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


 Dustin Williamson
 3 years ago
 Views:
Transcription
1 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
2
3 Contents. Introduction 4 Chapter. Dynamical systems and their dilations. Preliminaries 2. Stinespring Dilations for the cp map 2 3. NagyFoiaş Dilations Theory 8 4. Dilations Theory for Dynamical Systems 0 5. Spatial Morphism 7 Chapter 2. Towards the reversible dilations 23. Multiplicative dilation Ergodic property of the dilation 38 Chapter 3. C*Hilbert module and dilations 47. Definitions and notations Dilations constructed by using Hilbert modules Ergodic property 5 Appendix A. Algebraic formalism in ergodic theory 54 Appendix. Bibliography 56 3
4 . 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 VincentSmith [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 VincentSmith 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
5 . 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 NagyFoias 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 MuhlySolel 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].
6 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 NagyFoias 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 selfadjoint 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 npositive if the linear map Φ n is positive and we call Φ completely positive if Φ is npositive for all n N. WeobservethatifA and B are C*algebra, a linear map Φ : A B is cpmap 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 cpmap, 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 cpmaps 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 cpmap. Then there is a cpmap, Φ ar : A B (H), extendingφ. Briefly cpmap.
7 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 quasiconditional 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 quasiconditional 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 cpmap. 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 cpmap Φ : A A. By the Stinespring theorem for the cpmap 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].
8 2. STINESPRING DILATIONS FOR THE CP MAP 3 Proposition.4. The unital cpmap 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 wocontinuous to x and since for hypothesis Φ (x α ) Φ (x) inm + 2 and Φ (x α) Φ (x) inwocontinuous, 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) inwotopology. The Stinespring theorem admit the following extension: Theorem.. Let A beac*algebrawithunitandφ : A A an unital cpmap, 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
9 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 cpmaps 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, whilefortheviceversaforeacha, 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)).
10 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 cpmaps 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, ). 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 See [9] proposition.3..
11 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 cpmap, 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 cpmap 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.
12 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
13 3. NAGYFOIAŞ 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 cpmap. 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. NagyFoiaş 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.
14 3. NAGYFOIAŞ 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, ), 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]
15 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 cpmap 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 cpmap 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 cpmap 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 sotopology. (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
16 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 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 5tuple ba, Φ, b bϕ, i, E composed by a C*dynamical system ba, Φ, b bϕ and cpmaps 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 cpmap 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
17 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.
18 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
19 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 cpmap Φ : B (H) B (H) Φ (A) =V AV, A B (H), and ϕ is a Φinvariant 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 cpmap 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).
20 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 Thenthe5tuple(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 cpmap The ϕ Adjoint of morphism. Let (A, Φ,ϕ) be C*algebra dynamical system, a cp map Φ + : A A is said to be ϕadjoint 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 ϕadjoint where Φ + = Φ. If Φ admits a ϕadjoint, 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 ϕadjont Φ +.IfΦ + is a multiplivative map we have U ϕ U ϕ = I.
