Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations
We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on 𝓑(𝐺) (𝐺 a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is,...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2022 |
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Інститут математики НАН України
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| Цитувати: | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations. Michael Skeide. SIGMA 18 (2022), 071, 8 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859466281865969664 |
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| author | Skeide, Michael |
| author_facet | Skeide, Michael |
| citation_txt | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations. Michael Skeide. SIGMA 18 (2022), 071, 8 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on 𝓑(𝐺) (𝐺 a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of 𝐸₀-semigroups on 𝓑(𝐻) by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy. All other results that make similar statements (especially, [Mem. Amer. Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798]) for more general 𝐶*-algebras have been proved later by suitable adaptations of the methods exposed here. (They use Hilbert module techniques, which we carefully avoid here to make the result available without any appeal to Hilbert modules.)
|
| first_indexed | 2026-03-12T14:20:35Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 071, 8 pages
Spatial Markov Semigroups Admit
Hudson–Parthasarathy Dilations
Michael SKEIDE
Università degli Studi del Molise, Dipartimento di Economia,
Via de Sanctis, 86100 Campobasso, Italy
E-mail: skeide@unimol.it
URL: http://web.unimol.it/skeide/
Received February 23, 2022, in final form September 23, 2022; Published online October 03, 2022
https://doi.org/10.3842/SIGMA.2022.071
Abstract. We present, for the first time, the result (from 2008) that (normal, strongly
continuous) Markov semigroups on B(G) (G a separable Hilbert space) admit a Hudson–
Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if
the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The
proof is by general abstract nonsense (taken from Arveson’s classification of E0-semigroups
on B(H) by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing
the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson
and Parthasarathy. All other results that make similar statements (especially, [Mem. Amer.
Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798]) for more general C∗-algebras) have
been proved later by suitable adaptations of the methods exposed here. (They use Hilbert
module techniques, which we carefully avoid here in order to make the result available
without any appeal to Hilbert modules.)
Key words: quantum dynamics; quantum probability; quantum Markov semigroups; dila-
tions; product systems
2020 Mathematics Subject Classification: 46L55; 46L53; 81S22; 60J25
1 Introduction
(Quantum) Markov semigroups are models for irreversible evolutions of (quantum) physical
systems. Dilating a Markov semigroup, means embedding the irreversible system into a reversible
one in such a way that the original irreversible evolution can be recovered by projecting down
(via a conditional expectation) the reversible evolution to the subsystem.
Noises are models for reversible systems containing a subsystem. A noise is actually a re-
versible evolution on a “big” system with a conditional expectation onto a “small” subsystem
that leaves the small system invariant. One may think of a simultaneous description of a re-
versible system and the small system, but with the interaction switched off. When the interaction
is switched on, the dynamics of the compound system changes and leaves the small system no
longer invariant. The projection back to the small subsystem produces irreversible behavior.
Often, one tries to model the transition from the free dynamics (the noise) to the real dy-
namics by perturbation of the noise with a unitary cocycle. This is what we mean by a Hudson–
Parthasarathy dilation. In practically all known examples, such cocycles have been obtained by
means of a quantum stochastic calculus. (See Remark 2.7.) The stochastic generator of the co-
cycle, is composed from the generator of the Markov semigroup. Often, it may be interpreted in
terms of an interaction Hamiltonian. In these notes, we show in the case of B(G) (the algebra of
This paper is a contribution to the Special Issue on Non-Commutative Algebra, Probability and Analysis in Ac-
tion. The full collection is available at https://www.emis.de/journals/SIGMA/non-commutative-probability.html
mailto:skeide@unimol.it
http://web.unimol.it/skeide/
https://doi.org/10.3842/SIGMA.2022.071
https://www.emis.de/journals/SIGMA/non-commutative-probability.html
2 M. Skeide
all bounded operators on a Hilbert spaceG) without using any calculus, that a Markov semigroup
admits a Hudson–Parthasarathy dilation if and only if the Markov semigroup is spatial. The case
of a Markov semigroup on a general von Neumann algebra (in particular, also on a commutative
one, which corresponds to a classical dynamical system) is discussed, later, in Skeide [29].
