Quantum Markov semigroups
and quantum stochastic flows
– construction and perturbation
Alexander Belton
(Joint work with Martin Lindsay and Adam Skalski, and Stephen Wills)
Department of Mathematics and StatisticsLancaster UniversityUnited Kingdom
Noncommutative Geometry Seminar
Mathematical Institute of the Polish Academy of Sciences
25th November 2013
Structure of the talk
Construction
The classical theory
Non-commutative probability
Solving a quantum stochastic differential equation [BW]
Examples [BW]
Perturbation
The classical theory
Quantum Feynman–Kac on von Neumann algebras [BLS]
Quantum Feynman–Kac on C∗ algebras [BW]
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 2 / 67
Construction
Classical Markov semigroups
Markov processes
Markov semigroups Infinitesimal generators
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 4 / 67
Markov processes
Definition
A Markov process with state space S is a collection of S-valued randomvariables (Xt)t>0 on a common probability space such that
E[f (Xs+t)|σ(Xr : 0 6 r 6 s)
]= E
[f (Xs+t)|Xs
](s, t > 0)
for all f ∈ L∞(S).
Such a Markov process is time homogeneous if
E[f (Xs+t)|Xs = x
]= E[f (Xt)|X0 = x
](s, t > 0, x ∈ S)
for all f ∈ L∞(S).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 5 / 67
Markov semigroups
Given a time-homogeneous Markov process, setting
(Tt f )(x) = E[f (Xt)|X0 = x
](t > 0, x ∈ S)
defines a Markov semigroup on L∞(S).
Definition
A Markov semigroup on L∞(S) is a family (Tt)t>0 such that
1 Tt : L∞(S) → L∞(S) is a linear operator for all t > 0,
2 Ts Tt = Ts+t for all s, t > 0 and T0 = I (semigroup),
3 ‖Tt‖ 6 1 for all t > 0 (contraction),
4 Tt f > 0 whenever f > 0, for all t > 0 (positive).
If Tt1 = 1 for all t > 0 then T is conservative.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 6 / 67
Feller semigroups
Definition
Suppose the state space S is a locally compact Hausdorff space. TheMarkov semigroup T is Feller if
Tt
(C0(S)
)⊆ C0(S) (t > 0)
and‖Tt f − f ‖∞ → 0 as t → 0
(f ∈ C0(S)
).
Every sufficiently well-behaved time-homogeneous Markov process isFeller: Brownian motion, Poisson process, Levy processes, . . .
Theorem 1
If the state space S is separable then every Feller semigroup gives rise to a
time-homogeneous Markov process.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 7 / 67
Infinitesimal generators
Definition
Let T be a C0 semigroup on a Banach space E . Its infinitesimal generator
is the linear operator τ in E with domain
dom τ =f ∈ E : lim
t→0t−1(Tt f − f ) exists
and actionτ f = lim
t→0t−1(Tt f − f ).
The operator τ is closed and densely defined.
If T comes from a Markov process X then
E[f (Xt+h)− f (Xt)|Xt
]= (Thf − f )(Xt) = h(τ f )(Xt) + o(h),
so τ describes the change in X over an infinitesimal time interval.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 8 / 67
The Lumer–Phillips theorem
Theorem 2 (Lumer–Phillips)
A closed, densely defined operator τ in the Banach space E generates a
strongly continuous contraction semigroup on E if and only if
im(λI − τ) = E for some λ > 0
and (dissipativity)
‖(λI − τ)x‖ > λ‖x‖ for all λ > 0 and x ∈ dom τ.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 9 / 67
The Hille–Yosida–Ray theorem
Theorem 3 (Hille–Yosida–Ray)
A closed, densely defined operator τ in C0(S) is the generator of a Feller
semigroup on C0(S) if and only if
λI − τ : dom τ → X has bounded inverse for all λ > 0 and
τ satisfies the positive maximum principle.
Definition
Let S be a locally compact Hausdorff space. A linear operator τ in C0(S)satisfies the positive maximum principle if whenever f ∈ dom τ and s0 ∈ S
are such thatsups∈S
f (s) = f (s0) > 0
then (τ f )(s0) 6 0.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 10 / 67
Quantum Markov semigroups
Quantum Markov processes
Quantum Feller semigroups Infinitesimal generators
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 11 / 67
Quantum Feller semigroups
Theorem 4
Every commutative C ∗ algebra is isometrically isomorphic to C0(S), whereS is a locally compact Hausdorff space.
Definition
A quantum Feller semigroup on the C ∗ algebra A is a family (Tt)t>0 suchthat
1 Tt : A → A is a linear operator for all t > 0,
2 Ts Tt = Ts+t for all s, t > 0 and T0 = I ,
3 ‖Ttx − x‖ → 0 as t → 0 for all x ∈ A,
4 ‖Tt‖ 6 1 for all t > 0,
5 (Ttaij) ∈ Mn(A)+ whenever (aij) ∈ Mn(A)+, for all n > 1 and t > 0.
