UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA DILATION THEORY FOR C*-DYNAMICAL SYSTEMS

UNIVERSITÀ DEGLI STUDI DI ROMA LA SAPIENZA FACOLTÀ DI SCIENZE MATEMATICHE, FISICHE E NATURALI DILATION THEORY FOR C*-DYNAMICAL SYSTEMS CARLO PANDISCIA DOTTORATO DI RICERCA IN MATEMATICA XVIII CICLO Relatore Prof. László Zsidó

Contents. Introduction 4 Chapter. Dynamical systems and their dilations. Preliminaries 2. Stinespring Dilations for the cp map 2 3. Nagy-Foiaş Dilations Theory 8 4. Dilations Theory for Dynamical Systems 0 5. Spatial Morphism 7 Chapter 2. Towards the reversible dilations 23. Multiplicative dilation 23 2. Ergodic property of the dilation 38 Chapter 3. C*-Hilbert module and dilations 47. Definitions and notations 47 2. Dilations constructed by using Hilbert modules 49 3. Ergodic property 5 Appendix A. Algebraic formalism in ergodic theory 54 Appendix. Bibliography 56 3

. INTRODUCTION 4. Introduction In the operator framework of quantum mechanics we define a dynamical system by thetriple(a, Φ,ϕ), where A is a C -algebra, Φ is an unital completely positive map and ϕ is a state on A. In particular, if this map Φ is a *-automorphism, (A, Φ,ϕ)issaidbe a conservative dynamical system. The dilation problem for dynamical system (A, Φ,ϕ) is related with question wheter it is possible to interpret an irrevesible evolution of a physical system as the projection of a unitary reversible evolution of a larger system ba, Φ, b bϕ [9]. In [26] wefind a good description of what we intend for dilation of a dynamical system: The idea of dilation is to understand the dynamics Φ of A as projection from the dynamics Φ b of A. b In statistical physic the algebras A and A b may be considered as algebras of quantum mechanical observable so that A models the description of a small system embedded into a big one modelled by A. b In the classical example A is the algebra of random variables describing a brownian particle moving on a liquid in thermal equilibrium and A b is the algebra of random variables describing both the molecules of the liquid and particle. Many authors in the last years have studied the dilatative problem, we cite the pioneer works of Arveson [], Evans and Lewis [7], [8], and Vincent-Smith [3]. In absence of an invariant faithful state, Arveson, Evans and Lewis have verified that the dilations have been constructed for every completely positive map defined on W -algebra, while Vincent-Smith using a particular definition of dilation, shows that every W -dynamical system admits a reversible dilation. In our work we will assume the concept of dilation given by Kümmerer and Maassen in [2] and[3]. It is our opinion that this definition is that that describes better the physical processes. The statement of the problem is the following: Given a dynamical system (A, Φ,ϕ), to construct a conservative dynamical system ba, Φ, b bϕ containing it in the following sense. there is an injective linear *-multiplicative map i : A A b and a projection E of norm one of A b onto i (A) such that the diagram bφ n ba A b bϕ bϕ &. i C E ϕ ϕ % - A Φ n commutes for each n N. The ba, Φ, b bϕ, i, E is said to be a reversible dilation of the dynamical system (A, Φ,ϕ), furthermore an dilation is unital if the injective map i : A b A is unital. Kummer in [2] estabilishes that the existence of a reversible dilation depends on the existence of adjoint map in this sense: A completely positive map Φ + : A A is a ϕ adjoint of the completely positive map b A

. INTRODUCTION 5 Φ if for each a, b belongs to A we obtain that ϕ (b (Φ (a))) = ϕ (Φ + (b) a). The principal purpose of our work is to establish under which condition is possible to costruct a reversible dilation that keeps the ergodic and weakly mixing properties of the original dynamical system. An found difficulty has been that to determine the existence of the expectation conditioned as described in the preceding scheme (In fact generally, the exisistence of a conditional expectation between C*-algebras is fairly exceptional.) and the presence of an invariant state subsequently complicates the matters. This thesis is organized as follow. In chapter we introduce some preliminaries concept and we show the following generalization of the theorem of Stinespring: Gives an unital completely positive map Φ : A A on C*-algebra with unit A, there is a representation (H,π)ofA andanisometryv on the Hilbert space H such that π (Φ (a)) = Vπ (a) V for each element a belong to A. Subsequently we have used results contained in the paper [20] to show that all W -dynamical systems for which the dinamic Φ is a *-homomorphism with ϕ adjoint, admit an unital reversible dilation. In chapter 2 using the generalized Stinespring theorem and Nagy-Foias dilation theory for the linear contraction on Hilbert space, we proof that every dynamical system (A, Φ,ϕ) has a multiplicative dilations ba, Φ, b bϕ, i, E, that is a dilation in which the dynamic bφ : A b A b is not a *-automorphism of algebras, but an injective *-homomorphism. This dilation keeps ergodic and weakly mixing properties of the original dynamic system. We also recover a results on the existence of dilation for W -dynamical systems determined by Muhly-Solel their paper [6]. We make to notice that our proof differs for the method and the approach to that of the two preceding authors. For the methodologies applied by the authors, and relative results, the reader can see the further jobs [5] and[7]. In chapter 3 we apply Hilbert module methods to show the existence of a particular dilations cm, Φ, b bϕ, i, E of W*-dynamical system (M, Φ,ϕ)wherethedynamicΦ b is a completely positive map such that M is included in the multiplicative domains D bφ of Φ. b Also cm, Φ, b bϕ, i, E keeps the ergodic and mixing properties of the C*-dynamical system (M, Φ,ϕ). For the existence of expectation conditioned the reader can see Takesaki [29].

