M q ( C )) : f = e γ n ( f ) , n ∈ Z } . This algebra comes with a natural trace τ ( f ) := 1 q Z T tr( f ( t )) dt, where we are considering the Haar measure on T and tr( A ) = P i a ii . We observe that thefunction tr( f ( t )) is Z -periodic.Define L θ := (X r,s a rs U r V s : ( a rs ) ∈ S ( Z ) ) , where S ( Z ) is the set of rapidly decreasing sequences. It is clear that the derivations ∂ and ∂ , defined as follows on the generators, extend to L θ ∂ ( U h V k ) = 2 πihU h V k ∂ ( U h V k ) = 2 πikU h V k . Moreover, the above derivations extend to densely defined derivations both on A θ and L ( A θ , τ ).We still denote these extensions with the same symbols. We may consider the following spectraltriple (see [10])( L := L θ , H := C ⊗ L ( A θ , τ ) , D := − i ( ε ⊗ ∂ + ε ⊗ ∂ )) , where ε , ε denote the Pauli matrices. In order to fix the notation we recall that the Paulimatrices are self-adjoint, in particular they satisfy the condition ( ε k ) = I , k = 1 , B ∈ M ( Z ) such that det( B ) ≡ q
1, there is an associatedendomorphism α : A θ → A θ defined by α ( f )( t ) = f ( Bt ), [27]. Then, we consider the inductivelimit A ∞ = lim −→ A n as in (2.1). As in the case of the torus one can consider the followingisomorphic inductive family: A n consists of continuous B k Z -invariant matrix-valued functionson R , i.e A k := { f ∈ C ( R , M q ( C )) : f = e γ B k n ( f ) , n ∈ Z } , with trace τ k ( f ) = 1 q | det B k | Z T k tr( f ( t )) dt, and the embedding is unital inclusion α k +1 ,k : A k ֒ → A k +1 . In particular, A = A , and A = B .This means that A ∞ may be considered as a solenoid C ∗ -algebra (cf. [20], [18]).On the n -th noncommutative covering A n , the formula of the Dirac operator doesn’t changeand we can consider the following spectral triple( L ( n ) θ , C ⊗ L ( A n , τ ) , D = − i ( ε ⊗ ∂ + ε ⊗ ∂ )) . PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 21 In [1], we produced a semifinite spectral triple on A ∞ = lim −→ A n . More precisely, we constructeda Dirac operator D ∞ acting on H ∞ := C ⊗ L ( A , τ ) ⊗ L ( R , τ ) D ∞ := D ⊗ I − π X a =1 ε a ⊗ I ⊗ (cid:18) ∞ X h =1 I ⊗ h − ⊗ diag( s h ( · ) a ) (cid:19) , the algebra A ∞ embeds into the injective limitlim −→ B ( C ⊗ L ( A , τ )) ⊗ M r n ( C ) = B ( C ⊗ L ( A , τ )) ⊗ UHF( r ∞ )which in turn embeds into M ∞ := B ( C ⊗ L ( A , τ )) ⊗ R , which is endowed with the trace τ ∞ := T r ⊗ τ R . Then ( L ∞ , H ∞ , D ∞ ; M ∞ , τ ∞ ) is a finitely summable, semifinite, spectral tripleon lim −→ A n , with Hausdorff dimension 2 ([1], Theorem 3.7). Theorem 4.4.
Under the above hypotheses and with the notation of the former section, A θ ⋊ α N can be endowed with the finitely summable semifinite spectral triple ( L ⋊ , H ⋊ , D ⋊ ; M ⋊ , τ ⋊ ) ofTheorem 3.9, with Hausdorff dimension .Proof. According to Theorem 3.9 we only need to check that α ∞ is Lip-semibounded, that issup {k [ D ∞ , α − n ∞ ( f )] k , n ∈ N } < ∞ , ∀ f ∈ L ∞ . This is true because similar computations to those in the proof of Theorem 4.2 yieldsup {k [ D ∞ , α − n ∞ ( f )] k , n ∈ N } ≤ sup {k B − n kk [ D ∞ , f ] k , n ∈ N } The hypothesis of B being purely expanding ensures that sup {k [ D ∞ , α − n ∞ ( f )] k , n ∈ N } is finite. (cid:3) The crossed product for the shift-endomorphism of the UHF-algebra.
