Pure inductive limit state and Kolmogorov's property
arXiv:0704.1987v1  [math.OA]  16 Apr 2007
Pure inductive limit state and Kolmogorov’s property
Anilesh Mohari
S.N.Bose Center for Basic Sciences,
JD Block, Sector-3, Calcutta-98
E-mail:anilesh@boson.bose.res.in
Abstract
Let (B, λt, ψ) be a C
∗-dynamical system where (λt : t ∈ IT+) be a semi-
group of injective endomorphism and ψ be an (λt) invariant state on the C
subalgebra B and IT+ is either non-negative integers or real numbers. The
central aim of this exposition is to find a useful criteria for the inductive limit
state B →λt B canonically associated with ψ to be pure. We achieve this by
exploring the minimal weak forward and backward Markov processes associ-
ated with the Markov semigroup on the corner von-Neumann algebra of the
support projection of the state ψ to prove that Kolmogorov’s property [Mo2]
of the Markov semigroup is a sufficient condition for the inductive state to
be pure. As an application of this criteria we find a sufficient condition for a
translation invariant factor state on a one dimensional quantum spin chain to
be pure. This criteria in a sense complements criteria obtained in [BJKW,Mo2]
as we could go beyond lattice symmetric states.
http://arxiv.org/abs/0704.1987v1
1 Introduction:
Let τ = (τt, t ≥ 0) be a semigroup of identity preserving completely positive
maps [Da1,Da2,BR] on a von-Neumann algebra A0 acting on a Hilbert space
H0, where either the parameter t ∈ R+, the set of positive real numbers or
t ∈ Z+, the set of positive integers. We assume further that the map τt is
normal for each t ≥ 0 and the map t → τt(x) is weak
∗ continuous for each
x ∈ A0.
We say a projection p ∈ A0 is sub-harmonic and harmonic if τt(p) ≥ p
and τt(p) = p for all t ≥ 0 respectively. For a sub-harmonic projection p, we
define the reduced quantum dynamical semigroup (τ
t ) on the von-Neumann
algebra pA0p by τ
t (x) = pτt(x)p where t ≥ 0 and x ∈ A
0. 1 is an upper
bound for the increasing positive operators τt(p), t ≥ 0. Thus there exists an
operator 0 ≤ y ≤ 1 so that y = s.limt→∞τt(p). A normal state φ0 is called
invariant for (τt) if φ0τt(x) = φ0(x) for all x ∈ A0 and t ≥ 0. The support p of
a normal invariant state is a sub-harmonic projection and φ
0, the restriction
of φ0 to A
0 is a faithful normal invariant state for (τ
t ). Thus asymptotic
properties ( ergodic, mixing ) of the dynamics (A0, τt, φ0) is well determined
by the asymptotic properties (ergodic, mixing respectively ) of the reduced
dynamics (A
t , φ
0) provided y = 1. For more details we refer to [Mo1].
In case φ0 is faithful, normal and invariant for (τt), we recall [Mo1] that G =
{x ∈ A0 : τ̃tτt(x) = x, t ≥ 0} is von-Neumann sub-algebra of F = {x ∈ A0 :
∗)τt(x) = τt(x
∗x), τt(x)τt(x
∗) = τt(xx
∗) ∀t ≥ 0} and the equality G = IC is
a sufficient condition for φ0 to be strong mixing for (τt). Since the backward
process [AM] is related with the forward process via an anti-unitary operator
we note that φ0 is strongly mixing for (τt) if and only if same hold for (τ̃t). We
can also check this fact by exploring faithfulness of φ0 and the adjoint relation
[OP]. Thus IC ⊆ G̃ ⊆ F̃ and equality IC = G̃ is also a sufficient condition for
strong mixing where F̃ and G̃ are von-Neumann algebras associated with (τ̃t).