CHAPTER Dynamical systems and their dilations In this chapter using the results of Niculescu, Ströh and Zsido contained in their paper [20], we have show that a dynamical system with dynamics described by a homomorphism that admits adjoint as defined by Kummerer in [2], can be dilated to a minimal reversible dynamical system. Moreover this reversible system take the ergodic property of the original dynamical system. Fundamental ingredient of the proof is the the theory of the dilation of Nagy-Foias for the linear contractions on the Hilbert space. Preliminaries In this first section, we shortly introduce some results on the completely positive maps. For further details on the subject, the reader can see the Paulsen s books cited in the bibliography. A self-adjoint subspace S of a C*-algebra A that contains the unit of A is called operator system of A, while a linear map Φ : S B between the operator system S and the C*-algebra B is positive if it maps positive elements of S in positive elements of B. The set of all n n matrices, with entries from S, is denoted with M n (S). We define a new linear map Φ n : M n (S) M n (A) thusdefined: Φ n x i,j i,j = Φ (x i,j ) i,j, x i,j S, i, j =, 2...n. The linear map Φ is said be n-positive if the linear map Φ n is positive and we call Φ completely positive if Φ is n-positive for all n N. WeobservethatifA and B are C*-algebra, a linear map Φ : A B is cp-map if and only if P b i Φ (a i a j ) b j 0 i,j for each a,a 2,...a n A and b,b 2...b n B. Proposition.. If Φ : S B is a cp-map, then kφk = kφ ()k Proof. See [22] proposition 3.5. If Φ : A B is an unital cp map between C*-algebras, we have that Φ has norm. A fundamental result in the theory of the cp-maps is given by the extension theorem of Arveson []: Proposition.2. Let S be an operator system of the C*-algebra A, andφ : S B (H) a cp-map. Then there is a cp-map, Φ ar : A B (H), extendingφ. Briefly cp-map.

2. STINESPRING DILATIONS FOR THE CP MAP 2 Proof. See [22] proposition 6.5. Let us recall the fundamental definition of conditional expectation. Let B be a Banach algebra (in generally without unit) and let A be a subalgebra of Banach of B. We recal that a projection P is a continuous linear map from B onto A satisfying P (a) =a for each a A, while a quasi-conditional expectation Q is a projection from B onto A satisfying Q (xby) =xq (b) y for each x, y A, and b B. An conditional expectation is a quasi-conditional expectation of norm. InthecasethatA and B are C*-algebras there is the following result of the 957 of Tomiyama: Proposition.3. The linear map E : B A is a conditional expectation if and only if is a projection of norm. Proof. See [2], proposition 6.0. We observe that every conditional expectation is a cp-map. In fact for each a,a 2,...a n A and b,b 2...b n B, we obtain: Ã! P P a i E (b i b j ) a j = E a i b i b j a j 0. i,j i,j The multiplicative domains of the cp map Φ : A B is the set D (Φ) ={a A : Φ (a ) Φ (a) =Φ (a a)andφ (a) Φ (a )=Φ (aa )}, () furthermore we have the following relation (cfr.[22]): a D (Φ) if and only if Φ (a) Φ (b) =Φ (ab), Φ (b) Φ (a) =Φ (ba) for all b A. 2. Stinespring Dilations for the cp map We examine a concrete C*-algebra A of B (H) with unit and an unital cp-map Φ : A A. By the Stinespring theorem for the cp-map Φ, we can deduce a triple (V Φ,σ Φ, L Φ ) constituted by a Hilbert space L Φ, of the reprensentation σ Φ : A B (L Φ ) and a linear contraction V Φ : H L Φ such that Φ (a) =V Φσ Φ (a) V Φ, a A. (2) We recall to the reader 2 that the Hilbert space L Φ is the quotient space of A Φ H by the equivalence relation given by the linear space {a Φ Ψ : ka Ψk =0}, where ha Φ Ψ ; a 2 Φ Ψ 2 i LΦ = hψ ; Φ (a a 2 ) Ψ 2 i H and σ Φ (a) x Φ Ψ = ax Φ Ψ, for each x Φ Ψ L Φ with V Φ Ψ = Φ Ψ for each Ψ H. Since Φ is unital map the linear operator V Φ is an isometry whit adjoint VΦ defined by VΦa Φ Ψ = Φ (a) Ψ, for each a A and Ψ H. 2 For further details cfr.[22] and[23].