Considernow the case of the UHF-algebra of type r ∞ . This algebra is defined as the inductive limit ofthe following sequence of finite dimensional matrix algebras: M = M r ( C ) M n = M n − ⊗ M r ( C ) n ≥ , with maps φ ij : M j → M i given by φ ij ( a j ) = a j ⊗
1. We denote by A the C ∗ -algebra UHF( r ∞ )and set M − = C A in the inductive limit defining the above algebra. The C ∗ -algebra A has aunique normalized trace that we denote by τ .Consider the projection P n : L ( A , τ ) → L ( M n , Tr), where Tr : M n → C is the normalizedtrace, and define Q n := P n − P n − , n ≥ ,E ( x ) := τ ( x )1 A . For any s >
1, Christensen and Ivan [5] defined the following spectral triple for the algebraUHF( r ∞ ) def = A ( L , L ( A , τ ) , D = X n ≥ r ns Q n )where L is the algebra consisting of the elements of A with bounded commutator with D .It was proved that for any such value of the parameter s , this spectral triple induces a metricwhich defines a topology equivalent to the weak ∗ -topology on the state space ([5, Theorem3.1]). We consider the endomorphism of A given by the right shift, α ( x ) = 1 ⊗ x . Then as in (2.1)we may consider the inductive limit A ∞ = lim −→ A n . As in the previous sections, we have thefollowing isomorphic inductive family: A i is defined as A = A ; A n = M r ( C ) ⊗ n ⊗ A ; A ∞ = lim −→ A i and the embedding is the inclusion.In [1], we produced a semifinite spectral triple on lim −→ A n . More precisely, we defined thefollowing Dirac operator acting on H ∞ := L ( R , τ ) ⊗ L ( A , τ )(4.1) D ∞ = I −∞ , − ⊗ D + ∞ X k =1 r − sk I −∞ , − k − ⊗ F ⊗ E, where I −∞ ,k is the identity on the factors with indices in [ −∞ , k ], F : M r ( C ) → M r ( C ) ◦ isdefined as F ( x ) := x − tr( x )1 for x ∈ M r ( C ), and the algebra A ∞ embeds in the injective limitlim −→ B ( L ( A , τ )) ⊗ M r n ( C ) = B ( L ( A , τ )) ⊗ UHF( r ∞ )Set L ∞ = ∪ n L n , M ∞ = R ⊗ B ( L ( A , τ )), τ ∞ := τ R ⊗ T r . Then ( L ∞ , H ∞ , D ∞ ; M ∞ , τ ∞ ) is afinitely summable, semifinite, spectral triple, with Hausdorff dimension 2 /s ([1], Theorem 5.6). Theorem 4.5.
Under the above hypotheses and the notation of the former section,
UHF( r ∞ ) ⋊ α N can be endowed with the finitely summable semifinite spectral triple ( L ⋊ , H ⋊ , D ⋊ ; M ⋊ , τ ⋊ ) ofTheorem 3.9, with Hausdorff dimension /s .Proof. According to Theorem 3.9, in order to construct a spectral triple on A ⋊ α N we onlyneed to check that α ∞ is Lip-semibounded, that issup {k [ D ∞ , α − k ∞ ( f )] k , k ∈ N } < ∞ , ∀ f ∈ L ∞ . This is true because k [ D ∞ , α − k ∞ ( f )] k = r − ks k [ D ∞ , f ] k . In fact, let f = ( N − n − k = −∞ I ) ⊗ a ∈ A n , α k ∞ ( f ) = ( N − n + k − j = −∞ I ) ⊗ a ∈ A n − k for k ∈ Z .The Hilbert space on which D ∞ acts is the completion of A ∞ . On this Hilbert space, weconsider the right shift on the factors and we denote it by U α . We set Φ := ad( U α ). Then wehave that [ D ∞ , α − k ∞ ( f )] = X h ∈ Z r hs [ Q h , − n − k − O j = −∞ I ! ⊗ a ]= Φ − k X h ∈ Z r hs [ Q k + h , n − O j = −∞ I ! ⊗ a ] ! = r − ks Φ − k ([ D ∞ , f ])where we used that Φ( Q h ) = Q h +1 and Φ ↾ A ∞ = α ∞ . (cid:3) PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 23 v v v v x , x , w − v x , = v w − v R , R , Figure 1.