If A is unital and Tt1 = 1 for all t > 0 then T is conservative.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 12 / 67
From generators to semigroups
Problem
Given a densely defined operator τ on the C ∗ algebra A, show that τ isclosable and its closure that generates a quantum Feller semigroup T .
It can be very difficult to verify the conditions of the Lumer–Phillipstheorem.
Is there a non-commutative version of the positive maximumprinciple?
Strategy
Rather than construct the semigroup T directly, we shall instead constructa quantum Markov process which is a dilation of T .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 13 / 67
Non-commutative probability
Random variables
If X : Ω → S is a classical S-valued random variable then
A → M; f 7→ f X
is a unital ∗-homomorphism, where A = C0(S) and M = L∞(Ω,F,P).A non-commutative random variable is a unital ∗-homomorphism
j : A → M
from a unital C ∗ algebra A ⊆ B(h) to a von Neumann algebra M.Classical stochastic calculus has a universal sample space: cadlagfunctions.In quantum stochastic calculus, B(F) plays this role, where F is BosonFock space over L2(R+; k), and
A⊗m B(F) ⊆ M := B(h⊗F).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 14 / 67
Non-commutative probability
Adaptedness
L2(R+) ∼= L2[0, t)⊕ L2[t,∞)
=⇒ F ∼= Ft) ⊗F[t ; ε(f ) ∼= ε(f |[0,t))⊗ ε(f |[t,∞)),
where E := linε(f ) : f ∈ L2(R+; k), the linear span of the exponential
vectors (such that 〈ε(f ), ε(g)〉 = exp〈f , g〉) is dense in F .
A quantum flow is a family of unital ∗-homomorphisms
(jt : A → A⊗m B(F)
)t>0
which is adapted:
jt(a) = jt)(a)⊗ I[t ∈ A⊗m B(Ft))⊗ CI[t (t > 0, a ∈ A).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 15 / 67
Non-commutative probability
Dilation
A quantum flow (jt : A → M)t>0 is a dilation of the quantum Fellersemigroup T on A if there exists a conditional expectation E : M → Asuch that Tt = E jt for all t > 0.
Markovianity
In this framework, Markovianity is given by a cocycle property:
js+t = s σs jt (s, t > 0),
where
σs : A⊗m B(F)∼=−→ A⊗m B(F[s)
is the shift (CCR flow) and
s = js) ⊗m id[s : A⊗m B(F[s) → A⊗m B(F).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 16 / 67
Non-commutative probability
Cocycles produce semigroups on ALet EΩ : M → A be such that
〈u,EΩ(X )v〉 = 〈u ⊗ ε(0),Xv ⊗ ε(0)〉(u, v ∈ h, X ∈ M).
Then EΩ j is a semigroup on A if the quantum flow j is a Markoviancocycle.
If, further, t 7→ EΩ jt is norm continuous then the quantum flow j is aFeller cocycle and EΩ j is a quantum Feller semigroup.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 17 / 67
Non-commutative probability
Let A0 ⊆ A ⊆ B(h) be a norm-dense ∗-subalgebra of A whichcontains 1 = Ih.
Definition
A family of linear operators (Xt)t>0 in h⊗F with domains including h⊙Eis an adapted operator process if
〈uε(f ),Xtvε(g)〉 = 〈uε(1[0,t)f ),Xtvε(1[0,t)g)〉〈ε(1[t,∞)f ), ε(1[t,∞)g)〉
for all u, v ∈ h, f , g ∈ L2(R+; k) and t > 0.
An adapted mapping process on A0 is a family of linear maps
(jt : A0 → L(h⊙E ; h⊗F)
)t>0
such that(jt(x)
)t>0
is an adapted operator process for all x ∈ A0.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 18 / 67
A quantum stochastic differential equation
The generator
Let φ : A0 → A0⊙B be a linear map, where B = B(k) and k := C⊕ k.Distinguish the unit vector ω := (1, 0) ∈ k and let x := (1, x) for all x ∈ k.
For all z , w ∈ k, let φzw : A0 → A0 be the linear map such that
〈u, φzw (x)v〉 = 〈u ⊗ z , φ(x)v ⊗ w〉 (u, v ∈ h, x ∈ A0).
Definition 5
An adapted mapping process j on A0 satisfies the QSDE
j0(x) = x ⊗ IF , djt(x) = (t φ)(x) dΛt (x ∈ A0) (1)
if and only if
〈uε(f ), (jt(x) − x ⊗ IF )vε(g)〉 =∫ t
0〈uε(f ), js
(φf (s)
g(s)(x)
)vε(g)〉 ds
for all x ∈ A0 and all t > 0, u, v ∈ h and f , g ∈ L2(R+; k).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 19 / 67
Feller cocycles from QSDEs
Theorem 6
If the quantum flow j satisfies the QSDE (1) then j is a Feller cocycle and
the generator of the Feller semigroup EΩ j is an extension of φωω.