2. STINESPRING DILATIONS FOR THE CP MAP 3 Proposition.4. The unital cp-map Φ is a multiplicative if and only if V Φ is an unitary. Moreover for each x D (Φ) we have σ Φ (x) V Φ VΦ = V Φ VΦσ Φ (x) =σ Φ (x). Proof. For each Ψ H we obtain the follow implication: a Φ Ψ = Φ Φ (a) Ψ Φ (a a)=φ (a ) Φ (a), since ka Φ Ψ Φ Ψ (a) Ψk = hψ, Φ (a a) Ψi hψ, Φ (a ) Φ (a) Ψi. Furthermore, for each a A and Ψ H we have V Φ VΦ a Φ Ψ = Φ Φ (a) Ψ. Let Φ : A B an unital cp map between C*-algebra A and B, foreacha A we have: Φ (a a)=vφσ Φ (a ) σ Φ (a ) V Φ VΦσ Φ (a ) V Φ VΦσ Φ (a ) V Φ = Φ (a ) Φ (a), this shows that the Kadison inequality: Φ (a ) Φ (a) Φ (a a) (3) is satisfied. We now need a simple lemma: Lemma.. Let M i B (H i ) with i =, 2, are von Neumann algebra and the linear positive map Φ : M M 2 is wo continuous, then is w -continuous. Proof. Let {x α } an increasing net in M + with least upper bound x, wehavethat x α converges σ continuous to x, itfollowthatx α converges wo-continuous to x and since for hypothesis Φ (x α ) Φ (x) inm + 2 and Φ (x α) Φ (x) inwo-continuous, we have Φ (x) =lubφ (x α ), then Φ is w -continuous. A simple consequence of the lemma is the following proposition: Proposition.5. If M B (H) is a von Neumann algebra and Φ : M M is normal cp map, then the Stinespring representation σ Φ : M B (L Φ ) is normal. Proof. Let {x α } an increasing net in M + with least upper bound x, foreacha Φ Ψ L Φ we obtain: ha Φ Ψ; σ Φ (x α ) a Φ Ψi = hψ; Φ (ax α a) Ψi hψ; Φ (axa) Ψi and hψ; Φ (axa) Ψi = ha Φ Ψ; σ Φ (x) a Φ Ψi. Therefore σ Φ (x α ) σ Φ (x) inwo-topology. The Stinespring theorem admit the following extension: Theorem.. Let A beac*-algebrawithunitandφ : A A an unital cp-map, then there exists a faithful representation (π, H ) of A and an isometry V on Hilbert Space H such that: V π (a) V = π (Φ (a)) a A, (4) where σ 0 = id, Φ n = σ n Φ

2. STINESPRING DILATIONS FOR THE CP MAP 4 and (V n,σ n+, H n+ ) is the Stinespring dilation of Φ n for every n 0, L H = H j, H j = A Φj H j, for j and H 0 = H; (5) j=0 and V (Ψ 0, Ψ, Ψ 2,...)=(0, V 0 Ψ 0, V Ψ,...) for each (Ψ 0, Ψ, Ψ 2,...) H. Furthermore the map Φ is a homomorphism if and only if V V π (A) 0. Proof. By the Stinespring theorem there is triple (V 0,σ, H ) such that for each a A we have Φ (a) =V 0 σ (a) V 0. The application a A σ (Φ (a)) B (H ) is composition of cp-maps therefore also it is cp map. Set Φ (a) =σ (Φ (a)). By appling the Stinespring theorem to Φ,wehaveanewtriple(V,σ 2, H 2 ) such that Φ (a) =V σ 2 (a) V. By induction for n define Φ n (a) =σ n (Φ (a)) we have a triple (V n,σ n+, H n+ ) such that V n : H n H n+ and Φ n (a) =V nσ n+ (a) V n. We get the Hilbert space H defined in 5 and the injective reppresentation of the C*- algebra A on H : π (a) = L n 0σ n (a) (6) with σ 0 (a) =a, for each a A. Let V : H H be the isometry defined by V (Ψ 0, Ψ...Ψ n...) =(0, V 0 Ψ 0, V Ψ...V n Ψ n...), Ψ i H i. (7) The adjoint operator of V is V (Ψ 0, Ψ...Ψ n...) = V0Ψ, VΨ 2...Vn Ψ n..., Ψ i H i, (8) therefore V π L (a) V n = n 0Ψ n 0V L nσ n+ (a) V n Ψ n = L Φ n (a) Ψ n = n 0 = L σ n (Φ (a)) Ψ n = π (Φ (a)) L n. n 0 n 0Ψ We notice that let E n = V n Vn be the orthogonal projection of B (H n ), we have: E (Ψ 0, Ψ...Ψ n..) =(0, E 0 Ψ, E Ψ 2,...E n Ψ n+...). Let Φ be a multiplicative map then for each (Ψ 0, Ψ...Ψ n...) H we get: V V (Ψ 0, Ψ...Ψ n..) =(0, Ψ, Ψ 2,...Ψ n+...), (9) then V V π (a) =π (a) V V, whileforthevice-versaforeacha, b A we obtain: π (Φ (a)) π (Φ (b)) = V π (a) V V π (b) V = V π (a) π (b) V = = V π (ab) V = π (Φ (ab)).

