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, y � b . 1 If, in addition, X is complete w.r.t. the norm � x � = �� x, x �� 2 , then X is called a (right) Hilbert A -module .

Adjointable operator Let X and Y be Hilbert A -modules. • B A ( X, Y ): thes set of all bounded right A -module maps from X to Y • L A ( X, Y ): the set of all right A -module maps T : X → Y for which there is an operator T ∗ : Y → X , called the adjoint of T , such that � Tx, y � Y = � 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 = ( E 0 , E 1 , r, s ) be a directed graph where E 0 is a set of vertices, E 1 is a set of edges and r (resp. s ) is the range map (resp. the source map). • A : C ∗ -algebra C 0 ( E 0 ) of continuous functions f : E 0 → C vanishing at ∞ • C c ( E 1 ): space of continuous functions x : E 1 → C with finite support On the space C c ( E 1 ), we define the multiplication and the inner product ( x · f )( e ) = x ( e ) f ( s ( e )) and � x, y � ( v ) = � { e ∈ E 1 : s ( e )= v } x ( e ) y ( e ) . Then we get a Hilbert A -module X by completing the space C c ( E 1 ).

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 n � [ φ ( t − 1 t j )]( x ∗ for every n ∈ N , t 1 , . . . , t n ∈ G and x 1 , . . . , x n ∈ M , i x j ) ≥ 0 . i i,j =1 • 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 on F ( G, M ) × F ( G, M ) as follows: Let 1 be the unit element in M . n m � � For any f = x i δ t i and g = y j δ s j we define i =1 j =1 n m n m � � [ x ∗ i · φ ( t − 1 � � [ φ ( t − 1 s j )]( y j x ∗ � f, g � φ = s j ) · y j ](1) = i ). i i i =1 j =1 i =1 j =1 • 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 ) δ e : x ∈ M , t ∈ G } is dense in K . (iii) [ φ ( t )]( x ) = � ξ, π ( x ) U ( t ) ξ � K 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 ( τ h ( t )) = v h · φ ( t ) · v ∗ for all t ∈ G and all h ∈ H . h • This equality means [ φ ( τ h ( t ))]( x ) = [ φ ( t )]( v ∗ h xv h ) 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 on 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 ) δ e : x ∈ M , t ∈ G } is dense in K (iii) [ φ ( t )]( x ) = � ξ, π ( x ) U ( t ) ξ � K for all t ∈ G and x ∈ M τ ∗ (iv) ˜ τ h U ( t )˜ h = U ( τ h ( t )) for all t ∈ G and h ∈ H , i.e. U is ˜ τ -covariant , (v) π ( x ) U ( τ h ( t )) = π ( v h ) ∗ π ( x ) π ( v h ) U ( t ) for all h ∈ H , t ∈ G , x ∈ M (vi) ˜ τ h π ( x ) = π ( x )˜ τ h for all x ∈ M and h ∈ H .

Definition : G : locally compact group, K : Hilbert space. • A K -valued stochastic process { x t } t ∈ G over G is a map t �→ x t from G into K . • A K -valued stochastic process { x t } t ∈ G is stationary if the correlation function Γ( s, t ) = � x s , x t � depends only on s − 1 t . 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 { x t } . Denoting the induced action by the same τ , then τ h ( x t ) = x τ h ( t ) ∀ h ∈ H, t ∈ G . Definition : A K -valued stochastic process { x t } t ∈ G is called τ -invariant if Γ( τ h ( s ) , τ h ( 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 { x t } t ∈ G s.t. { x t } 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 { x t } 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 → L A ( Y ) and a vector ξ in X s.t. Γ( s, t ) = � x s , x t � = � π ( s ) ξ, π ( t ) ξ � and Y is isomorphic to the closed linear span of the set { x t · 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 { x t } t ∈ G is called v -covariant if the correlation function Γ is v -covariant, that is, if Γ( τ h ( s ) , τ h ( t )) = v ∗ h Γ( s, t ) v h 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 { x t } t ∈ G such that { x t } 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 { J t : A → B} t ∈ I between C ∗ -algebras. • A (quantum) dynamical semigroup on a unital C ∗ -algebra A is a semigroup { φ t : t ≥ 0 } of completely positive linear maps of A into A s.t. (i) φ 0 = id A where id A is the identity map on A , (ii) φ t (1) = 1 for all t ≥ 0.

Parthasarathy: M : von Neumann algebra on a Hilbert space H 0 { φ t : t ≥ 0 } : one-parameter semigroup of unital c.p. maps from M into B ( H 0 ) ⇒ there exist (i) family of representations ( π t , H t ) ( t ≥ 0) of M (ii) and isometries V ( s, t ) : H s → H t (0 ≤ s < t ) V ∗ ( s, t ) π t ( a ) V ( s, t ) = π s ( φ t − s ( a )) such that 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.