Theorem 7
Let the quantum flow j satisfy (1). Then φ : A0 → A0⊙B is such that
φ is ∗-linear,φ(1) = 0 and
φ(xy) = φ(x)(y ⊗ Ik) + (x ⊗ Ik)φ(y) + φ(x)∆φ(y) for all x, y ∈ A0,
where ∆ := Ih ⊗ Pk =[ 0 00 Ih⊗k
]∈ A0⊙B(k) and Pk := |ω〉〈ω|⊥ ∈ B(k) is
the orthogonal projection onto k ⊂ k.
Question
When are these necessary conditions sufficient for the solution of (1) to bea quantum flow?
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 20 / 67
Flow generators
Lemma 8
The map φ : A0 → A0⊙B is a flow generator if and only if
φ(x) =
[τ(x) δ†(x)
δ(x) π(x)− x ⊗ Ik
]for all x ∈ A0, (2)
where
π : A0 → A0⊙B(k) is a unital ∗-homomorphism,
δ : A0 → A0⊙B(C; k) is a π-derivation, i.e., a linear map such that
δ(xy) = δ(x)y + π(x)δ(y) (x , y ∈ A0)
δ† : A0 → A0⊙B(k;C) is such that δ†(x) = δ(x∗)∗ for all x ∈ A0,
τ : A0 → A0 is ∗-linear and such that
τ(xy)− τ(x)y − xτ(y) = δ†(x)δ(y) (x , y ∈ A0).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 21 / 67
Quantum Wiener integrals
Theorem 9
For all n ∈ N and T ∈ B(h⊗ k⊗n) there exists a family(Λnt (T )
)t>0
of
linear operators in h⊗F , with domains including h⊙E , that is adaptedand such that
〈uε(f ),Λnt (T )vε(g)〉 =
∫
Dn(t)〈u ⊗ f ⊗n(t),Tv ⊗ g⊗n(t)〉 dt 〈ε(f ), ε(g)〉
for all u, v ∈ h, f , g ∈ L2(R+; k) and t > 0, where the simplex
Dn(t) := t := (t1, . . . , tn) ∈ [0, t]n : t1 < · · · < tn
and
f ⊗n(t) := f (t1)⊗ · · · ⊗ f (tn) et cetera.
We include n = 0 by setting Λ0t (T ) := T ⊗ IF for all t > 0.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 22 / 67
An estimate for quantum Wiener integrals
Proposition 10
If n ∈ Z+, t > 0, T ∈ B(h⊗ k⊗n) and f ∈ L2(R+; k) then
‖Λnt (T )uε(f )‖ 6
Knf ,t√n!
‖T‖ ‖uε(f )‖,
where Kf ,t :=√
(2 + 4‖f ‖2)(t + ‖f ‖2).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 23 / 67
A quantum random walk
Definition
Given a flow generator φ, the family of linear maps
φn : A0 → A0 ⊙B⊙ n
is defined by setting
φ0 := idA0 and φn+1 :=(φn ⊙ idB
) φ for all n ∈ Z+,
where idA0 is the identity map on A0 and similarly for idB.
Let
Aφ := x ∈ A0 : ∃Cx ,Mx > 0 with ‖φn(x)‖ 6 CxMnx ∀ n ∈ Z+
be those elements of A0 for which(φn(x)
)n>0
has polynomial growth.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 24 / 67
Solutions to the QSDE
Theorem 11
If x ∈ Aφ then the series
jt(x) :=∞∑
n=0
Λnt
(φn(x)
)
is strongly absolutely convergent on h⊙E for all t > 0. The family of
operators(jt(x)
)t>0
is an adapted mapping process on Aφ which satisfies
the QSDE (1) on Aφ.
Questions1 When does Aφ = A0?
2 When does the adapted mapping process j given by Theorem 11extend to a quantum flow?
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 25 / 67
Answers
If Aφ = A0 then the adapted mapping process j given by Theorem 11extends to a quantum flow as long as A is sufficiently well behaved.
In practice it will be difficult to find directly constants Cx and Mx
such that ‖φn(x)‖ 6 CxMnx for all x ∈ A0.
Key to both: the higher-order Ito formula.
Two easy cases when Aφ = A0
If k is finite dimensional and φ is bounded then so is each φn, with
‖φn‖ 6
(dim k
)n−1‖φ‖n (n ∈ N).
If φ is completely bounded then so is each φn, with
‖φn‖ 6 ‖φ‖ncb (n ∈ Z+).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 26 / 67
The higher-order Ito formula
Notation
Let α ⊆ 1, . . . , n, with elements arranged in increasing order andcardinality |α|. The unital ∗-homomorphism
A0⊙B⊙|α| → A0⊙B⊙ n; T 7→ T (n, α)
is defined by linear extension of the map
A⊗ B1 ⊗ · · · ⊗ B|α| 7→ A⊗ C1 ⊗ · · · ⊗ Cn,
where
Ci :=
Bj if i is the jth element of α,Ik
if i is not an element of α.