2. STINESPRING DILATIONS FOR THE CP MAP 5 Remark.. Let M be a von Neumann algebra and Φ is normal, then the representation (π, H ) of M on H is normal, since the Stinespring representations (V n,σ n+, H n+ ) of the cp-maps Φ n = M B (H n ), are normal representations. We observe that V / π (A) andv V / π (A). Indeed if x is an element x A such that π (x) =V,wehavefordefinition that for every (Ψ 0, Ψ,...Ψ n...) H (xψ 0,σ (x) Ψ,...σ n (x) Ψ n...) =(0, V 0 Ψ 0, V Ψ,...V n Ψ n...), therefore x =0. If exists a A such that V V = π (a) thenforeach(ψ 0, Ψ...Ψ n..) H we have π (a)(ψ 0, Ψ,...Ψ n...) =(0, V 0 V0Ψ 0, V VΨ,...V n VnΨ n...) it follows that a =0. Remark.2. If x belong to multiplicative domains D (Φ) we have π (x) V V = V V π (x) =π (x). Moreover let F = I V V, we have Fπ (A) V =0if and only if the cp map Φ is multplicative. In fact for each a, b A we get (Fπ (a) V) Fπ (b) V = π (Φ (ab) Φ (a) Φ (b)). We study some simple property of the linear contraction V. Proposition.6. The linear contraction V satisfies the relation ker (I V )=ker(i V )=0. Moreover for each Ψ H, we have np lim V k np n n + Ψ = lim V n n + Ψ k =0, with D E lim Ψ, V k n Ψ =0. Moreover for each A B (H ) we obtain: lim n Vk A AV Ψ k =0. Proof. Let (Ψ 0, Ψ,...Ψ n...) H with V (Ψ 0, Ψ,...Ψ n...) =(Ψ 0, Ψ,...Ψ n...). For definition (0,V 0 Ψ 0,V Ψ, V n Ψ n...) =(Ψ 0, Ψ,...Ψ n...) it follow that (Ψ 0, Ψ,...Ψ n...) =(0, 0,...0...). It is well known that the relation ker (I V )=ker(i V )isalwaystrueforlinear contraction on the Hilbert spaces 3. np The relation lim V Ψ k = 0 follow by the mean ergodic theory of von Neumann. n n+ For the second relation we get: V k V k Ψ = 0,,0...0, J k,0 J k,0 Ψ k, J k, J k, Ψ k+, J k+,2 J k+,2 Ψ k+2... 3 See [9] proposition.3..

2. STINESPRING DILATIONS FOR THE CP MAP 6 where for each h, k N with h>kwe set: J k,h = V h V h+ V k. V k V Ψ k 2 P = n J k +α,k α J k +α,k α Ψ α 2 P n kψ α k 2 α=k α=k since J k +α,k α J k +α,k α np Then lim kψ α k 2 = 0 it follow that lim V k n α=k n V Ψ k =0. Furthermore we get: Ψ, V k A AV Ψ k kak 2 Ψ, V V k Ψ k. np Since n+ Ψ, V k Ψ 0wehaveD lim Ψ, V k n Ψ =0 4 but we get Ψ, V Ψ k P = n P hψ α, J k +α,k α Ψ α i n kψ α k 2 α=k α=k then lim Ψ, V k n Ψ =0. Proposition. leads to the following definition: Definition.. Let Φ : A A be a cp-map, a triple (π, H, V) costitued by a faithful representation π : A B (H) on the Hilbert space H and by a linear isometry V, such that for each a A we get: π (Φ (a)) = V π (a) V (0) is a isometric covariant representation of the cp map Φ. For our purposes it will be necessary to find an isometric covariant representation of appropriate dimensions, this is possible for the following theorem: Proposition.7. Let Φ : A A be cp-map with isometric representation (π, H, V), if Φ isn t an automorphism, for each cardinal number c there exist an isometric covariant representation (π c, H c, V c ) with the following property: Representation π is an equivalent subrepresentation of π c with dim H c dim (H) and dim ker (Vc ) c; Moreover there is a cp map E o : B (H c ) B (H) such that for each a A, T B (H c ) we have E o (π c (a) T )=π (a) E o (T ), with E o (Vc T V c )=V E o (T ) V; () Proof. Let c be a cardinal number and L a Hilbert space with dim (L) =c, since Φ isn t automorphism we have dim (ker V ), then there is a vector ξ ker V of one norm. We set with H c the Hilbert space H c = H L and with V c the linear isometry V c = V I L. 4 Cfr. appendix.

