Dynamical Semigroups and Stochastic Processes Heo, Jaeseong - - PowerPoint PPT Presentation

Dynamical Semigroups and Stochastic Processes

Heo, Jaeseong (Hanyang University) 許宰誠 (漢陽大學校) Symposium on Probability & Analysis 2010 August 10, 2010

Reference: [1] J. Heo, Hilbert C∗-module representation on Haagerup tensor products and group systems, Publ. Res. Inst. Math. Sci. (Kyoto Univ.) 35 (1999), pp. 757–768. [2] J. Heo, Stationary stochastic processes in a group system, J. Math. Phys. 48 (2007), no.10, 103502 [3] J. Heo, Reproducing kernel Hilbert C∗-modules and kernels associated with cocycles, J. Math. Phys. 49 (2008), no.10, 103507. [4] V. Belavkin, J. Heo and U. C. Ji, Reconstruction Theorem for Stationary Monotone Quantum Markov Processes, Preprint, 2010.

Hilbert C∗-module

Definition: Let A be a C∗-algebra. A right A-module X is called a (right) pre- Hilbert A-module if there is an A-valued mapping ·, · : X ×X → A which is linear in the second variable and has the following properties: (i) |x|2 := x, x ≥ 0, and the equality holds only if x = 0. (ii) x, y = y, x∗. (iii) x, y · b = x, yb. If, in addition, X is complete w.r.t. the norm x = x, x

1 2, then X is called a

(right) Hilbert A-module.

Adjointable operator

Let X and Y be Hilbert A-modules.

• BA(X, Y ): thes set of all bounded right A-module maps from X to Y
• LA(X, Y ): the set of all right A-module maps T : X → Y for which there is an
• perator T ∗ : Y → X, called the adjoint of T, such that

Tx, yY = x, T ∗y for x ∈ X, y ∈ Y . Some Remarks: Let X and Y be Hilbert A-modules.

• All bounded A-module maps between X and Y do not have always adjoints.
• Closed submodules of X need not be complemented.
• In general, the parallelogram law is not satisfied in a Hilbert C∗-module setting.

Why Hilbert C∗-module?

• KK-theory for operator algebras
• C∗-algebraic quantum group theory
• Quantum electro-dynamics (interaction between matter and radiation)
• Hilbert module over the momentum algebra of the electron
• Quantum instrument, quantum measurement theory
• Quantum probability theory, quantum white noise
• quantum stochastic calculus over a Hilbert module

Examples of Hilbert C∗-module

Let E = (E0, E1, r, s) be a directed graph where E0 is a set of vertices, E1 is a set of edges and r (resp. s) is the range map (resp. the source map).

• A: C∗-algebra C0(E0) of continuous functions f : E0 → C vanishing at ∞
• Cc(E1): space of continuous functions x : E1 → C with finite support

On the space Cc(E1), we define the multiplication and the inner product (x · f)(e) = x(e)f(s(e)) and x, y(v) =

{e∈E1:s(e)=v} x(e)y(e).

Then we get a Hilbert A-module X by completing the space Cc(E1).

Stochastic processes on a continuous parameter is frequently given in terms of a physical system or other entity which depends on the parameter t (time) and whose state is specified by the position of a point Q = Q(t) varying in some space in accordance with a given probability law. We will consider a locally compact group as a parameter and Hilbert spaces or Hilbert C∗-modules as spaces with some property.

Positive definite function

Definition: G: locally compact group, M: vN algebra with predual M∗

• A function φ : G → M∗ is positive definite if

for every n ∈ N, t1, . . . , tn ∈ G and x1, . . . , xn ∈ M,

n

• i,j=1

[φ(t−1

i

tj)](x∗

i xj) ≥ 0.

• A function φ : G → M∗ is weakly continuous if

the map given by t → [φ(t)](x) is continuous on G for every x ∈ M.

F(G, M): vector space of all finitely supported fts from G into M

• A positive definite function φ : G → M∗ induces a sesquilinear form ·, ·φ
• n F(G, M) × F(G, M) as follows: Let 1 be the unit element in M.

For any f =

n

• i=1

xiδti and g =

m

• j=1

yjδsj we define f, gφ =

n

• i=1

m

• j=1

[x∗

i · φ(t−1 i

sj) · yj](1) =

n

• i=1

m

• j=1

[φ(t−1

i

sj)](yjx∗

i ).

• This sesquilinear form ·, ·φ is positive semi-definite.