Thus we find two competing criteria for strong mixing. However it is straight
forward whether F = F̃ or G = G̃. Since given a dynamics it is difficult to
describe (τ̃t) explicitly and thus this criterion G = IC is rather non-transparent.
We prove in section 2 that G = {x ∈ F : τtσs(x) = σsτt(x), ∀t ≥ 0. s ∈R}
where σ = (σs : s ∈R) is the Tomita’s modular auto-morphism group [BR,OP]
associated with φ0. So G is the maximal von-Neumann sub-algebra ofA0, where
(τt) is an ∗-endomorphism [Ar], invariant by the modular auto-morphism group
(σs). Moreover σs(G) = G for all s ∈R and τ̃t(G) = G for all t ≥ 0. Thus by a
theorem of Takesaki [OP], there exists a norm one projection IEG from A0 onto
G which preserves φ0 i.e. φ0IE = φ0. Exploring the fact that τ̃t(G) = G, we
also conclude that the conditional expectation IEG commutes with (τt). This
enables us to prove that (A0, τt, φ0) is ergodic (strongly mixing) if and only if
(G, τt, φ0) is ergodic (strongly mixing). Though τt(G) ⊆ G for all t ≥ 0, equality
may not hold in general. However we have
τt(G) =
τ̃t(G̃)
where G̃ = {x ∈ A0 : τt(τ̃t(x)) = x, t ≥ 0}. G = G̃ holds if and only if
τt(G) = G, τ̃t(G̃) = G̃ for all t ≥ 0. Thus G0 =
t≥0 τt(G) is the maximal
von-Neumann sub-algebra invariant by the modular automorphism so that
(G0, τt, φ0) is an ∗−automorphisms with (G0, τ̃t, φ0) as it’s inverse dynamics.
Once more there exists a conditional expectation IEG0 : A0 → A0 onto G0
commuting with (τt). This ensures that (A0, τt, φ0) is ergodic (strongly mixing)
if and only if (G0, τt, φ0) is ergodic (strongly mixing). It is clear now that
G0 = G̃0, thus G0 = IC, a criterion for strong mixing, is symmetric or time-
reversible. As an application in classical probability we can find an easy criteria
for a stochastically complete Brownian flows [Mo5] on a Riemannian manifold
driven by a family of complete vector fields to be strong mixing.
Exploring the criterion G0 = IC we also prove that for a type-I factor A0
with center completely atomic, strong mixing is equivalent to ergodicity when
the time variable is continuous i.e. R+ (Theorem 3.4). This result in particular
extends a result proved by Arveson [Ar] for type-I finite factor. In general,
for discreet time dynamics (A0, τ, φ0), ergodicity does not imply strong mixing
property (not a surprise fact since we have many classical cases). We also prove
that τ on a type-I von-Neumann algebra A0 with completely atomic center is
strong mixing if and only if it is ergodic and the point spectrum of τ in the
unit circle i.e. {w ∈ S1 : τ(x) = wx for some non zero x ∈ A0} is trivial.
The last result in a sense gives a direct proof of a result obtained in section 7
of [BJKW] without being involved with Popescu dilation.
In section 3 we consider the unique up to isomorphism minimal forward
weak Markov [AM,Mo1,Mo4] stationary process {jt(x), t ∈ IT, x ∈ A0} asso-
ciated with (A0, τt, φ0). We set a family of isomorphic von-Neumann algebras
{A[t : t ∈ IT} generated by the forward process so that A[t ⊆ A[s whenever
s ≤ t. In this framework we construct a unique modulo unitary equivalence
minimal dilation (A[0, αt, t ≥ 0, φ), where α = (αt : t ≥ 0) is a semigroup
of ∗−endomorphism on a von-Neumann algebra A[0 acting on a Hilbert space
H[0 with a normal invariant state φ and a projection P in A[0 so that
(a) PA[0P = π(A0)
(b) Ω ∈ H[0 is a unit vector so that φ(X) =< Ω, XΩ >;
(b) Pαt(X)P = π(τt(PXP )) for t ≥ 0, X ∈ A[0;
(c) {αtn(PXnP ).....αt3(PX3P )αt2(PX2P )αt1(PX1P )Ω : 0 ≤ t1 ≤ t2.. ≤
tn, n ≥ 1}, Xi ∈ A[0} is total in H[0,
where π is the GNS representation of A0 associated with the state φ0. In
case φ0 is also faithful, we consider the backward process (j
t ) defined in [AM]
associate with the KMS adjoint Markov semigroup and prove that commutant
of A[t is equal to A
t] = {j
s(x) : x ∈ A0, s ≤ t}
′′ for any fix t ∈ IT .