2. STINESPRING DILATIONS FOR THE CP MAP 7 Let {e i } i J be a orthonormal base of the Hilbert space L, wehavecard(j )=c and Since for each j J we obtain: ξ e j ker V c j J. V c (ξ e j )=(V I L )(ξ e j )=V ξ e j = 0 e j =0, it follow that dim (ker V c ) c. The faithfull *-representation π c : A B (H c )defined by satisfies the relation 0. In fact for each a A we obtain: π c (a) =π (a) I L, a A Vc π c (a) V c =(V I L )(π (a) I L )(V I L )=V π (a) V I L = = π (Φ (a)) I L = π c (Φ (a)). Let l o L vector of one norm and Π lo : H c H the linear isometry Π lo h = h l o, h H, with adjoint Π l o h l = hl, l o i h, h H, l L. The cp map E o : B (H c ) B (H) sodefined: E o (T )=Π l o T Π lo, T B (H c ) (2) for each a A, T B (H c ) enjoys of the following property: E o (π c (a) T )=π (a) E o (T ). In fact for each h,h 2 H we obtain hh 2, E o (π c (a) T ) h i = hπ c (a ) Π lo h 2,TΠ lo h i = hπ (a ) h 2 l o,tπ lo h i = = hπ (a ) h 2, Π lo T Π lo h i = hπ (a ) h 2, E o (T ) h i = hh 2,π(a) E o (T ) h i. We now verify the relation. For each h,h 2 H we have: hh 2, E o (V c T V c ) h i = hv c Π lo h 2,TV c Π lo h i = hvh 2 l o,tvh l o i = = hπ lo Vh 2,TΠ lo Vh i = Vh 2, Π l o T Π lo Vh = hvh2, E o (T ) Vh i = = hh 2, V E o (T ) Vh i. Lemma.2. Let A be an unit C*-algebra and θ o : A B (H o ) representation of A, then for every infinite cardinal number c dim (H o ) there is a representation θ : A B (H) such that θ (a) = L j Jθ o (a) with and card(j) =c. H = L j JH o

3. NAGY-FOIAŞ DILATIONS THEORY 8 Proof. Let H be an any Hilbert space with dim (H) =c with {e i } i I and {f j } j J orthonormal bases of H o and of L respectively. For definition we have that card {J} = c while card {I} =dim(h o ). The cardinal number c isn t finte then for the notes rules of the cardinal arithmetic it results that card {I J} = card {J}. Then we can write that J = {I j : j J} = {I j : j J} with card (I j )=dim(h o ). In fact for every j J the norm closure of the span {f k : k I j } is isomorphic to the Hilbert space H o. We get H = L span {f k : k I j } = L o, j J j JH and for each a A, Ψ j H o we define θ (a) L Ψ j = L o (a) Ψ j. j J j Jθ We now have a further generalization of the theorem.: Corollary.. Let Φ : A A be a cp-map. if Φ isn t an automorphism, there exists an isometric covariant representation (π, H, V) and a representation θ : A B (ker (V )) such that θ (a) = L j Jπ (a), a A, where J is a set of cardinalty dim (H) card (J) dim (H ), and H is the Hilbert space 5. Proof. Let c be the infinite cardinal number with c dim H, for the proposition.7 there is an isometric covariant representation (π c, H c, V c ) subequivalent to π with dim (ker V c ) c. Then for the preceding lemma there is a *-representation θ = L j Jπ with card(j) =dim(kerv c ). 3. Nagy-Foiaş DilationsTheory Let T and S be operators on the Hilbert spaces H and K respectively. We call S adilationoft if H is a subspace of K and the following condition is satisfied for each n N: T n Ψ = P H S n Ψ, Ψ H, where P H denotes the orthogonal projection from K onto H. Given a contraction operator T on the Hilbert space H, the defect operator D T is defined by D T = 2 I T T.

3. NAGY-FOIAŞ DILATIONS THEORY 9 Moreover we define the following operator T b on the Hilbert space K = H l 2 D T H 5 : bt = T 0 C T W, (3) where the operators W : l 2 D T H l 2 D T H and C T : H l 2 D T H are so defined: W (ξ 0,ξ...ξ n..) =(0,ξ 0,ξ...ξ n..), ξ l 2 D T H and C T h =(D T h, 0,...0...), h H, D T is the defect operator of T. Moreover for each (ξ 0,ξ,..ξ n...) l 2 D T H we have: C T (ξ 0,ξ,...ξ n...) =D T ξ 0, and C TC T = I T T. We observe that for each ξ l 2 D T H : W (ξ 0,ξ...ξ n..) =(ξ...ξ n..), and D W (ξ 0,ξ...ξ n..) =(ξ 0, 0, 0,...0..) where D W is the defect operator of the contraction W, therefore D W is the orthogonal projection of the space D T H. Obviously T b is a dilation of T and a simple calculation shows that T b is an isometric, therefore T b is an isometric dilation of T. An isometric dilation T b on K of T is minimal if H is cyclic for T; thatis K = _ bt n H, n N moreover it is shown that the 3 is the only, up to unitary equivalences, minimal dilation of T. The dilations bt, K and bt2, K 2 of T are equivalent if exists an unitary operator U : K K 2 such that UT b = T b 2 U and U H = id. We recall the following proposition: Proposition.8. Every contraction operator T on the Hilbert space H has a unitary dilation b T on a Hilbert space K such that (minimal property) K = _ n Zb T n H. The operator b T is then determined by T uniquely (up to unitary equivalences). Proof. See [8] theorem.. 5 For further details cfr.[8] and[9]