Example
(A⊗ B1 ⊗ B2 ⊗ B3)(5, 1, 3, 4
)= A⊗ B1 ⊗ I
k⊗ B2 ⊗ B3 ⊗ I
k.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 27 / 67
More notation
For all n ∈ Z+ and α ⊆ 1, . . . , n, let
φ|α|(x ; n, α) :=(φ|α|(x)
)(n, α) for all x ∈ A0.
Also let∆(n, α) := (Ih ⊗ P
⊗|α|k )(n, α),
so that Pk acts on the components of k⊗n which have indices in α and Ik
acts on the others.
Theorem 12
Let φ be a flow generator. For all n ∈ Z+ and x, y ∈ A0,
φn(xy) =∑
α∪β=1,...,n
φ|α|(x ; n, α)∆(n, α ∩ β)φ|β|(y ; n, β),
where the summation is taken over all α and β with union 1, . . . n.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 28 / 67
Two corollaries
Corollary 13
If φ : A0 → A0⊙B is a flow generator then Aφ is a unital ∗-subalgebraof A0, which is equal to A0 if Aφ contains a ∗-generating set for A0.
Corollary 14
Let φ : A0 → A0⊙B be a flow generator and let j be the adapted
mapping process on Aφ given by Theorem 11. If x, y ∈ Aφ then
x∗y ∈ Aφ, with
〈jt(x)uε(f ), jt(y)vε(g)〉 = 〈uε(f ), jt(x∗y)vε(g)〉
for all u, v ∈ h and f , g ∈ L2(R+; k).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 29 / 67
The first dilation theorem
Theorem 15
Let φ : A0 → A0⊙B be a flow generator and suppose A0 contains its
square roots: for all non-negative x ∈ A0, the square root x1/2 lies in A0.
If Aφ = A0 then there exists a quantum flow such that
t(x) = jt(x) on h⊙E (x ∈ A0),
where j is the adapted mapping process given by Theorem 11.
Remark
If A is an AF algebra, i.e., the norm closure of an increasing sequence offinite-dimensional ∗-subalgebras, then its local algebra A0, the union ofthese subalgebras, contains its square roots.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 30 / 67
The second dilation theorem
Theorem 16
Let A be the universal C ∗ algebra generated by isometries si : i ∈ I, andlet A0 be the ∗-algebra generated by si : i ∈ I.If φ : A0 → A0⊙B is a flow generator such that Aφ = A0 then there
exists a quantum flow such that
t(x) = jt(x) on h⊙E (x ∈ A0),
where j is the adapted mapping process given by Theorem 11.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 31 / 67
Random walks on discrete groups
The group algebra
Let G is a discrete group and set A = C0(G )⊕C1 ⊆ B(ℓ2(G )
), where
x ∈ C0(G ) acts on ℓ2(G ) by multiplication.
Let A0 = lin1, eg : g ∈ G, where eg (h) := 1g=h for all h ∈ G .
Permitted moves
Let H be a non-empty finite subset of G \ e and let the Hilbert space khave orthonormal basis fh : h ∈ H; the maps
λh : G → G ; g 7→ hg (h ∈ H)
correspond to the permitted moves in the random walk to be constructedon G .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 32 / 67
Random walks on discrete groups
Lemma 17
Given a transition function
t : H × G → C; (h, g) 7→ th(g),
the map φ : A0 → A0⊙B such that
x 7→[ ∑
h∈H |th|2(x λh − x)∑
h∈H th(x λh − x)⊗ 〈fh|∑
h∈H th(x λh − x)⊗ |fh〉∑
h∈H(x λh − x)⊗ |fh〉〈fh|
]
is a flow generator with φn(eg ) equal to
∑
h1∈H∪e
· · ·∑
hn∈H∪e
eh−1n ···h−1
1 g⊗mhn(h
−1n · · · h−1
1 g)⊗ · · · ⊗mh1(h−11 g)
for all n ∈ N and g ∈ G.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 33 / 67
Random walks on discrete groups
One-step matrices
For all g ∈ G and h ∈ H, let
me(g) :=
[−∑
h∈H |th(g)|2 −∑h∈H th(g)〈fh|
−∑
h∈H th(g)|fh〉 −Ik
]
and
mh(g) :=
[|th(g)|2 th(g)〈fh|th(g)|fh〉 |fh〉〈fh|
].
Then‖me(g)‖ = 1 +
∑
h∈H
|th(g)|2
and‖mh(g)‖ = 1 + |th(g)|2.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 34 / 67
Random walks on discrete groups
A sufficient condition for Aφ = A0
If
Mg := limn→∞
sup|th(h−1
n · · · h−11 g)| : h1, . . . , hn ∈ H ∪ e, h ∈ H
<∞
(3)then
‖φn(eg )‖ 6(1 + |H|+ 2|H|M2
g
)n(n ∈ Z+),
where |H| denotes the cardinality of H.