The Sierpinski gasket K = K and the covering map p = p : K → K .4.4. The crossed product for the self-coverings of the Sierpiski gasket.
We concludethis paper with the case of a self-covering of the Sierpinski gasket that was studied by theauthors in [2]. The Sierpinski gasket is the self-similar fractal determined by 3 similarities withscaling parameter 1/2 centered in the vertices v = (0 , v = (1 / , √ / v = (1 , K such that K = [ j =0 , , w j ( K ) , where w j is the dilation around v j with contraction parameter 1 / V ( K ) the set { v , v , v } , and let E ( K ) := { ( p, q ) : p, q ∈ V , p = q } . We call an element ofthe family { w i ◦ · · · ◦ w i k ( K ) : k ≥ } a cell , and call its diameter the size of the cell. We callan element of the family E ( K ) = { w i ◦ · · · ◦ w i k ( e ) : k ≥ , e ∈ E ( K ) } an (oriented) edge of K and we denote by e − (resp. e + ) the source (resp. the target) of the oriented edge e . Note thata cell C := w i ◦ · · · ◦ w i k ( K ) has size( C ) = 2 − k and, if e ∈ E ( K ), then e = w i ◦ · · · ◦ w i k ( e )has length 2 − k .In the following we shall consider K := K , E := E ( K ), K n := w − n K . Let us now considerthe middle point x i,i +1 of the segment ( w − v i , w − v i +1 ), i = 0 , ,
2, the map R i +1 ,i : w − w i K → w − w i +1 K consisting of the rotation of π around the point x i,i +1 , i = 0 , , p : K → K and φ : K → K given by p ( x ) = x, x ∈ K,R , ( x ) , x ∈ w − w K,R , ( x ) , x ∈ w − w K, and φ ( x ) = w − x if x ∈ C R , ( w − ( x )) if x ∈ C R , ( w − ( x )) if x ∈ C Note that p ( x ) = φ ( w ( x )) for all x ∈ K (see Figure 4.4). Similarly, for every n ≥
0, we define a family of coverings p n : K n +1 → K n and φ n : K n → K n by p n +1 := w − n ◦ p ◦ w n and φ n := w − n ◦ φ ◦ w n . Proposition 4.6.
The following diagrams are commutative K K p o o K p o o · · · p o o K φ O O K p o o φ O O K p o o φ O O · · · p o o Proof.
