Post on 11-Jul-2020
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
a.belton@lancaster.ac.uk
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