Hence Aφ = A0 if (3) holds for all g ∈ G .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 35 / 67
Random walks on discrete groups
Examples
1 If t is bounded then (3) holds for all g ∈ G .
2 If G = (Z,+), H = ±1 and the transition function t is bounded,with t+1(g) = 0 for all g < 0 and t−1(g) = 0 for all g 6 0, then theFeller semigroup T which arises corresponds to the classicalbirth-death process with birth and dates rates |t+1|2 and |t−1|2,respectively.
3 If G = (Z,+), H = +1 and t+1 : g 7→ 2g then Mg = 2g and thecondition (3) holds for all g ∈ G . Thus the construction applies toexamples where the transition function t is unbounded.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 36 / 67
The symmetric quantum exclusion process
The CAR algebra
For a non-empty set I, the CAR algebra is the unital C ∗ algebra A withgenerators bi : i ∈ I, subject to the anti-commutation relations
bi , bj = 0 and bi , b∗j = 1i=j (i , j ∈ I).
Let A0 be the unital algebra generated by bi , b∗i : i ∈ I.
Lemma 18
For each x ∈ A0 there exists a finite subset I0 ⊆ I such that x lies in the
finite-dimensional ∗-subalgebra
AI0 := linb∗j1 · · · b
∗jqbi1 · · · bip : distinct i1, . . . , ip ∈ I0, j1, . . . , jq ∈ I0.
Consequently, A is an AF algebra and A0 contains its square roots.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 37 / 67
The symmetric quantum exclusion process
Amplitudes
Let αi ,j : i , j ∈ I ⊆ C be a fixed collection of amplitudes, so that(I, αi ,j) is a complex digraph. For all i ∈ I, let
supp(i) := j ∈ I : αi ,j 6= 0 and supp+(i) := supp(i) ∪ i.
Thus supp(i) is the set of sites with which site i interacts and | supp(i)| isthe valency of the vertex i . Suppose
| supp(i)| <∞ (i ∈ I).
The transport of a particle from site i to site j with amplitude αi ,j isdescribed by the operator
ti ,j := αi ,j b∗j bi .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 38 / 67
The symmetric quantum exclusion process
Energies
Let ηi : i ∈ I ⊆ R be fixed. The total energy in the system is given by
h :=∑
i∈I
ηi b∗i bi ,
where ηi gives the energy of a particle at site i .
Lemma 19
Setting
τ(x) := i∑
i∈I
ηi [b∗i bi , x ]− 1
2
∑
i ,j∈I
τi ,j(x)
defines a ∗-linear map τ : A0 → A0, where
τi ,j(x) := t∗i ,j [ti ,j , x ] + [x , t∗i ,j ]ti ,j .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 39 / 67
Lemma 20
Let k be a Hilbert space with orthonormal basis fi ,j : i , j ∈ I. Setting
δ(x) :=∑
i ,j∈I
[ti ,j , x ]⊗ |fi ,j〉 (x ∈ A0),
where
|fi ,j〉 : C 7→ k; λ 7→ λfi ,j ,
defines a linear map δ : A0 → A0⊙B(C; k) such that
δ(xy) = δ(x)y + (x ⊗ Ik)δ(y)
and δ†(x)δ(y) = τ(xy)− τ(x)y − xτ(y) (x , y ∈ A0),
with τ defined as in Lemma 19. Hence
φ : A0 → A0 ⊙B; x 7→[τ(x) δ†(x)δ(x) 0
]
is a flow generator.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 40 / 67
The symmetric quantum exclusion process
Lemma 21
If the amplitudes satisfy the symmetry condition
|αi ,j | = |αj ,i | for all i , j ∈ I (4)
then
φn(bi0) =∑
i1∈supp+(i0)
· · ·∑
in∈supp+(in−1)
bin ⊗ Bin−1,in ⊗ · · · ⊗ Bi0,i1
for all n ∈ N and i0 ∈ I, where
Bi ,j := 1j=iλi |ω〉〈ω| + |ω〉〈αi ,j fi ,j | − |αj ,i fj ,i〉〈ω| (i , j ∈ I)
and
λi := −iηi − 12
∑
j∈supp(i)
|αj ,i |2 (i ∈ I).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 41 / 67
The symmetric quantum exclusion process
Example
Suppose that the amplitudes satisfy the symmetry condition (4), andfurther that there are uniform bounds on the amplitudes, valencies andenergies, so that
M := supi ,j∈I
|αi ,j |, V := supi∈I
| supp(i)| and H := supi∈I
|ηi |
are all finite. Then
|λi | 6 |ηi |+ 12VM
2 and ‖Bi ,j‖ 6 |λi |+ 2M 6 H + 12VM
2 + 2M
for all i , j ∈ I. Hence
‖φn(bi )‖ 6 (V + 1)n(H + 1
2VM2 + 2M
)n(n ∈ Z+, i ∈ I)
and Aφ = A0.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 42 / 67
The universal rotation algebra
Definition
Let A be the universal rotation algebra: this is the universal C ∗ algebrawith unitary generators U, V and Z satisfying the relations
UV = ZVU, UZ = ZU and VZ = ZV .