The necessary notions, spatial Markov semigroup, noise, Hudson–Parthasarathy dilation, and
so forth, are explained in Section 2 and then used to formulate the result. In Section 3 we review
the basic results about spatial E0-semigroups and spatial product systems needed in the proof
of the result, and we prove the result. In Section 4, finally, we list several natural questions.
Our proof makes heavy use of Arveson’s results [2] on the classification of E0-semigroups (in
particular, spatial ones) by tensor product systems of Hilbert spaces (Arveson systems). (We
should emphasize that the main scope of [29] is not to generalize the present result to general
von Neumann algebras – which is easy –, but to fill a long standing gap, namely, to answer
the question how Arveson’s classification of E0-semigroups by product systems generalizes to
Hilbert modules.) We shall assume that the reader is familiar with Arveson’s results at least
in the spatial case, and we shall also assume that the reader knows the works by Bhat [6] and
Arveson [3] on the relation with Markov semigroups via the so-called minimal weak dilation
(once more, in particular in the spatial case).
2 Notations and statement of the result
2.1. E0-semigroups, in these notes,1 are strongly continuous one-parameter semigroups of nor-
mal unital endomorphisms of B(H) where H is some separable infinite-dimensional Hilbert
space.2 Arveson [2] associated with every E0-semigroup a tensor product system of Hilbert
spaces (or, for short, an Arveson system). He showed that two E0-semigroups, ϑ1 on B
(
H1
)
and ϑ2 on B
(
H2
)
say, have isomorphic Arveson systems if and only if they are cocycle conjugate.
That is, there exist a unitary u : H1 → H2 and a strongly (and, therefore, ∗-strongly) continuous
family of unitaries ut ∈ B
(
H2
)
fulfilling:
1. The ut form a left cocycle with respect to ϑ2, that is, us+t = usϑ
2
s(ut) for all s, t ∈ R+.
2. uϑ1
t (u
∗au)u∗ = utϑ
2
t (a)u
∗
t for all t ∈ R+, a ∈ B
(
H2
)
.
2.2. Markov semigroups, in these notes, are strongly continuous one-parameter semigroups of
normal unital completely positive maps (CP-maps) on B(G) where G is some separable Hilbert
space. Bhat [6, Theorem 4.7] states that every Markov semigroup T on B(G) admits a weak
dilation to an E0-semigroup ϑ on B(H) in the following sense: There is an isometry ξ : G → H
such that
Tt(b) = ξ∗ϑt(ξbξ
∗)ξ
for all t ∈ R+, b ∈ B(G).3 This weak dilation can be chosen minimal in the sense that H has no
proper subspace containing ξG and invariant under ϑR+(ξB(G)ξ∗). The minimal weak dilation
1By “in these notes” we refer to the fact that in most other of our papers we do not require any continuity
of our semigroups. (Almost all constructions preserve strong continuity, in the sense that if we input a strongy
continuous Markov (or CP) semigroup then the dilations we construct turn out to be strongly continuous, too.
But continuity is not a necessary prerequisite for the constructions to work. And here, we require our Hilbert
spaces separable, only because we wish to quote theorems from Arveson’s theory.
2Once for all other cases, strongly continuous for a von Neumann algebra B ⊂ B(H) refers to the strong
operator topology (or the point strong topology). That is, a strongly continuous family Tt of operators on B
satisfies that t 7→ Tt(b)h is norm continuous for all b ∈ B, h ∈ H.
3Frequently, G is identified with the subspace ξG of H, so that B(G) would be identified with the corner
ξB(G)ξ∗ = pB(H)p of B(H), where p denotes the projection in B(H) onto ξG. That would lead to slightly more
readable formulae. But, in these notes, we will, have – by definition (see Section 2.3) – sitting B(G) as a unital
subalgebra of B(H) so that idH ∈ B(G) ⊂ B(H) needs to be carefully distinguished from the corner pB(H)p (of
course, isomorphic to B(G)).
Spatial Markov Semigroups Admit Hudson–Parthasarathy Dilations 3
is unique up to suitable unitary equivalence, namely, by a unique unitary that sends one ξ to
the other. Bhat [6] defines the Arveson system associated with the Markov semigroup to be
the Arveson system of its minimal weak dilation ϑ.4 Here we need only know that the Arveson
system of any weak dilation of T contains the Arveson system of T .