Indeed, first note that φ ◦ p = φ ◦ p = p ◦ φ and w ◦ φ = p , which implies that p ◦ φ = φ ◦ p . Then, for any n ≥ p n ◦ φ n = w − n +10 ◦ p ◦ w n − ◦ w − n ◦ φ ◦ w n = w − n +10 ◦ p ◦ w − ◦ φ ◦ w ◦ w n − = w − n +10 ◦ p ◦ φ ◦ w n − = w − n +10 ◦ φ ◦ p ◦ w n − = φ n − ◦ p n . (cid:3) It follows that the maps { φ n } n ≥ induce a map in the projective limit and by functoriality amap on lim −→ C ( K i ) which we denote by α ∞ . An element f ∈ C ( K n ) can be seen in lim −→ C ( K i ) asthe sequence [ f ] = (0 n , f, f ◦ p n +1 , f ◦ p n,n +2 , . . . ), where p n,n + k := p n +1 ◦ · · · ◦ p n + k . Accordinglythe map α ∞ reads as α ∞ [ f ] := (0 n , f ◦ φ n , f ◦ p n +1 ◦ φ n +1 , f ◦ p n,n +2 ◦ φ n +2 , . . . ) . By functoriality each ( φ n ) ∗ : C ( K n ) → C ( K n ) is a proper endomorphism, that is, it is injective,but not surjective. With the notation of Proposition 4.1, we set β i equal to w ∗ for all i ≥ α ∞ is invertible and its inverse is given by α − ∞ [ f ] := (0 n +1 , f ◦ w , f ◦ p n +1 ◦ w , f ◦ p n,n +2 ◦ w , . . . ) . Denote by E n := { w − n e, e ∈ E ( K ) } , E ∞ := ∪ n ≥ E n , E n := { e ∈ E ∞ , length( e ) = 2 n } , P n theprojection of ℓ ( E ∞ ) onto ℓ ( E n ). It was shown in [2, Sec. 6] that A ∞ := lim −→ C ( K n ) supportsa semifinite spectral triple ( L ∞ , H ∞ , D ∞ ; M ∞ , τ ∞ ), where M ∞ := π τ ( B ∞ ) ′′ is a suitable closureof the geometric operators (see [2, Sec. 5] for a precise definition), D ∞ := F | D | : ℓ ( E ∞ ) → ℓ ( E ∞ ), F is the orientation reversing operator on edges and | D ∞ | := X n ∈ Z − n P n . Theorem 4.7.
Under the above hypotheses and with the notation of the former section, C ( K ) ⋊ α N can be endowed with the finitely summable semifinite spectral triple ( L ⋊ , H ⋊ , D ⋊ ; M ⋊ , τ ⋊ ) ofTheorem 3.9, with Hausdorff dimension log .Proof. According to Theorem 3.9, in order to construct a spectral triple on C ( K ) ⋊ α N we onlyneed to check that α ∞ is Lip-semibounded, that issup k ≥ k [ D ∞ , α − k ∞ ( f )] k < ∞ , ∀ f ∈ L ∞ := ∪ n ≥ Lip( K n ) . We are going to show that for any f ∈ C ( K n ) it holds k [ D ∞ , α − k ∞ ( f )] k = k [ D ∞ , f ] k k k ∈ N . PECTRAL TRIPLES ON IRREVERSIBLE C ∗ -DYNAMICAL SYSTEMS 25 Indeed, since both p n and φ n are isometries, we have that k [ D ∞ , α − k ∞ ( f )] k = ⊕ e ∈ E ∞ α − k ∞ ( f )( e + ) − α − k ∞ ( f )( e − ) l ( e ) F = ⊕ e ∈ E ∞ f ( w k ( e + )) − f ( w k ( e − )) l ( e ) F = ⊕ e ∈ E ∞ f ( w k ( e + )) − f ( w k ( e − ))2 k l ( w k ( e )) F = ⊕ e ′ ∈ E ∞ f ( e ′ + ) − f ( e ′− )2 k l ( e ′ ) F = k [ D ∞ , f ] k k . (cid:3) Acknowledgement
This work was supported by the following institutions: the ERC Advanced Grant 669240QUEST ”Quantum Algebraic Structures and Models”, the MIUR PRIN “Operator Algebras,Noncommutative Geometry and Applications”, the INdAM-CNRS GREFI GENCO, and theINdAM GNAMPA. V. A. acknowledges the support by the Swiss National Science foundationthrough the SNF project no. 178756 (Fibred links, L-space covers and algorithmic knot the-ory). D. G. and T. I. acknowledge the MIUR Excellence Department Project awarded to theDepartment of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.
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