It is the group C ∗ algebra corresponding to the discrete Heisenberg groupΓ := 〈u, v , z | uv = zvu, uz = zu, vz = zv〉.
A pair of derivations
Letting A0 denote the ∗-subalgebra generated by U, V and Z , there areskew-adjoint derivations
δ1 : A0 → A0; UmV nZ p 7→ mUmV nZ p
and δ2 : A0 → A0; UmV nZ p 7→ nUmV nZ p.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 43 / 67
The universal rotation algebra
Theorem 22
Fix c1, c2 ∈ C, let δ = c1δ1 + c2δ2 and define the Bellissard map
τ : A0 → A0;
UmV nZ p 7→
−(12 |c1|
2m2 + 12 |c2|
2n2 + c1c2mn + (c1c2 − c1c2)p)UmV nZ p.
Then
φ : A0 → A0 ⊙B(C2); x 7→[τ(x) δ†(x)
δ(x) 0
]
is a flow generator. Furthermore, U, V , Z ∈ Aφ and Aφ = A0.
Remark
If c1c2 = c1c2 then this construction specialises to the non-commutativetorus.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 44 / 67
The non-commutative torus
Definition
Let A be the non-commutative torus with parameter λ ∈ T, so that A isthe universal C ∗ algebra with unitary generators U and V subject to therelation
UV = λVU,
and letA0 := 〈U,V 〉 = linUmV n : m, n ∈ Z.
An automorphism
For each (µ, ν) ∈ T2, let πµ,ν be the automorphism of A such that
πµ,ν(UmV n) = µmνnUmV n for all m, n ∈ Z.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 45 / 67
The non-commutative torus
Theorem 23
Fix (µ, ν) ∈ T2 with µ 6= 1. There exists a flow generator
φ : A0 → A0⊙B(C2); x 7→[τ(x) −µδ(x)δ(x) πµ,ν(x)− x
],
where the πµ,ν-derivation
δ : A0 → A0; UmV n 7→ 1− µmνn
1− µUmV n
is such that δ† = −µδ and the map
τ :=µ
1− µδ : A0 → A0; UmV n 7→ µ(1− µmνn)
(1− µ)2UmV n.
Furthermore, U, V ∈ Aφ and so Aφ = A0.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 46 / 67
Perturbation
Classical Feynman–Kac perturbation
Ito’s formula for Brownian motion
If f : R → R is twice continuously differentiable then
f (Bt) = f (B0) +
∫ t
0f ′(Bs) dBs +
1
2
∫ t
0f ′′(Bs) ds (t > 0).
Corollary
Define a Feller semigroup (Tt)t>0 on C0(R) by setting
(Tt f )(x) := E[f (Bt)|B0 = x
](t > 0, f ∈ C0(R), x ∈ R).
Then1
t
[f (Bt)− f (B0)|B0 = x
]→ 1
2f ′′(x) as t → 0,
so the generator of this semigroup extends f 7→ 12 f
′′.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 48 / 67
Classical Feynman–Kac perturbation
Proposition
If v : R → R is well behaved and Yt := exp(∫ t
0 v(Bs) ds)then
Yt f (Bt) = f (B0) +
∫ t
0Ys f
′(Bs) dBs
+
∫ t
0
(v(Bs)Ys f (Bs) +
1
2Ys f
′′(Bs))ds.
Proof
This follows from the classical Ito product formula:
d(Yt f (Bt)
)= (dYt)f (Bt) + Yt d
(f (Bt)
)+ dYt d
(f (Bt)
).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 49 / 67
Classical Feynman–Kac perturbation
Theorem (F–K)
Let
(St f )(x) := E
[exp
(∫ t
0v(Bs) ds
)f (Bt)|B0 = x
].
Then (St)t>0 is a Feller semigroup on C0(R) with generator which extendsf 7→ 1
2 f′′ + vf .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 50 / 67
Introducing non-commutativity
Idea
There are two ways in which non-commutativity can appear:
1 quantum stochastic noises replace Brownian motion;
2 C0(R) is replaced by a general C ∗ algebra.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 51 / 67
Feynman–Kac perturbation (von Neumann)
The free flow
Until further notice, A is a von Neumann algebra, so ⊗m = ⊗, and j is anultraweakly continuous, normal quantum flow and a Feller cocycle.
Multipliers
A multiplier for j is an adapted operator process Y ⊆ A⊗ B(F) such that
Y0 = Ih⊗F and Ys+t = Js(Yt)Ys (s, t > 0).
Theorem 24
If Y and Z are multipliers for j and
kt : A → A⊗ B(F); a 7→ Y ∗t jt(a)Zt (t > 0)
then k is a Feller cocycle and EΩ k is a semigroup on A.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 52 / 67
Some questions
Generators1 How is the generator of EΩ k related to that of EΩ j?