2.3. By a noise over B(G) we understand an E0-semigroup S on some B(H) that contains B(G)
as a unital W ∗-subalgebra5 and an isometry ω : G → H, fulfilling the following:
1. S leaves B(G) invariant, that is, St(b) = b for all t ∈ R+, b ∈ B(G) ⊂ B(H).
2. ω is B(G)-C-bilinear, that is, bω = ωb for all b ∈ B(G).
3. S leaves E : a 7→ ω∗aω invariant, that is, E ◦ St = E for all t ∈ R+.
By (2), E is a conditional expectation (onto B(G) sitting as unital subalgebra of B(H)). By (3)
(applied to a = ωbω∗), S with the isometry ω is a weak dilation of the identity Markov semigroup
on B(G). Therefore, the projection p := ωω∗ ∈ B(H) is increasing, that is, St(p) ≥ p for all
t ∈ R+. A noise is reversible, if all St are automorphisms. In this case, St(p) = p for all t ∈ R+.
6
Remark 2.4. This definition of noise is from Skeide [25]. In the scalar case (that is, G = C) it
corresponds to noises in the sense of Tsirelson [31, 32]. A reversible noise is close to a Bernoulli
shift in the sense of Hellmich, Köstler and Kümmerer [13].
Definition 2.5. A Hudson–Parthasarathy dilation (HP-dilation, for short) of a Markov semi-
group T on B(G) is a noise (S, ω) and a unitary left cocycle ut with respect to S, such that the
cocycle conjugate E0-semigroup ϑ defined by setting ϑt(a) = utSt(a)u
∗
t , fulfills
E ◦ ϑt ↾ B(G) = Tt
for all t ∈ R+.
The HP-dilation is reversible, if the underlying noise is reversible. In this case, also ϑ is an
automorphism semigroup.
The HP-dilation is weak if ϑ with the isometry ω is also a weak dilation of T .
Observation 2.6. Recall that being a weak HP-dilation is stronger a condition than being
an HP-dilation. Since reversible noises have the trivial (that is, the one-dimensional) Arveson
system, not much remains for what a Markov semigroup T can be if it admits a weak and
reversible HP-dilation. (One can show, that a Markov semigroup with trivial Arveson system has
the form Tt = w∗
t •vt for a semigroup vt of isometries in B(G).7 Taking any unitary dilation ut ∈
B(H) and the trivial (= constant) noise on B(H) (so that the unitary semigroup is also a cocycle)
we get an HP-dilation.) However, since any E0-semigroup on B(H) extends to an automorphism
4Strictly speaking, for an E0-semigroup as defined in the first paragraph, H should be infinite-dimensional.
However, the missing case, where the H of the minimal weak dilation is finite-dimensional, happens if and only
if G is finite-dimensional and T consists of automorphisms. In this case, for our theorem below there is nothing
to prove, and we will tacitly exclude it from the subsequent discussion.
5That is, we identify elements of B(G) with their image under a(n unnamed) normal unital representation
B(G) → B(H).
6In fact, suppose S is implemented as St = vt • v∗t by a unitary semigroup vt. One easily checks that p is
increasing if and only if ω∗v∗t ω is an isometry. Since St(b) = b we have bvt = vtb. Since also bω = ωb, it follows
that ω∗v∗t ω is in the center of B(G) and, therefore, a unitary. So, p is also decreasing, thus, constant.
7This is a statement that follows most easily using Hilbert (or, better, von Neumann) modules. (The GNS-
product system of von Neumann B(G)-correspondences of a normal CP-semigroup on B(G) is B(G)⊗H⊗ where H⊗
is the Arveson system of that CP-semigroup; this first in Bhat and Skeide [9, Section 13] though there not very
explicit. If H⊗ is the trivial Arveson system, then the units of the GNS-product system are just the semigroups
in B(G). Conversely, a CP-semigroup of the form c∗t • ct has the trivial product system of B(G)-correspondences
as GNS-product system (see Shalit and Skeide [22, Observation 7.2]) and, therefore, the trivial Arveson system.)
The argument does not fit into this note that avoids module techniques, so we omit a formal proof.