4. DILATIONS THEORY FOR DYNAMICAL SYSTEMS 0 4. Dilations Theory for Dynamical Systems We define a C -dynamical systems a couple (A, Φ) constituted by an unital C*- algebra A and an unital cp-map Φ : A A. Astateϕ on A is say be Φ invariant if for each a A we have ϕ (Φ (a)) = ϕ (a). (4) The C dynamical systems with invariant state ϕ is a triple (A, Φ,ϕ)whereϕ is a Φ invariant state on A. A W dynamical systems is a couple (M, Φ) constituted by a von Neumann Algebra M and an unital normal cp-map Φ : M M. The W dynamical systems with invariant state ϕ is a triple (M, Φ,ϕ)whereϕ is a faithful normal Φ invariant state on M. A C dynamical systems (A, Φ) issaybemultiplicative if Φ is a homomorphism, while is say be invertible if the cp-map Φ is invertible. We have a reversible C -dynamical systems (A, Φ) ifφ is an automorphism of C algebras. Remark.3. We observe that from the Kadison inequality 3, for every a A we have: ϕ (Φ (a ) Φ (a)) ϕ (a a). Let (A, Φ,ϕ)beaC -dynamical systems with invariant state ϕ and (H ϕ,π ϕ, Ω ϕ )its GNS. We define for each a A, the following operator of B (H ϕ ): U ϕ π ϕ (a) Ω ϕ = π ϕ (Φ (a)) Ω ϕ. (5) For definition, for each a A we have kπ ϕ (Φ (a)) Ω ϕ k 2 = ϕ (Φ (a ) Φ (a)) ϕ (a a)=kπ ϕ (a) Ω ϕ k 2. Then U ϕ : H ϕ H ϕ is linear contraction of Hilbert spaces. Example (Commutative case). Let (M,ϕ,Φ) be a abelian W - dynamical system, as well known, the commutative algebra M can be represented in the form L (X) for some classic probability space (X, Σ,µ) where ϕ (f) = R fdµfor each f L (X). The GNS of ϕ is costitued by L 2 (X),π ϕ, Ω ϕ whit πϕ (f) Ψ = f Ψ for each f L (X) and Ψ L 2 (X). Moreover for the linear contraction U ϕ we get U ϕ Ψ = Φ (f) Ψ for each f L (X) and Ψ L 2 (X). We have the following result for the ergodic theory: Proposition.9. Let (A, Φ,ϕ) be a dynamical system and (H ϕ,π ϕ, Ω ϕ ) the GNS of the state ϕ. There exists a unique linear contraction U Φ on the H ϕ where the relation 5 holds and denoting the orthogonal projection on the linear space ker (I U ϕ )= ker I U ϕ by Pϕ, we have U ϕ P ϕ = P ϕ U ϕ = P ϕ and n + nx U k ϕ P ϕ in so-topology. (6) If the application Φ is homomorphism, then U ϕ is an isometry on H ϕ such that U ϕ U ϕ π ϕ (Φ (A)) 0 B (H ϕ ) (7) ϕ

and 4. DILATIONS THEORY FOR DYNAMICAL SYSTEMS U ϕ π ϕ (a) =π ϕ (Φ (a)) U ϕ, a A. (8) Proof. See [20] lemma 2.. 4.. Dilations for Dynamical Systems. We now give the fundamental definition of dilation of a dynamical system. Definition.2. Let (A, Φ,ϕ) be a C*-dynamical system. The 5-tuple ba, Φ, b bϕ, i, E composed by a C*-dynamical system ba, Φ, b bϕ and cp-maps E : A b A, i: A A,is b say be a dilation of (A, Φ,ϕ) if for each a A and n N we have E bφ n (i (a)) = Φ n ((a)), and for each x A b bϕ (x) =ϕ (E (x)). Two dilations ba, Φ b, bϕ,i, E and ba2, Φ b 2, bϕ 2,i 2, E 2 of the C -dynamical system (A, Φ,ϕ)areequivalent if exists an automorphism Λ : A b A b 2 such that Λ Φ b = Φ b 2 Λ, bϕ 2 = bϕ Λ and E 2 Λ = E, Λ i = i 2. (9) The dilation ba, Φ, b bϕ, i, E of the C*-dynamical system (A,ϕ,Φ), is say be a reversible [multiplicative] dilation if ba, Φ, b bϕ is a reversible [multiplicative] C*-dynamical system. The dilation ba, Φ, b bϕ, i, E of the C*-dynamical system (A,ϕ,Φ), is say be a unital dilation if the cp-map i is unital, i.e. i ( A )= ba. Remark.4. Let ba, Φ, b bϕ, i, E be a reversible dilation of (A,ϕ,Φ), for definition we have that E i = id A where i is injective map while E is surjective map. We have a first proposition that affirms that the map E is a conditional expectation. Proposition.0. Let ba, Φ, b bϕ, i, E be a reversible dilation of (A,ϕ,Φ), foreach a, b A, x A b we have: E (i (a) xi (b)) = ae (x) b. Proof. For each a A we obtain E (i (a ) i (a)) = a a, since a a = E (i (a a)) E (i (a ) i (a)) E (i (a )) E (i (a)) = a a. Then for each a A, the element i (a) is in the multiplicative domains of E, it follow by the relation that E (i (a) X) =E (i (a)) E (X) ande (Xi(a)) = E (X) E (i (a)) for each X A. b We observe that if ba, Φ, b bϕ, i, E be a reversible dilation of (A,ϕ,Φ) wehave E (i (a ) i (a 2 ) i (a n )) = a a 2 a n