Proposition: G: a locally compact group, M: a vN algebra with predual M∗ If φ is a weakly continuous positive definite function from G into M∗, then there exist a Hilbert space K, a ∗-repn π of M on K, a unitary repn U of G on K and a vector ξ in K such that (i) U(t)π(x) = π(x)U(t) for all t ∈ G and x ∈ M. (ii) linear span of the set {π(x)U(t)δe : x ∈ M, t ∈ G} is dense in K. (iii) [φ(t)](x) = ξ, π(x)U(t)ξK for all t ∈ G and x ∈ M.

G, H: locally compact groups Aut(G): group of all automorphisms endowed with pointwise convergence top. τ: action of H on G, i.e. continuous homomorphism of H into Aut(G). v: unitary repn of H into the unitary group U(M) of M φ : G → M∗: positive definite function. Definition: The function φ is called v-covariant if φ(τh(t)) = vh · φ(t) · v∗

h

for all t ∈ G and all h ∈ H.

• This equality means [φ(τh(t))](x) = [φ(t)](v∗

hxvh) for all x ∈ M.

Theorem: G: locally compact group, M: vN algebra with separable predual M∗ If φ is a weakly continuous v-covariant positive definite function from G into M∗, then there exist a Hilbert space K, a ∗-repn π of M on K, a unitary repn U of G

• n K, a vector ξ in K and a unitary repn ˜

τ of H on K s.t. (i) U(t)π(x) = π(x)U(t) for all t ∈ G and x ∈ M (ii) linear span of the set {π(x)U(t)δe : x ∈ M, t ∈ G} is dense in K (iii) [φ(t)](x) = ξ, π(x)U(t)ξK for all t ∈ G and x ∈ M (iv) ˜ τhU(t)˜ τ ∗

h = U(τh(t)) for all t ∈ G and h ∈ H, i.e. U is ˜

τ-covariant, (v) π(x)U(τh(t)) = π(vh)∗π(x)π(vh)U(t) for all h ∈ H, t ∈ G, x ∈ M (vi) ˜ τhπ(x) = π(x)˜ τh for all x ∈ M and h ∈ H.

Definition: G: locally compact group, K: Hilbert space.

• A K-valued stochastic process {xt}t∈G over G is a map t → xt from G into K.
• A K-valued stochastic process {xt}t∈G is stationary if the correlation function

Γ(s, t) = xs, xt depends only on s−1t. Let H be a locally compact group with an action τ on the group G where the action τ of H on G means a continuous homomorphism of H into Aut(G). This action naturally induces an action on a stochastic process {xt}. Denoting the induced action by the same τ, then τh(xt) = xτh(t) ∀ h ∈ H, t ∈ G. Definition: A K-valued stochastic process {xt}t∈G is called τ-invariant if Γ(τh(s), τh(t)) = Γ(s, t) for all s, t ∈ G and h ∈ H.

Theorem: G: locally compact group, M: vN algebra with separable predual M∗ Let v be a unitary repn of a locally compact group H into the unitary group U(M). If φ is a weakly continuous v-covariant positive definite function from G into M∗, then there exist a Hilbert space K and a K-valued stationary stochastic process {xt}t∈G s.t. {xt}t∈G is τ-invariant. We can also get the repn from a Hilbert C∗-module valued stochastic process. Let X be a Hilbert A-module. If {xt}t∈G is an X-valued stationary stochastic process with the correlation function Γ, then there exist a Hilbert A-module Y , a repn π : G → LA(Y ) and a vector ξ in X s.t. Γ(s, t) = xs, xt = π(s)ξ, π(t)ξ and Y is isomorphic to the closed linear span of the set {xt · a : t ∈ G, a ∈ A}.

Definition: Let X be a Hilbert A-module. Let H be a locally compact group with an action τ on G and v be a strongly unitary repn from H into U(A). An X-valued stochastic process {xt}t∈G is called v-covariant if the correlation function Γ is v-covariant, that is, if Γ(τh(s), τh(t)) = v∗

hΓ(s, t)vh

for all s, t ∈ G and h ∈ H. Theorem: Let G be a locally compact group and let A be a C∗-algebra. Let v be a unitary repn of a locally compact group H into the unitary group U(A). If φ is a weakly continuous v-covariant positive definite function from G into A, then there exist a Hilbert A-module X and an X-valued stationary stochastic process {xt}t∈G such that {xt}t∈G is v-covariant.

Quantum dynamical semigroup

Definition: Let A, B be C∗-algebras.

• A stochastic process in quantum probability is a time-indexed family of *-

homomorphisms {Jt : A → B}t∈I between C∗-algebras.

• A (quantum) dynamical semigroup on a unital C∗-algebra A is

a semigroup {φt : t ≥ 0} of completely positive linear maps of A into A s.t. (i) φ0 = idA where idA is the identity map on A, (ii) φt(1) = 1 for all t ≥ 0.