4 M. Skeide
semigroup on some B
(
Ĥ
)
⊃ B(H) ∋ id
Ĥ
(Arveson and Kishimoto [5] or Skeide [27]), existence of
a weak HP-dilation grants existence of a reversible HP-dilation – a reversible HP-dilation having
sitting inside a weak HP-dilation in a specific way. In general, it is still unknown whether
existence of a reversible or general HP-dilation implies existence of a weak HP-dilation. In
particular, since by our main result, Theorem 2.9 below, spatial Markov semigroups admit weak
HP-dilations, it is unknown whether there are non-spatial Markov semigroups that admit other
types of HP-dilations. So, the only-if part of the theorem might fail, if we drop weak from the
hypotheses on the HP-dilation.
Remark 2.7. Since the seminal work of Hudson and Parthasarathy [14, 15], the cocycles of
Hudson–Parthasarathy dilations have been obtained with the help of quantum stochastic calculi
as solutions of quantum stochastic differential equations. [14] dealt with a Lindblad generator
with finite degree of freedom, while [15] considers a general (bounded) Lindblad generator. Cheb-
otarev and Fagnola [10] deal with a large class of unbounded generators. Versions for general
von Neumann algebras (Goswami and Sinha [12], Köstler [16]) or C∗-algebras (Skeide [23]) re-
quire Hilbert modules. (Apart from the fundamental monograph [19] by Parthasarathy, a still
up-to-date reference for everything that has to do with calculus based on Boson Fock spaces or
modules are Lindsay’s lecture notes [18]. Results that use other types of Fock constructions or
abstract representation spaces are scattered over the literature.)
We get our cocycle in an entirely different way, appealing to granted cocycle conjugacy of
E0-semigroups having the same Arveson system as discussed in Section 2.1.
To, finally, formulate our main result, we need a last notion.
Definition 2.8 (Arveson [3, Definition 2.1]). A unit for a Markov semigroup T on B(G) is
a strongly continuous semigroup c in B(G) such that T dominates the elementary CP-semigroup
St(b) := c∗t bct, that is, the difference Tt − St is completely positive for all t ∈ R+. A Markov
semigroup on B(G) is spatial, if it admits units.
And, now, here is our main result:
Theorem 2.9. A Markov semigroup on B(G) is spatial if and only if it admits a weak Hudson–
Parthasarathy dilation. Such a Hudson–Parthasarathy dilation may be extended to a reversible
Hudson–Parthasarathy dilation.
3 Proof
We said, a Markov semigroup is spatial if it admits a unit. Of course, also an E0-semigroup is
a Markov semigroup. For E0-semigroups there is Powers’ definition [20] of spatiality in terms
of intertwining semigroups of isometries, also referred to as (isometric) units. Also for Arveson
systems there is the concept of units and an Arveson system is spatial, if it admits a unit. We
do not repeat here the definition of Arveson system nor that of a unit for an Arveson system,
but simply will put together known statements about them. (The discussion for general von
Neumann algebras in Skeide [29] is much more self-contained, and many of the statements
for B(G), we simply quote here, will drop out there very naturally without any effort.) Bhat [7,
Section 6] compared several notions of units. What is important to us is that all concepts of
spatiality in the sense of existence of units coincide: A semigroup, Markov or E0, is spatial in
whatsoever sense if and only if its associated Arveson system is spatial; moreover, whether a
weak dilation is spatial or not, does not depend on whether the dilation is minimal, but only on
whether the dilated Markov semigroup is spatial or not; see Bhat [7, Section 6] or Arveson [4,
Sections 8.9 and 8.10].
Spatial Markov Semigroups Admit Hudson–Parthasarathy Dilations 5
Let us start with the more obvious parts of the theorem. It is the word weak that guarantees
that the Markov semigroup that admits a weak HP-dilation has to be spatial. (A noise has
a spatial Arveson system, so that also the Arveson system of the cocycle conjugate HP-dilation
ϑ is spatial. Since ϑ is also a weak dilation, by the discussion in Section 2.2, its Arveson system
contains the Arveson system of T . This is enough to see that T is spatial; see Bhat [7, Section 6]
or, more recently, Bhat, Liebscher, and Skeide [8], but see also the discussion below of the
relation between noises over B(G) and noises over C.) We mentioned already in Observation 2.6
that a weak HP-dilation embeds into a reversible one.