4. DILATIONS THEORY FOR DYNAMICAL SYSTEMS 2 for each a,a 2,...a n A, since E (i (a ) i (a 2 ) i (a n )) = a E (i (a 2 ) i (a n )). Then E ((i (a) i (b) i (ab)) (i (a) i (b) i (ab))) = 0 and bϕ ((i (a) i (b) i (ab)) (i (a) i (b) i (ab))) = 0. From this last relation we have the following remark: Remark.5. Let cm, bϕ, Φ,i,E b be a reversible dilation of the W*-dynamical system (M, Φ,ϕ), then the map i is multiplicative (but is not necessarily unital) and i E : M c cm is (unique) conditional expectation on von Neumann algebra i (M) 006. We have now an important definition: Definition.3. The reversible dilation ba, Φ, b bϕ, i, E (A, Φ,ϕ) is to said be minimal if Ã! [ ba = C k ZbΦ k (i (A)) of the C -dynamical system whileistosaidbemarkovif Ã! [ ba = C Φ k Nb k (i (A)). We study now the relation between the representations GNS of the C*-dynamical system (A, Φ,ϕ) and one its possible dilation ba, Φ, b bϕ, i, E. Let Z : H ϕ H bϕ be the linear operator thus defined: Zπ ϕ (a) Ω ϕ = π bϕ (i (a)) Ω bϕ, a A (20) The operator is an isometry since kz π ϕ (a) Ω ϕ k 2 = bϕ (i (a ) i (a)) = bϕ (i (a a)) = ϕ (a A)=kπ ϕ (a) Ω ϕ k 2. Moreover for each x A b we have: Z π bϕ (x) Ω bϕ π ϕ (a) Ω ϕ = bϕ (x i (a)) = ϕ (E (x ) a) = π ϕ (E (x)) Ω bϕ,π ϕ (a) Ω ϕ. Then Z π bϕ (x) Ω bϕ = π ϕ (E (x)) Ω ϕ, (2) and a simple calculation shows that for each a A and x A b we obtain: Zπ ϕ (a) =π bϕ (i (a)) Z (22) and Z π bϕ (x) Z = π ϕ (E (x)). (23) 6 Cfr.[22] Proposition 3.5.

4. DILATIONS THEORY FOR DYNAMICAL SYSTEMS 3 We notice that the operator Q = ZZ is the ortogonal projection on the Hilbert space generated by the vectors π bϕ (i (a)) Ω bϕ : a A ª with Qπ bϕ (x) Ω bϕ = π bϕ (i (E (x))) Ω bϕ, x A. b (24) For all n N we have U n ϕ = Z U n bϕ Z, (25) since for each a A : Z U n bϕ Zπ ϕ (a) Ω ϕ = Z π bϕ bφ n (i (a)) Ω bϕ = π bϕ E bφ n (i (a)) Ω bϕ = = π bϕ (Φ n (a)) Ω ϕ = U n ϕπ ϕ (a) Ω ϕ. We study now the relation between the orthogonal projections P ϕ =[ker(i U ϕ )] and P bϕ = ker I U bϕ. From the relation 25 for each N N we have the relation Ã! NX U k ϕ = Z NX U k bϕ Z N + N + it follow that P ϕ = Z P bϕ Z. (26) Proposition.. Let ba, Φ, b bϕ, i, E be a dilation of the C -dynamical system (A,ϕ,Φ) the unitary operator U bϕ is a dilation of the contraction ZU ϕ Z. Moreover to equivalent dilations of the C -dynamical system corresponds equivalent dilations of the linear contraction U ϕ. Proof. We observe that for each a A and n N we have: (ZU ϕ Z ) n π bϕ (a) Ω bϕ = QU n bϕ Zπ bϕ (a) Ω bϕ = Qπ bϕ bφ n (i (a)) Ω bϕ = = π bϕ (i (Φ n (a))) Ω bϕ = ZU n ϕπ bϕ (a) Ω bϕ =(ZU ϕ Z ) n Zπ bϕ (a) Ω bϕ, consequently for each Ψ H ϕ we have QU n bϕ Zh=(ZU ΦZ ) n Ψ. Let ba, Φ b, bϕ,i, E and ba2, Φ b 2, bϕ 2,i 2, E 2 are two equivalent dilations of the C*- dynamical system (A, Φ,ϕ) with automorphism Λ : A b A b 2 definedin9. We set for each a A Λ π bϕ (a) Ω bϕ = π bϕ2 (Λ (a)) Ω bϕ2, we have an unitary operator Λ : H bϕ H bϕ2 such that Λ U bϕ = U bϕ2 Λ. We have the following remark: Remark.6. If ba, Φ, b bϕ is a minimal dilation, in general, it is not said that the operator U bϕ is minimal unitary dilation of U ϕ. In fact the Hilbert space H bϕ is the norm closed linear space generate by the set of elements n o U n bϕ π bϕ (i (a )) U n k bϕ π bϕ (i (a k )) Ω bϕ : a i A, n i N