2 How is the generator of k related to that of j?
Existence1 How do we produce multipliers?
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 53 / 67
The multiplier QSDE
Theorem 25
Let F ∈ A⊗ B(k) be such that
q(F ) := F + F ∗ + F∆F ∗6 0,
where ∆ :=[0 0
0 Ih⊗k
].
There exists a unique operator process Y ⊆ A⊗m B(F) such that
Yt = Ih⊗F +
∫ t
0s(F )Ys dΛs (t > 0),
where s := js ⊗m idB(k)
and Ys := Ys ⊗ Ik.
Each Yt is a contraction, and is an isometry if and only if q(F ) = 0.
The adapted operator process Y is a multiplier for j with generator F .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 54 / 67
Classical multipliers
Feynman–Kac
Yt = exp(∫ t
0v(Bs) ds) ⇒ Yt = 1 +
∫ t
0v(Bs)Ys ds = 1 +
∫ t
0js(v)Ys ds.
Cameron–Martin
Yt = exp(Bt −
1
2t)
⇒ Yt = 1 +
∫ t
0Ys dBs = 1 +
∫ t
0s
([0 1
1 0
])Ys dΛs .
Then EΩ k has generator f 7→ 12 f
′′ + f ′.
C–M: Setting (Tt f )(x) := E[f (Bt + t)|B0 = x
]gives the same generator.
See Pinsky (1972), Stroock (1970).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 55 / 67
A QSDE for the free flow
Assumption
Suppose the free flow j satisfies the QSDE (1) on A0 ⊆ A, i.e.,
djt(x) = (t φ)(x) dΛt (x ∈ A0),
where φ : A0 → A⊗m B is the generator of j .
Example: Brownian motion
If jt : f 7→ f (Bt) then A0 ⊆ C 2(R) and
jt(g) = g(Bt) = g(B0) +
∫ t
0g ′(Bs) dBs +
1
2
∫ t
0g ′′(Bs) ds
= j0(g) +
∫ t
0s
([ 12δ2(g) δ(g)
δ(g) 0
])dΛs (g ∈ A0),
where δ : h 7→ h′.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 56 / 67
The flow generator
Structure of φ
As in Lemma 8, the generator φ has the form
φ =
[L δ†
δ π − ι
],
where
π : A0 → A⊗m B(k) is a unital ∗-homomorphism,
ι : A0 → A0⊙B(k); x 7→ x ⊗ Ik,
δ : A0 → A⊗m B(C; k) is a π-derivation: δ(xy) = δ(x)y + π(x)δ(y)
δ† : x 7→ δ(x∗)∗
L “generates” EΩ j , L(xy)− L(x)y − xL(y) = δ†(x)δ(y).
The Brownian case
(fg)′′ − f ′′g − fg ′′ = 2f ′g ′.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 57 / 67
Quantum Feynman–Kac perturbation
Theorem 26
If the multipliers Y and Z have generators F and G respectively then
dkt(x) = (kt ψ)(x) dΛt (x ∈ A0),
where
ψ(x) = φ(x)+F ∗(∆φ(x)+ι(x)
)+F ∗∆
(φ(x)+ι(x)
)∆G+
(φ(x)∆+ι(x)
)G .
Corollary
The semigroup EΩ k has generator which extends
x 7→ L(x) + ℓ∗F δ(x) + ℓ∗F π(x) ℓG + δ†(x) ℓG +m∗F x + x mG ,
where
F =
[mF ∗ℓF ∗
]and G =
[mG ∗ℓG ∗
].
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 58 / 67
Quantum Feynman–Kac perturbation – proof
Quantum Ito product formula
If X =∫F dΛ and Y =
∫G dΛ then
XtYt =
∫ t
0Xs dYs + (dXs)Ys + dXs dYs
=
∫ t
0
(XsGs + FsYs + Fs∆Gs
)dΛs (t > 0).
Key step in proving of Theorem 26
d(jt(x)Zt) = ( ˜jt(x)t(G )Zt + t(φ(x)
)Zt + t
(φ(x)
)∆t(G )Zt
)dΛt
= t(ι(x)G + φ(x) + φ(x)∆G )Zt dΛt .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 59 / 67
Quantum Feynman–Kac for C ∗-algebras
The matrix-space product
A⊗m B(F) := T ∈ B(h⊗F) : T xy ∈ A for all x , y ∈ F,
where T xy ∈ B(h); 〈u,T x
y v〉 = 〈u ⊗ x ,Tv ⊗ y〉.
If k : A1 → A2 is completely bounded then
k ⊗m idB(H) : A1 ⊗m B(H) → A2 ⊗m B(H); (k ⊗m idB(H))(T )xy = k(T xy ).