So, what remains to be shown is the construction of a weak HP-dilation for any spatial
Markov semigroup. Therefore, for the balance, let us fix a spatial Markov semigroup T .
Every weak dilation of T is spatial. This has important consequences: For a spatial Arveson
system it is easy to construct an E0-semigroup S on some B(H) that has as associated Arveson
system the one we started with; see [2, Appendix]. Moreover, it is easy to see that for this E0-
semigroup there exists a unit vector Ω ∈ H such that S leaves the state φ := ⟨Ω, •Ω⟩ invariant
(that is, φ◦St = φ for all t ∈ R+); in other words, S is a noise over C.8 Two E0-semigroups are
cocycle conjugate if and only they have isomorphic Arveson systems; see the corollary of Arveson
[2, Definition 3.20]. So, a Markov semigroup is spatial if and only if one (and, therefore, all)
weak dilation(s) is (are) cocycle conjugate to an E0-semigroup with an invariant vector state.
From E0-semigroups with invariant vector states to noises and back, there is only a small
step. Suppose we have a noise (S, ω) over B(G) on B(H). Clearly, a unital W ∗-subalgebra
B(G) of B(H) decomposes H into H“=”G ⊗ H for some multiplicity space H, and S leaves
B(G)“=”B(G) ⊗ idH invariant if and only if St = idB(G)⊗St for a unique E0-semigroup S
on B(H).9 For that the isometry ω intertwines the actions of B(G), it necessarily has the
form ω = idG⊗Ω: g 7→ g ⊗ Ω for a unique unit vector Ω ∈ H. Clearly, S leaves the conditional
expectation E invariant if and only ifS leaves the vector state φ := ⟨Ω, •Ω⟩ invariant. Conversely,
if S is an E0-semigroup with an invariant vector state φ induced by a unit vector Ω ∈ H, then
the E0-semigroup S = idB(G)⊗S on B(G ⊗ H) with the isometry ω := idG⊗Ω is a noise
over B(G). On the other hand, S is just a multiple of S, and multiplicity does not change the
Arveson system; see Arveson [2, Poposition 3.15]. Therefore, the Arveson system of a noise S
is spatial. Consequently, a Markov semigroup is spatial if and only if one (and, therefore, all)
weak dilation(s) is (are) cocycle conjugate to a noise.
To construct a weak Hudson–Parthasarathy dilation for T , we need to choose the noise (to
which a weak dilation is cocycle conjugate) in such a way that it admits a Hudson–Parthasarathy
cocycle (which turns the noise by cocycle perturbation into a dilation). The Arveson system
of the weak dilation ϑ on B(H) of T with the isometry ξ is spatial. To that Arveson system
construct an E0-semigroup S on a Hilbert space H with an invariant vector state φ = ⟨Ω, •Ω⟩.
We tensor it with the identity on B(G) as described before to obtain a noise (S, ω) with the
same Arveson system as ϑ. We wish to identify the two Hilbert spaces (infinite-dimensional
and separable, unless T is an automorphism semigroup on Mn) by a unitary u : H → G⊗ H in
such a way that uξ = ω. But this is easy. If T is an E0-semigroup, then, since T is its own
weak dilation, there is nothing to show. If T is not an E0-semigroup, then both E0-semigroups,
ϑ and S, are proper. We simply fix a unitary u : H → G⊗ H that takes ξg to g ⊗ Ω = ωg and
is arbitrary on the (infinite-dimensional!10) complements of ξG and G⊗ Ω. There exists, then,
8Arveson’s construction is the core of any forthcoming construction of an E0-semigroup from a unital (and
central for modules) unit in a product system, by an inductive limit. It is easy to see that the elements of the
unit are, under the inductive limit, sent to a fixed unit vector Ω and that this Ω does the job. Moreover, vector
states are pure, so the S is an E0-semigroup in standard form; see Powers [21] and Alevras [1], but also Bhat and
Skeide [9, 24, 25] for the module case.
9St(idG ⊗a) is in the commutant of St(B(G)) = B(G). So St leaves idG ⊗B(H) invariant.