Parthasarathy: M: von Neumann algebra on a Hilbert space H0 {φt : t ≥ 0}: one-parameter semigroup of unital c.p. maps from M into B(H0) ⇒ there exist (i) family of representations (πt, Ht) (t ≥ 0) of M (ii) and isometries V (s, t) : Hs → Ht (0 ≤ s < t) such that V ∗(s, t)πt(a)V (s, t) = πs(φt−s(a)) and V (t, u)V (s, t) = V (s, u) for 0 ≤ s < t < u < ∞. Remark: This may be considered as a continuous time version of Stinespring’s theorem for a family of unital completely positive maps.

We can get the Hilbert C∗-module version of Parthasarathy’s theorem using slightly different construction. Theorem: If {φt : t ≥ 0} is a (quantum) dynamical semigroup on a unital C∗- algebra A, then there exist (1) a family {Xt : t ≥ 0} of Hilbert A-modules, (2) a family {Jt : t ≥ 0} of *-homomorphisms Jt : A → LA(Xt), (3) a family {V (s, t) : 0 ≤ s < t < ∞} of isometries V (s, t) : Xs → Xt such that V (s, t)∗Jt(a)V (s, t) = Js(φt−s(a)) and V (t, u)V (s, t) = V (s, u) for all 0 ≤ s < t < u < ∞.

Definition: Let A, B be (unital) C∗-algebras.

• A (unital) C∗-dynamical system is a triple (A, G, α) where α is an action of a

locally compact group G on a (unital) C∗-algebra A.

• A linear map φ : A → B is called to be (α, u)-covariant if

φ(αg(a)) = ugφ(a)u∗

g

(a ∈ A, g ∈ G) where u : G → U(B) is a *-homo. of G into a unitary group U(B).

• A covariant representation of (A, G, α) on a Hilbert B-module X is a pair (π, σ)

where π is a repn of A on X and σ is a unitary repn of G into LB(X) s.t. π(αg(a)) = σ(g)π(a)σ(g)∗ (a ∈ A, g ∈ G).

Theorem: Let (A, G, α) be a unital C∗-dynamical system and u : G → U(B) be a unitary repn of G into a unital C∗-algebra B. If φ : A → B is a (α, u)-covariant completely positive linear map, then there exist a Hilbert B-module X with a generating vector e, a covariant repn (π, σ) of (A, G, α) on X and an element v ∈ LB(B, X) s.t. (i) φ(a) = e, π(a)e, v∗π(a)v = mφ(a) for all a ∈ A, (ii) σ(g)v = vmug for all g ∈ G, (iii) {π(a)(e · b) : a ∈ A, b ∈ B} spans a dense subspace of X, where m is a left multiplication operator on B.

The following is the covariant version of KSGNS construction. Theorem: Let (A, G, α) be a unital C∗-dynamical system and u : G → U(LB(X)) be a unitary repn of G where X is a Hilbert B-module. If φ : A → LB(X) is (α, u)-covariant completely positive linear, then there exist a Hilbert B-module Y , a *-repn π : A → LB(Y ), an isometry V ∈ LB(X, Y ) and a unitary repn β : G → U(LB(Y )) s.t. (i) φ(a) = V ∗π(a)V for all a ∈ A, (ii) {π(a)V (x) : a ∈ A, x ∈ X} is dense in Y , (iii) π(αg(a)) = β(g)π(a)β(g)∗ for all a ∈ A and g ∈ G, (iv) ug = V ∗β(g)V for all g ∈ G.

Definition: (A, G, α) : C∗-dynamical system, u : G → U(A) unitary repn A (quantum) dynamical semigroup {φt : t ≥ 0} on A is (α, u)-covariant if φt(αg(a)) = ugφt(a)u∗

g

for every t ≥ 0. Let {φt : t ≥ 0} be a (α, u)-covariant dynamical semigroup on A. ⇒ There exist a Hilbert A-module X, a covariant repn (π, σ) of (A, G, α) on X and an operator v ∈ LA(A, X) s.t. φ(a) = e, π(a)e, v∗π(a)v = mφ(a), σ(g)v = vmug where m is a left multiplication operator on B.

For 0 ≤ t1 < t2 < ∞, the map πt1 ◦ φt2−t1 is unital completely positive linear and satisfies the equality πt1 ◦ φt2−t1(αg(a)) = πt1(ug)πt1 ◦ φt2−t1(a)πt1(ug)∗ (a ∈ A, g ∈ G). If we denote βt(g) = πt(ug) for every t ≥ 0 and g ∈ G, then πt1 ◦ φt2−t1 is a (α, βt1)-covariant completely positive linear map from A into LA(Xt1).