Problems
Henceforth A is only a C ∗ algebra. Lack of ultraweak continuity/closuremeans
1 A⊗m B(F) is not an algebra ⇒ kt(a) = Y ∗t jt(a)Zt 6∈ A ⊗m B(F),
2 t := jt ⊗m idB(k)
is not multiplicative ⇒ key step for Theorem 26
fails.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 60 / 67
Semigroup decomposition for a Markovian cocycle
Theorem 27
A mapping process(kt : A → B(h⊗F)
)t>0
of completely bounded maps
is a Feller cocycle if
1 im kt ⊆ A⊗m B(F) ( t > 0 );
2 ks+t = ks σs kt ( s, t > 0 );
3 EΩ k0 = idA.
A mapping process k is a Feller cocycle if and only if there exists a total
subset T ⊆ k such that 0 ∈ T,
t 7→ Pz ,wt : A → B(h); a 7→ kt(a)
(1[0,t)z)
(1[0,t)w)
is a semigroup on A for all z, w ∈ T,whenever f and g are step functions subordinate to some partition
0 = t0 < t1 < · · · ,
kt(a)(1[0,t)f )
(1[0,t)g)= P f (t0),g(t0)
t1−t0 · · · P f (tn),g(tn)
t−tn
(t ∈ [tn, tn+1)
).
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 61 / 67
C∗ Feller cocycles from QSDEs
Theorem 28
Let k be a mapping process with locally bounded norm and such that
k0(x) = x ⊗ IF , dkt(x) =(kt ψ)(x) dΛt on E(T) (x ∈ A0)
for a norm-densely defined linear map ψ : A0 → A⊗m B(k) and a total
set 0 ∈ T ⊆ k, where
E(T) := linε(f ) : f is a T-valued, right-continuous step function.
If, for all z, w ∈ T, there exists a C0-semigroup generator ηz ,w with a core
Az ,w ⊆ A0 such that
ψ(x)zw = ηz ,w (x) (x ∈ Az ,w )
then k is a Feller cocycle.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 62 / 67
Existence of multipliers
Obstruction
Let F ∈ A⊗m B(k). The construction of Theorem 25 applies, but theproof of contractivity fails as t is not multiplicative.
Solution
Choose the generator F such that q(F ) 6 0 and
t(F )∗∆t(F ) = t(F
∗∆F ) (t > 0). (5)
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 63 / 67
A QSDE for the perturbed process
Theorem 29
If Y and Z are contractive multipliers with generators F and G
respectively such that
∆F ,∆G ∈T ∈ B
(h⊗ (k)
): Ty ∈ A⊗ B(C; k) for all y ∈ k
then the mapping process
kt : A → B(h⊗F); a 7→ Y ∗t jt(a)Zt (t > 0)
is such that
k0(x) = x ⊗ IF , dkt(x) =(kt ψ)(x) dΛt on E(T) (x ∈ A0)
where ψ is as in Theorem 26.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 64 / 67
Simplifying the generator
Bounded parts
Note that
ψ(x) = φ(x) + F ∗(∆φ(x) + ι(x)
)+ F ∗∆
(φ(x) + ι(x)
)∆G
+(φ(x)∆ + ι(x)
)G
= (Ih⊗k
+∆F )∗φ(x)(Ih⊗k
+∆G )
+F ∗ι(x) + F ∗∆ι(x)∆G + ι(x)G .
Let ∆F =[
0 0
ℓF wF−Ih⊗k
]and ∆G =
[0 0
ℓG wG−Ih⊗k
]. Then
(Ih⊗k
+∆F )∗φ(x)(Ih⊗k
+∆G )
=
[L(x) + ℓ∗F δ(x) + δ†(x) ℓG δ†(x)wG
w∗F δ(x) 0
]+
[ℓ∗F
w∗F
](π−ι)(x)
[ℓG wG
].
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 65 / 67
Applying Theorem 28
Conclusion
If, for all z , w ∈ k, there exist a subspace Az ,w ⊆ A0 such that
Az ,w → A; x 7→ L(x) + ℓ∗F δ(x) + δ†(x) ℓG +(w∗F δ(x)
)z+
(δ†(x)wG )w
is closable with closure which generates a C0 semigroup then k is a Fellercocycle and EΩ k is a semigroup on A with generator which extends
x 7→ L(x) + ℓ∗F δ(x) + ℓ∗F π(x) ℓG + δ†(x) ℓG +m∗F x + x mG .
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 66 / 67
References
Construction
A. C. R. Belton & S. J. Wills, An algebraic construction of quantum flows
with unbounded generators, Annales de l’Institut Henri Poincare (B)Probabilites et Statistiques, to appear. arXiv:math/1209.3639
Perturbation
A. C. R. Belton, A. G. Skalski & J. M. Lindsay, Quantum Feynman–Kac
perturbations, Journal of the London Mathematical Society, to appear.arXiv:math/1202.6489
A. C. R. Belton & S. J. Wills, Feynman–Kac perturbation of C ∗ quantum
flows, in preparation.
Alexander Belton (Lancaster University) Quantum semigroups and stochastic flows IMPAN, 25xi13 67 / 67
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