10Clear is that the scalar noise S lives on an infinite-dimensional H so that ∞ = dimΩ⊥ ≤ dim(G⊗Ω)⊥. It is
a bit of folklore that also (ξG)⊥ in a weak dilation of a non-E0-Markov semigroup has to be infinite-dimensional,
6 M. Skeide
a left cocycle ut with respect to S that fulfills
uϑt(u
∗au)u∗ = utSt(a)u
∗
t .
We find
Tt(b) = ξ∗ϑt(ξbξ
∗)ξ = ξ∗u∗uϑt(u
∗uξbξ∗u∗u)u∗uξ
= ω∗uϑt(u
∗ωbω∗u)u∗ω = ω∗utSt(ωbω
∗)u∗tω,
so that utSt(•)u∗t with the isometry ω is a weak dilation of T . In particular, the projection ωω∗
must be increasing, that is, utSt(ωω
∗)u∗tωω
∗ = ωω∗ or utSt(ωω
∗)u∗tω = ω. Now, by the special
property of ω, we have ωbω∗ = (ωω∗)b(ωω∗). It follows
Tt(b) = ω∗utSt(ωω
∗)u∗tutSt(b)u
∗
tutSt(ωω
∗)u∗tω = ω∗utSt(b)u
∗
tω,
that is, the cocycle perturbation of the noise (S, ω) by the cocycle ut is a Hudson–Parthasarathy
dilation of T .
4 Remarks and outlook
Our construction of a Hudson–Parthasarathy dilation is by completely abstract means. This
leaves us with a bunch of natural questions.
Is our cocycle in any way adapted? Hudson–Parthasarathy cocycles obtained with quantum
stochastic calculus on the Boson Fock space are adapted in the sense that ut is in the commutant
of St(1⊗B(H)) for each t ∈ R+. Is our cocycle possibly adapted in the sense of [9, Definition 7.4]?
(Today, we would prefer to say weakly adapted. Roughly, this means St(ωω
∗)u∗tωω
∗ = u∗tωω
∗,
so that the u∗tωω
∗ form a partially isometric cocycle.) Instead of the minimal Arveson system
of the minimal weak dilation, we could have started the constructive part with the Arveson
system associated with the free flow generated by the spatial minimal Arveson system in a sense
to be worked out in Skeide [30]. (This has been outlined in Skeide [25].) These free flows come
along with their own notion of adaptedness (see Kümmerer and Speicher [17], Fowler [11], and
Skeide [23]), and we may ask whether the cocycle is adapted in this sense.
In any case, quantum stochastic calculi, also the abstract one in Köstler [16], provide a relation
between additive cocycles and multiplicative (unitary) cocycles. Differentials of additive cocycles
are, roughly, the differentials of the quantum stochastic differential equation to be resolved. We
may ask, whether this relation holds for all spatial Markov semigroups, also if they are not
realized on the Fock spaces, that is, on type I or completely spatial noises. The additive cocycles,
usually, take their ingredients from the generator of the Markov semigroup. If that generator
is bounded, then one may recognize the constituents of the Christensen–Evans generator, or,
in the B(G)-case, of the Lindblad generator. This raises the problem to characterize spatial
Markov semigroups in terms of their generators. Do they have generators that resemble in
some sense the Lindblad form? Apparently the most general form of unbounded Lindblad type
but it is not so easy to find a reference for that. It is a bit inconvenient that for seeing that, we would have to
enter the construction of the minimal dilation (contained in any other weak dilation). It is enough to mention that
the construction of the minimal dilation as in Bhat and Skeide [9, Section 5] (for modules but only over B(G),
for which the “translation” in [9, Section 13] can be applied) is also (unlike Bhat’s in [6]!) by an inductive limit,
and that it is easy to see that in each step of the inductive limit “new space” is added, if the Markov semigroup
is nonendomorphic. (By ‘step’, we mean that we restrict the parameter set R+ to the discrete case N0 ⊂ R+
and the isometry embedding the space at time n into that at time n + 1 is always proper. (The latter part is
a retraction of the fact that the Stinespgring isometry of the Stingespring construction of a proper Markov map is
always a proper isometry. This implies the former part, because the isometry going from n to n+ 1 is (for B(G),
at least!) just an amplification of the Stinestpring isometry.)
Spatial Markov Semigroups Admit Hudson–Parthasarathy Dilations 7
generators of Markov semigroups on B(G) has been discussed in Chebotarev and Fagnola [10].