By applying the covariant version of the KS GNS construction to πt1 ◦ φt2−t1, we obtain a quadruple (Xt1,t2, πt1,t2, Vt1,t2, βt1,t2) where Xt1,t2 is a Hilbert A-module, πt1,t2 : A → LA(Xt1,t2) is a *-repn, Vt1,t2 ∈ LA(Xt1, Xt1,t2) is an isometry and βt1,t2 is a unitary repn of G into LA(Xt1,t2) such that (i) πt1 ◦ φt2−t1(a) = V ∗

t1,t2πt1,t2(a)Vt1,t2 for all a ∈ A,

(ii) the set {πt1,t2(a)Vt1,t2(x) : a ∈ A, x ∈ Xt1} is dense in Xt1,t2, (iii) πt1,t2(αg(a)) = βt1,t2(g)πt1,t2(a)βt1,t2(g)∗ for all a ∈ A and g ∈ G, (iv) βt1(g) = V ∗

t1,t2βt1,t2(g)Vt1,t2 for all g ∈ G.

By continuing this process, we get a family of quadruples {(Xt1,...,tn, πt1,...,tn, Vt1,...,tn, βt1,...,tn) : 0 ≤ t1 < · · · < tn < ∞, n ≥ 2} where (1) Xt1,...,tn is a Hilbert A-module, (2) πt1,...,tn : A → LA(Xt1,...,tn) is a *-homomorphism, (3) Vt1,...,tn ∈ LA(Xt1,...,tn−1, Xt1,...,tn) is an isometry, (4) βt1,...,tn : G → LA(Xt1,...,tn) is a unitary repn such that (i) πt1...,tn−1 ◦ φtn−tn−1(a) = V ∗

t1,...,tnπt1,...,tn(a)Vt1,...,tn for all a ∈ A,

(ii) the set {πt1,...,tn(a)Vt1,...,tn(x) : a ∈ A, x ∈ Xt1...,tn−1} is dense in Xt1,...,tn, (iii) πt1,...,tn(αg(a)) = βt1,...,tn(g)πt1,...,tn(a)βt1,...,tn(g)∗ for all a ∈ A and g ∈ G, (iv) βt1...,tn−1(g) = V ∗

t1,...,tnβt1,...,tn(g)Vt1,...,tn for all g ∈ G.

Theorem: (A, G, α) : C∗-dynamical system, u : G → U(A) unitary repn. If {φt : t ≥ 0} is a (α, u)-covariant dynamical system on A, then there exist (1) a family {Xt : t ≥ 0} of Hilbert A-modules, (2) a family {Jt : t ≥ 0} of *-homomorphisms Jt : A → LA(Xt), (3) a family {V (s, t) : 0 ≤ s < t < ∞} of isometries V (s, t) : Xs → Xt, (4) a unitary repn γ : G → LA(Xt) (t ≥ 0) such that (i) Js(φt−s(αg(a))) = Js(ug)V (s, t)∗Jt(a)V (s, t)Js(ug)∗ (0 ≤ s < t < ∞, a ∈ A), (ii) V (t, u)V (s, t) = V (s, u) (0 ≤ s < t < u < ∞), (iii) the dynamical semigroup {Jt : t ≥ 0} is (α, γ)-covariant.

Symmetry semigroup C∗-system

We assume that a symmetry semigroup S acts on a (unital) C∗-algebra B by a semigroup of endomorphisms s → θs ∈ End(B) such that for each s ∈ S the map θs : B → B is an injective endomorphism (shift) on B, and θs ◦ θr = θs·r. Definition: Let (us) := {us ∈ U(A) : s ∈ S} be a repn of S.

• The triple (B, S, θ) is called a symmetry semigroup C∗-system.
• A C∗-subalgebra C ⊆ B is consistent with a semigroup (θs)s∈S if Cs := θs(C)

contains C for any s ∈ S.

Definition: Let (B, S, θ) be a symmetry semigroup C∗-system.

• A linear map φ : B → A is called (θ, u)-covariant if

φ(θs(b)) = usφ(b)u∗

s for all b ∈ B, s ∈ S.

• A covariant representation of (B, S, θ) is a triple (π, V, X), where π is a unital

*-morphism of B on a Hilbert A-module X and V is an isometric A-modular repn from S into BA(X) s.t. VrVs = Vr·s, Vsx, Vsx = x, x and π(θs(b))Vs = Vsπ(b).