Inner Product Spaces and Orthogonality
Inner Product Spaces and Orthogonality week 34 Fall 2006 Dot product of R n The inner product or dot product of R n is a function, defined by u, v a b + a 2 b 2 + + a n b n for u a, a 2,, a n T, v b,
More informationTensor product of vector spaces
Tensor product of vector spaces Construction Let V,W be vector spaces over K = R or C. Let F denote the vector space freely generated by the set V W and let N F denote the subspace spanned by the elements
More informationGROUP ALGEBRAS. ANDREI YAFAEV
GROUP ALGEBRAS. ANDREI YAFAEV We will associate a certain algebra to a finite group and prove that it is semisimple. Then we will apply Wedderburn s theory to its study. Definition 0.1. Let G be a finite
More informationOPERATOR AND SPECTRAL THEORY
Lecture 1 OPERATOR AND SPECTRAL THEORY Stéphane ATTAL Abstract This lecture is a complete introduction to the general theory of operators on Hilbert spaces. We particularly focus on those tools that are
More informationMODULES OVER A PID KEITH CONRAD
MODULES OVER A PID KEITH CONRAD Every vector space over a field K that has a finite spanning set has a finite basis: it is isomorphic to K n for some n 0. When we replace the scalar field K with a commutative
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More informationFinite dimensional C algebras
Finite dimensional C algebras S. Sundar September 14, 2012 Throughout H, K stand for finite dimensional Hilbert spaces. 1 Spectral theorem for selfadjoint opertors Let A B(H) and let {ξ 1, ξ 2,, ξ n
More informationNOTES ON LINEAR TRANSFORMATIONS
NOTES ON LINEAR TRANSFORMATIONS Definition 1. Let V and W be vector spaces. A function T : V W is a linear transformation from V to W if the following two properties hold. i T v + v = T v + T v for all
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More information7. Operator Theory on Hilbert spaces
7. Operator Theory on Hilbert spaces In this section we take a closer look at linear continuous maps between Hilbert spaces. These are often called bounded operators, and the branch of Functional Analysis
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 22 RAVI VAKIL CONTENTS 1. Discrete valuation rings: Dimension 1 Noetherian regular local rings 1 Last day, we discussed the Zariski tangent space, and saw that it
More informationIRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL. 1. Introduction
IRREDUCIBLE OPERATOR SEMIGROUPS SUCH THAT AB AND BA ARE PROPORTIONAL R. DRNOVŠEK, T. KOŠIR Dedicated to Prof. Heydar Radjavi on the occasion of his seventieth birthday. Abstract. Let S be an irreducible
More informationCONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation
Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in
More informationFactoring of Prime Ideals in Extensions
Chapter 4 Factoring of Prime Ideals in Extensions 4. Lifting of Prime Ideals Recall the basic AKLB setup: A is a Dedekind domain with fraction field K, L is a finite, separable extension of K of degree
More informationNOTES ON CATEGORIES AND FUNCTORS
NOTES ON CATEGORIES AND FUNCTORS These notes collect basic definitions and facts about categories and functors that have been mentioned in the Homological Algebra course. For further reading about category
More information6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a nonempty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points
More informationLet H and J be as in the above lemma. The result of the lemma shows that the integral
Let and be as in the above lemma. The result of the lemma shows that the integral ( f(x, y)dy) dx is well defined; we denote it by f(x, y)dydx. By symmetry, also the integral ( f(x, y)dx) dy is well defined;
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationThe cover SU(2) SO(3) and related topics
The cover SU(2) SO(3) and related topics Iordan Ganev December 2011 Abstract The subgroup U of unit quaternions is isomorphic to SU(2) and is a double cover of SO(3). This allows a simple computation of
More informationMatrix Representations of Linear Transformations and Changes of Coordinates
Matrix Representations of Linear Transformations and Changes of Coordinates 01 Subspaces and Bases 011 Definitions A subspace V of R n is a subset of R n that contains the zero element and is closed under
More informationLectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain
Lectures notes on orthogonal matrices (with exercises) 92.222  Linear Algebra II  Spring 2004 by D. Klain 1. Orthogonal matrices and orthonormal sets An n n realvalued matrix A is said to be an orthogonal
More informationAn Advanced Course in Linear Algebra. Jim L. Brown
An Advanced Course in Linear Algebra Jim L. Brown July 20, 2015 Contents 1 Introduction 3 2 Vector spaces 4 2.1 Getting started............................ 4 2.2 Bases and dimension.........................