But only a subclass of these generators could be dilated by using Hudson–Parthasarathy calculus
on the Boson Fock space. It seems natural to expect that Markov semigroups having this sort of
generators are all spatial. Is it possibly that they could not be dilated on the Fock space because
their product systems are type II (non-Fock) and not completely spatial (Fock)? In this case, can
the solution be obtained with a calculus on noises emerging from type II product systems that
are no longer CCR-flows (Fock noises)? In any case, whenever for an example a solution of the
problem has been obtained with calculus, then we may ask, whether our abstract cocycle (which,
of course, can be written down explicitly; see Skeide [29]) can be related to the concrete cocycle
emerging from calculus. Generally, we may ask, how two possible cocycles with respect to the
same noise (adapted in some sense or not) are related. (This question connects to properties of
the automorphism group of spatial Arveson systems such as discussed in Tsirelson [33].)
Last but not least, we ask, if there exist nontrivial examples of nonspatial Markov semigroups.
By this we mean Markov semigroups that are not type III E0-semigroups, or tensor products
of such with a spatial Markov semigroup. A big step towards answering this question has been
taken in Skeide [28], where (starting from our construction Skeide [26] of an E0-semigroup for
every Arveson system) we constructed for each Arveson system a non-endomorphic Markov
semigroup. If we find a type III Arveson system that does not factor into a tensor product with
a spatial product system, then we would find a properly non-spatial Markov semigroup.
Acknowledgements
I would like to thank Rajarama Bhat, Malte Gerhold, Martin Lindsay, and Orr Shalit for several
valuable comments and bibliographical hints, which improved the final version in several ways.
In particular, I wish to express my gratitude to the referees. One of them spotted a seriously
unclear point in a definition and a small gap in an argument. This work was supported by
research funds of University of Molise and Italian MIUR under PRIN 2007.
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1 Introduction
2 Notations and statement of the result
3 Proof
4 Remarks and outlook
References
|
| id | nasplib_isofts_kiev_ua-123456789-211717 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T14:20:35Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Skeide, Michael 2026-01-09T12:43:12Z 2022 Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations. Michael Skeide. SIGMA 18 (2022), 071, 8 pages 1815-0659 2020 Mathematics Subject Classification: 46L55; 46L53; 81S22; 60J25 arXiv:0809.3538 https://nasplib.isofts.kiev.ua/handle/123456789/211717 https://doi.org/10.3842/SIGMA.2022.071 We present, for the first time, the result (from 2008) that (normal, strongly continuous) Markov semigroups on 𝓑(𝐺) (𝐺 a separable Hilbert space) admit a Hudson-Parthasarathy dilation (that is, a dilation to a cocycle perturbation of a noise) if and only if the Markov semigroup is spatial (that is, if it dominates an elementary CP-semigroup). The proof is by general abstract nonsense (taken from Arveson's classification of 𝐸₀-semigroups on 𝓑(𝐻) by Arveson systems up to cocycle conjugacy) and not, as usual, by constructing the cocycle as a solution of a quantum stochastic differential equation in the sense of Hudson and Parthasarathy. All other results that make similar statements (especially, [Mem. Amer. Math. Soc. 240 (2016), vi+126 pages, arXiv:0901.1798]) for more general 𝐶*-algebras have been proved later by suitable adaptations of the methods exposed here. (They use Hilbert module techniques, which we carefully avoid here to make the result available without any appeal to Hilbert modules.) I would like to thank Rajarama Bhat, Malte Gerhold, Martin Lindsay, and Orr Shalit for several valuable comments and bibliographical hints, which improved the final version in several ways. In particular, I wish to express my gratitude to the referees. One of them spotted a seriously unclear point in a definition and a small gap in an argument. This work was supported by research funds of the University of Molise and the Italian MIUR under PRIN 2007. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations Article published earlier |
| spellingShingle | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations Skeide, Michael |
| title | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_full | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_fullStr | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_full_unstemmed | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_short | Spatial Markov Semigroups Admit Hudson-Parthasarathy Dilations |
| title_sort | spatial markov semigroups admit hudson-parthasarathy dilations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211717 |
| work_keys_str_mv | AT skeidemichael spatialmarkovsemigroupsadmithudsonparthasarathydilations |