Theorem: Let (B, S, θ) be a symmetry semigroup C∗-system and let u be a repn

• f S into the unitary group U(A) of a C∗-algebra A.

If φ : B → A is a unital (θ, u)-covariant completely positive linear map, then (1) ∃ a covariant repn (π, V, X) of (B, S, θ) with a generating vector e ∈ X s.t. φ(b) = e | π(b)e, (2) for any φ-stationary process {bt : t ∈ S} ⊂ B, i.e., φ(b∗

r·tbs·t) = φ(b∗ rbs),

the X-valued process {xt ≡ π(bt)eut : t ∈ S} is u-stationary, that is, Cr·t,s·t(x) = u∗

tCr,s(x)ut for all r, s, t ∈ S.

Definition: Let T = [0, ∞) be an additive semigroup and let {Xt : t ∈ T} be a family of Hilbert A-modules. A family {Fr(t) : r < t ∈ T} of operators Fr(t) : Xt → Xr (r < t) is called a hemigroup if the backward dynamical condition Fr(s)Fs(t) = Fr(t), (t > s > r) holds and Fr(r) = I for all r ∈ T. Recall that a quantum dynamical conservative semigroup over a unital C∗-algebra A is a one-parameter family φ = {φt : t ∈ T} of unital completely positive (UCP) linear maps φt : A → A satisfying the dynamical condition: φs ◦ φt = φs+t, (s, t ∈ T) and is conservative in the sense that φt(1A) = 1A for all t ∈ T.

Theorem: Let {φt : t ∈ T} be a quantum dynamical conservative semigroup over a unital C∗-algebra A with a unit 1. Then there exist (1) an increasing family X = {Xt : t ∈ T} of Hilbert A-modules, (2) a family π = {πt : t ∈ T} of ∗-repns πt of A on Xt with π0 = idA, (3) a hemigroup F = {Fr(t) : r < t ∈ T} of co-isometries Fr(t) : Xt → Xr s.t.

• πt(a)F †

t (tn)πtn(an) · · · F † t2(t1)πt1(a1)et1

• 2

= κτ

∅(a∗, a),

(a = (a1, . . . , an, a)) for the state vector-martingale et = F †

t (0)1, where

κτ

0(a∗, b) = φt1(a∗ 1φt2−t1(· · · a∗ nφt−tn(a∗b)bn · · · )b1).

In particular, the process π satisfies the strong Markov property Fr(t)πt(Bt)Fr(t)† ⊆ πr(Br) for all t > r ∈ T.

Let T = [0, ∞), S a symmetry semigroup and A a unital C∗-algebra. Let (us) := {us ∈ U(A) : s ∈ S} be a repn of S in the unitary group U(A). Let (A, S, θ) be a symmetry semigroup C∗-system. Recall that a quantum dynamical semigroup {φt : t ≥ 0} on A is (θ, u)-covariant if φt is (θ, u)-covariant for any t ≥ 0, that is, φt(θs(a)) = usφt(a)u∗

s

for all a ∈ A, s ∈ S, t ∈ T. We shall assume that for any s ∈ S, θs : A → A is unitarily implemented as θs(a) = usau∗

s,

(a ∈ A).

Theorem: Let (A, S, θ) be a symmetry semigroup C∗-system. If φ = {φt : t ∈ T} is a (θ, u)-covariant dynamical semigroup over A, then there exist (1) an increasing Hilbert A-module system X = (Xt), (2) a stochastic process π = (πt) given by unital ∗-repns πt : A → LA(Xt), (3) a hemigroup F = {Fr(t) : r < t ∈ T} of co-isometries Fr(t) : Xt → Xr, (4) a family V = {V(·)(t) : t ∈ T} of semigroups {Vs(t) : s ∈ S} of unitaries Vs(t) ∈ LA(Xt) satisfying Vr(t)Vs(t) = Vr·s(t) s.t. the cocycle conditions hold; πt(θs(a))Vs(t) = Vs(t)πt(a), (a ∈ A, s ∈ S, t ∈ T), and Ft0(t)Vs(t) = Vs(t0)Ft0(t), (s ∈ S, t > t0 ∈ T).

For each fixed t ∈ T and a ∈ A, consider the quantum stochastic process {πt(θs(a)) : s ∈ S} and its correlation function C(t)

r,s (a) = πt(θs(a))et | πt(θs(a))et.

We get the following for the stationarity of the quantum stochastic process π. Proposition: The quantum stochastic process {πt(θs(a)) : s ∈ S} is u-stationary in the sense that C(t)

v·r,v·s(a) = uvC(t) r,s (a)u∗ v

for all r, s, v ∈ S.

謝謝 ! Thank you for your attention !