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More informationFactorization Theorems
Chapter 7 Factorization Theorems This chapter highlights a few of the many factorization theorems for matrices While some factorization results are relatively direct, others are iterative While some factorization
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationLinear Maps. Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007)
MAT067 University of California, Davis Winter 2007 Linear Maps Isaiah Lankham, Bruno Nachtergaele, Anne Schilling (February 5, 2007) As we have discussed in the lecture on What is Linear Algebra? one of
More informationOn the structure of C algebra generated by a family of partial isometries and multipliers
Armenian Journal of Mathematics Volume 7, Number 1, 2015, 50 58 On the structure of C algebra generated by a family of partial isometries and multipliers A. Yu. Kuznetsova and Ye. V. Patrin Kazan Federal
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
More informationRINGS WITH A POLYNOMIAL IDENTITY
RINGS WITH A POLYNOMIAL IDENTITY IRVING KAPLANSKY 1. Introduction. In connection with his investigation of projective planes, M. Hall [2, Theorem 6.2]* proved the following theorem: a division ring D in
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More information1 Sets and Set Notation.
LINEAR ALGEBRA MATH 27.6 SPRING 23 (COHEN) LECTURE NOTES Sets and Set Notation. Definition (Naive Definition of a Set). A set is any collection of objects, called the elements of that set. We will most
More informationSets of Fibre Homotopy Classes and Twisted Order Parameter Spaces
Communications in Mathematical Physics ManuscriptNr. (will be inserted by hand later) Sets of Fibre Homotopy Classes and Twisted Order Parameter Spaces Stefan BechtluftSachs, Marco Hien Naturwissenschaftliche
More informationin quantum computation
in computation Kestrel Institute and Oxford University  Oxford, August 2008 Outline Quantum programming λ Abstraction with pictures Consequences  Abstraction in Category of measurements  Outline Quantum
More informationBILINEAR FORMS KEITH CONRAD
BILINEAR FORMS KEITH CONRAD The geometry of R n is controlled algebraically by the dot product. We will abstract the dot product on R n to a bilinear form on a vector space and study algebraic and geometric
More informationRow Ideals and Fibers of Morphisms
Michigan Math. J. 57 (2008) Row Ideals and Fibers of Morphisms David Eisenbud & Bernd Ulrich Affectionately dedicated to Mel Hochster, who has been an inspiration to us for many years, on the occasion
More informationT ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
More informationSOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS
SOME PROPERTIES OF FIBER PRODUCT PRESERVING BUNDLE FUNCTORS Ivan Kolář Abstract. Let F be a fiber product preserving bundle functor on the category FM m of the proper base order r. We deduce that the rth
More informationSection 6.1  Inner Products and Norms
Section 6.1  Inner Products and Norms Definition. Let V be a vector space over F {R, C}. An inner product on V is a function that assigns, to every ordered pair of vectors x and y in V, a scalar in F,
More informationQuadratic Equations in Finite Fields of Characteristic 2
Quadratic Equations in Finite Fields of Characteristic 2 Klaus Pommerening May 2000 english version February 2012 Quadratic equations over fields of characteristic 2 are solved by the well known quadratic
More informationLecture 6: The Group Inverse
Lecture 6: The Group Inverse The matrix index Let A C n n, k positive integer. Then R(A k+1 ) R(A k ). The index of A, denoted IndA, is the smallest integer k such that R(A k ) = R(A k+1 ), or equivalently,
More informationMICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000 000 S 00029939(XX)00000 MICROLOCAL ANALYSIS OF THE BOCHNERMARTINELLI INTEGRAL NIKOLAI TARKHANOV AND NIKOLAI VASILEVSKI
More informationMATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets.