4. DILATIONS THEORY FOR DYNAMICAL SYSTEMS 4 while the space W n N Un bϕ ZH ϕ is generate by the set of elements n o U n bϕ π bϕ (i (a)) Ω bϕ : a A, n Z. We see now an example of as the Nagy dilation for the contraction on the Hilbert space is applied to the dilation theory of dynamical systems. Example 2. Let H be a Hilbert space and V an isometry on H, we get the unital cp-map Φ : B (H) B (H) Φ (A) =V AV, A B (H), and ϕ is a Φ-invariant state of B (H). In this way we get the C*-dynamic system (B (H), Φ,ϕ). Let K, V b be the Nagy dilation of the isometry V : bv= V 0 C W, and Hilbert space K = H l 2 (I). We have an auntomorphims Φ b : B (K) B (K) bφ (X) = VX b V b, X B (H), such that for each A B (H) we have: J Φ b n (JAJ ) J = Φ (A). The C*-dynamical systems B (K), Φ, b bϕ with bϕ (X) =ϕ (J XJ), is a reversible dilation of (B (H), Φ,ϕ), since X B (K) B (K) i B (H) bφ n Φ n B (K) E B (H) is a commutative diagram, where: the application E : B (K) B (H) is the unital cp-map E (X) =J XJ, X B (K) while i : B (H) B (K) is the *-multiplicative map (non unital) i (A) =JAJ, X B (K). We observe that bϕ is a Φ invariant b state, since bϕ bφ (X) = ϕ J Φ b (X) J = ϕ J VX b V b J = ϕ (V J XJV) =ϕ (J XJ) = bϕ (X) for all X B (K).

4. DILATIONS THEORY FOR DYNAMICAL SYSTEMS 5 We now study the problem list that we have with the dilations of composition. Let (A, Φ,ϕ) be a C*-dynamical system and (A o, Φ o,ϕ o, E o,i o ) a its Markov multiplicative dilation. If the C*-dynamical system (A o, Φ o,ϕ o ) admits a minimal reversible dilation (A, Φ,ϕ, E,i ), we have the follow diagram: A Φ n oo A i E Φ n o A o i o A Φ n A o E o A Ã [ A o = C k NΦ k o (i o (A)), ϕ o = ϕ E o Ã! [ A = C k ZΦ k (i (A)), ϕ = ϕ o E Thenthe5-tuple(A, bϕ, Φ, E,i)withE = E o E and i = i i o with bϕ = bϕ E, is a reversible dilation of the C*-dynamical system (A, Φ,ϕ), but in generally it is not minimal. Weobservethatifϕ is faithful state on A then ϕ o is faithful state on A if and only if E o is a faithful cp-map. 4.2. The ϕ Adjoint of morphism. Let (A, Φ,ϕ) be C*-algebra dynamical system, a cp map Φ + : A A is said to be ϕ-adjoint of Φ, if for each a a we have ϕ (Φ (a) b) =ϕ aφ + (b). We observe that (Φ + ) + = Φ. Moreover every reversible C*-dynamical system admits a ϕ-adjoint where Φ + = Φ. If Φ admits a ϕ-adjoint, for each a A we have U ϕπ ϕ (a) Ω ϕ = π ϕ Φ + (a) Ω ϕ, since for each a, b A, we get: U ϕ π ϕ (b) Ω ϕ,π ϕ (a) Ω ϕ = ϕ (b Φ (a)) = ϕ Φ + (b ) a = π ϕ Φ + (b) Ω,π ϕ (a) Ω ϕ. We introduce a necessary condition for the existence of a reversible dilation (cfr.[2] proposition 2..8). Proposition.2. Let (A, Φ,ϕ) be a C*-dynamical system with a reversible dilation ba, Φ, b bϕ, E,i. Then Φ has a ϕ adjoint Φ + and ba, Φ b, bϕ, E,i is a dilation of the C*-dynamical system (A, Φ +,ϕ). Proof. For a, b A and n N we have: ϕ (aφ n (b)) = ϕ ae bφ n (i (b)) = ϕ E = bϕ bφ n (i (a)) i (b) = ϕ aφ b n (i (b)) E bφ n (i (a)) b! = bϕ. i (a) Φ b n (i (b)) = Then the ϕ adjoint of Φ results to be Φ + = E b Φ i. Remark.7. Let (A, Φ,ϕ) be a C*-dynamical system with a ϕ-adjont Φ +.IfΦ + is a multiplivative map we have U ϕ U ϕ = I.