MATH 304 Linear Algebra Lecture 20: Inner product spaces. Orthogonal sets. Norm The notion of norm generalizes the notion of length of a vector in R n. Definition. Let V be a vector space. A function α
More informationThree observations regarding Schatten p classes
Three observations regarding Schatten p classes Gideon Schechtman Abstract The paper contains three results, the common feature of which is that they deal with the Schatten p class. The first is a presentation
More informationRevision of ring theory
CHAPTER 1 Revision of ring theory 1.1. Basic definitions and examples In this chapter we will revise and extend some of the results on rings that you have studied on previous courses. A ring is an algebraic
More information16.3 Fredholm Operators
Lectures 16 and 17 16.3 Fredholm Operators A nice way to think about compact operators is to show that set of compact operators is the closure of the set of finite rank operator in operator norm. In this
More informationLinear Algebra. A vector space (over R) is an ordered quadruple. such that V is a set; 0 V ; and the following eight axioms hold:
Linear Algebra A vector space (over R) is an ordered quadruple (V, 0, α, µ) such that V is a set; 0 V ; and the following eight axioms hold: α : V V V and µ : R V V ; (i) α(α(u, v), w) = α(u, α(v, w)),
More informationISOMETRIES OF R n KEITH CONRAD
ISOMETRIES OF R n KEITH CONRAD 1. Introduction An isometry of R n is a function h: R n R n that preserves the distance between vectors: h(v) h(w) = v w for all v and w in R n, where (x 1,..., x n ) = x
More informationEXERCISES FOR THE COURSE MATH 570, FALL 2010
EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime
More informationFIBER PRODUCTS AND ZARISKI SHEAVES
FIBER PRODUCTS AND ZARISKI SHEAVES BRIAN OSSERMAN 1. Fiber products and Zariski sheaves We recall the definition of a fiber product: Definition 1.1. Let C be a category, and X, Y, Z objects of C. Fix also
More informationSummary of week 8 (Lectures 22, 23 and 24)
WEEK 8 Summary of week 8 (Lectures 22, 23 and 24) This week we completed our discussion of Chapter 5 of [VST] Recall that if V and W are inner product spaces then a linear map T : V W is called an isometry
More informationGROUPS ACTING ON A SET
GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for
More informationGroup Theory. Contents
Group Theory Contents Chapter 1: Review... 2 Chapter 2: Permutation Groups and Group Actions... 3 Orbits and Transitivity... 6 Specific Actions The Right regular and coset actions... 8 The Conjugation
More informationModule MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions
Module MA1S11 (Calculus) Michaelmas Term 2016 Section 3: Functions D. R. Wilkins Copyright c David R. Wilkins 2016 Contents 3 Functions 43 3.1 Functions between Sets...................... 43 3.2 Injective
More informationMATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix.
MATH 304 Linear Algebra Lecture 4: Matrix multiplication. Diagonal matrices. Inverse matrix. Matrices Definition. An mbyn matrix is a rectangular array of numbers that has m rows and n columns: a 11
More informationMath 550 Notes. Chapter 7. Jesse Crawford. Department of Mathematics Tarleton State University. Fall 2010
Math 550 Notes Chapter 7 Jesse Crawford Department of Mathematics Tarleton State University Fall 2010 (Tarleton State University) Math 550 Chapter 7 Fall 2010 1 / 34 Outline 1 SelfAdjoint and Normal Operators
More information6 Commutators and the derived series. [x,y] = xyx 1 y 1.
6 Commutators and the derived series Definition. Let G be a group, and let x,y G. The commutator of x and y is [x,y] = xyx 1 y 1. Note that [x,y] = e if and only if xy = yx (since x 1 y 1 = (yx) 1 ). Proposition
More information1 VECTOR SPACES AND SUBSPACES
1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such
More informationA MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASEFIELD MODEL WITH MEMORY
Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASEFIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,
More informationG = G 0 > G 1 > > G k = {e}
Proposition 49. 1. A group G is nilpotent if and only if G appears as an element of its upper central series. 2. If G is nilpotent, then the upper central series and the lower central series have the same
More informationUNIT 2 MATRICES  I 2.0 INTRODUCTION. Structure
UNIT 2 MATRICES  I Matrices  I Structure 2.0 Introduction 2.1 Objectives 2.2 Matrices 2.3 Operation on Matrices 2.4 Invertible Matrices 2.5 Systems of Linear Equations 2.6 Answers to Check Your Progress
More informationMATH PROBLEMS, WITH SOLUTIONS
MATH PROBLEMS, WITH SOLUTIONS OVIDIU MUNTEANU These are free online notes that I wrote to assist students that wish to test their math skills with some problems that go beyond the usual curriculum. These
More informationOn linear isometries on nonarchimedean power series spaces
On linear isometries on nonarchimedean power series spaces Wies law Śliwa and Agnieszka Ziemkowska Abstract. The nonarchimedean power series spaces A p (a, t) are the most known and important examples
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationRate of convergence towards Hartree dynamics
Rate of convergence towards Hartree dynamics Benjamin Schlein, LMU München and University of Cambridge Universitá di Milano Bicocca, October 22, 2007 Joint work with I. Rodnianski 1. Introduction boson
More informationMAT188H1S Lec0101 Burbulla
Winter 206 Linear Transformations A linear transformation T : R m R n is a function that takes vectors in R m to vectors in R n such that and T (u + v) T (u) + T (v) T (k v) k T (v), for all vectors u
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More informationCHAPTER 5: MODULAR ARITHMETIC
CHAPTER 5: MODULAR ARITHMETIC LECTURE NOTES FOR MATH 378 (CSUSM, SPRING 2009). WAYNE AITKEN 1. Introduction In this chapter we will consider congruence modulo m, and explore the associated arithmetic called
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More information3. Prime and maximal ideals. 3.1. Definitions and Examples.
COMMUTATIVE ALGEBRA 5 3.1. Definitions and Examples. 3. Prime and maximal ideals Definition. An ideal P in a ring A is called prime if P A and if for every pair x, y of elements in A\P we have xy P. Equivalently,
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationChapter 13: Basic ring theory
Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring
More informationThe Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression
The Singular Value Decomposition in Symmetric (Löwdin) Orthogonalization and Data Compression The SVD is the most generally applicable of the orthogonaldiagonalorthogonal type matrix decompositions Every
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationSolutions to TOPICS IN ALGEBRA I.N. HERSTEIN. Part II: Group Theory
Solutions to TOPICS IN ALGEBRA I.N. HERSTEIN Part II: Group Theory No rights reserved. Any part of this work can be reproduced or transmitted in any form or by any means. Version: 1.1 Release: Jan 2013
More information0.1 Linear Transformations
.1 Linear Transformations A function is a rule that assigns a value from a set B for each element in a set A. Notation: f : A B If the value b B is assigned to value a A, then write f(a) = b, b is called
More informationOn the Spaces of λconvergent and Bounded Sequences
Thai Journal of Mathematics Volume 8 2010 Number 2 : 311 329 www.math.science.cmu.ac.th/thaijournal Online ISSN 16860209 On the Spaces of λconvergent and Bounded Sequences M. Mursaleen 1 and A.K. Noman
More informationWeak topologies. David Lecomte. May 23, 2006
Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationClassification of Cartan matrices
Chapter 7 Classification of Cartan matrices In this chapter we describe a classification of generalised Cartan matrices This classification can be compared as the rough classification of varieties in terms
More informationLinear Algebra Done Wrong. Sergei Treil. Department of Mathematics, Brown University
Linear Algebra Done Wrong Sergei Treil Department of Mathematics, Brown University Copyright c Sergei Treil, 2004, 2009, 2011, 2014 Preface The title of the book sounds a bit mysterious. Why should anyone
More informationClassification of Bundles
CHAPTER 2 Classification of Bundles In this chapter we prove Steenrod s classification theorem of principal G  bundles, and the corresponding classification theorem of vector bundles. This theorem states
More information4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION
4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. Review 1 2. Diagonal Matrices 1 3. Eigenvectors and Eigenvalues 2 4. Characteristic Polynomial 4 5. Diagonalizability 6 6. Appendix:
More information18.06 Problem Set 4 Solution Due Wednesday, 11 March 2009 at 4 pm in 2106. Total: 175 points.
806 Problem Set 4 Solution Due Wednesday, March 2009 at 4 pm in 206 Total: 75 points Problem : A is an m n matrix of rank r Suppose there are righthandsides b for which A x = b has no solution (a) What
More information1 Orthogonal projections and the approximation
Math 1512 Fall 2010 Notes on least squares approximation Given n data points (x 1, y 1 ),..., (x n, y n ), we would like to find the line L, with an equation of the form y = mx + b, which is the best fit
More informationHomological Algebra  Problem Set 3
Homological Algebra  Problem Set 3 Problem 1. Let R be a ring. (1) Show that an Rmodule M is projective if and only if there exists a module M and a free module F such that F = M M (we say M is a summand
More informationOperatorvalued version of conditionally free product
STUDIA MATHEMATICA 153 (1) (2002) Operatorvalued version of conditionally free product by Wojciech Młotkowski (Wrocław) Abstract. We present an operatorvalued version of the conditionally free product
More informationFacts About Eigenvalues
Facts About Eigenvalues By Dr David Butler Definitions Suppose A is an n n matrix An eigenvalue of A is a number λ such that Av = λv for some nonzero vector v An eigenvector of A is a nonzero vector v
More informationThe determinant of a skewsymmetric matrix is a square. This can be seen in small cases by direct calculation: 0 a. 12 a. a 13 a 24 a 14 a 23 a 14
4 Symplectic groups In this and the next two sections, we begin the study of the groups preserving reflexive sesquilinear forms or quadratic forms. We begin with the symplectic groups, associated with
More informationMATH 240 Fall, Chapter 1: Linear Equations and Matrices
MATH 240 Fall, 2007 Chapter Summaries for Kolman / Hill, Elementary Linear Algebra, 9th Ed. written by Prof. J. Beachy Sections 1.1 1.5, 2.1 2.3, 4.2 4.9, 3.1 3.5, 5.3 5.5, 6.1 6.3, 6.5, 7.1 7.3 DEFINITIONS
More informationCOMMUTATIVE RINGS. Definition: A domain is a commutative ring R that satisfies the cancellation law for multiplication:
COMMUTATIVE RINGS Definition: A commutative ring R is a set with two operations, addition and multiplication, such that: (i) R is an abelian group under addition; (ii) ab = ba for all a, b R (commutative
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 16 RAVI VAKIL Contents 1. Valuation rings (and nonsingular points of curves) 1 1.1. Completions 2 1.2. A big result from commutative algebra 3 Problem sets back.
More informationCourse 421: Algebraic Topology Section 1: Topological Spaces
Course 421: Algebraic Topology Section 1: Topological Spaces David R. Wilkins Copyright c David R. Wilkins 1988 2008 Contents 1 Topological Spaces 1 1.1 Continuity and Topological Spaces...............
More informationChapter 7. Induction and Recursion. Part 1. Mathematical Induction
Chapter 7. Induction and Recursion Part 1. Mathematical Induction The principle of mathematical induction is this: to establish an infinite sequence of propositions P 1, P 2, P 3,..., P n,... (or, simply
More informationRieszFredhölm Theory
RieszFredhölm Theory T. Muthukumar tmk@iitk.ac.in Contents 1 Introduction 1 2 Integral Operators 1 3 Compact Operators 7 4 Fredhölm Alternative 14 Appendices 18 A AscoliArzelá Result 18 B Normed Spaces
More informationEigenvalues, Eigenvectors, Matrix Factoring, and Principal Components
Eigenvalues, Eigenvectors, Matrix Factoring, and Principal Components The eigenvalues and eigenvectors of a square matrix play a key role in some important operations in statistics. In particular, they
More informationThe Ideal Class Group
Chapter 5 The Ideal Class Group We will use Minkowski theory, which belongs to the general area of geometry of numbers, to gain insight into the ideal class group of a number field. We have already mentioned
More information2 Polynomials over a field
2 Polynomials over a field A polynomial over a field F is a sequence (a 0, a 1, a 2,, a n, ) where a i F i with a i = 0 from some point on a i is called the i th coefficient of f We define three special
More informationTHREE DIMENSIONAL GEOMETRY
Chapter 8 THREE DIMENSIONAL GEOMETRY 8.1 Introduction In this chapter we present a vector algebra approach to three dimensional geometry. The aim is to present standard properties of lines and planes,
More informationChapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.
Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize
More information1 Limiting distribution for a Markov chain
Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easytocheck
More information