As an application of our result on asymptotic behavior of a Markov semi-
group, we also study a family of endomorphism (B, λt) on a von-Neumann
algebra. Following Powers [Po2] an endomorphism αt : B0 → B0 is called shift
t≥0 αt(B) is trivial. In general such a shift may not admit an invariant state
[BJP]. Here we assume that λt admits an invariant state ψ and address how the
shift property is related with Kolmogorov’s property of the canonical Markov
semigroup (A0, τt, ψ) on the support projection on the von-Neumann algebra
πψ(B)
′′ of the state vector state in the GNS space (Hπ, π,Ω) associated with
(B, ψ). As a first step here we prove that Powers’s shift property is equivalent
to Kolmogorov’s property of the adjoint Markov semigroup (τ̃t). However in
the last section we show that Kolmogorov’s property of a Markov semigroup
need not be equivalent to Kolmogorov’s property of the KMS adjoint Markov
semigroup. Thus Powers’s shift property in general is not equivalent to Kol-
mogorov’s property of the associated Markov semigroup.
Section 4 includes the main mathematical result by proving a criteria for
the inductive limit state, associated with an invariant state of an injective
endomorphism on a C∗ algebra, to be pure. To that end we explore the minimal
weak Markov process associated with the reduced Markov semigroup on the
corner algebra of the support projection and prove that the inductive limit
state is pure if the Markov semigroup satisfies Kolmogorov’s property. Further
for a lattice symmetric factor state, Kolmogorov’s property is also necessary
for purity of the inductive limit state.
The last section deals with an application of our main results on translation
invariant state on quantum spin chain. We give a simple criteria for such a
factor state to be pure and find its relation with Kolmogorov’s property. Here
we also deal with the unique temperature state i.e. KMS state on Cuntz alge-
bra to illustrate that Powers shift property is not equivalent to Kolmogorov’s
property of the associated canonical Markov map on the support projection. In
fact this shows that Kolmogorov’s property is an appropriate notion to describe
purity of the inductive state.
2 Time-reverse Markov semigroup and
asymptotic properties:
In this section we will deal will a von-Neumann algebra A and a completely
positive map τ or a semigroup τ = (τt, t ≥ 0} of such maps on A. We
assume further that there exists a normal invariant state φ0 for τ and aim
to investigate asymptotic properties of the Markov map. We say (A0, τt, φ0)
is ergodic if {x : τt(x) = x, t ≥ 0} = {zI, z ∈ IC} and we say mixing if
τt(x) → φ0(x) in the weak
∗ topology as t→ ∞ for all x ∈ A0.
For the time being we assume φ0 is faithful and recall following [OP,AM],
the unique Markov map τ̃ on A0 which satisfies the following adjoint relation
φ0(σ1/2(x)τ(y)) = φ0(τ̃(x)σ−1/2(y)) (2.1)
for all x, y ∈ A0 analytic elements for the Tomita’s modular automorphism
(σt : t ∈ IR) associated with a faithful normal invariant state for a Markov
map τ on A0. For more details we refer to the monograph [OP]. We also quote
now [OP, Proposition 8.4 ] the following proposition without a proof.
PROPOSITION 2.1: Let τ be an unital completely positive normal maps
on a von-Neumann algebra A0 and φ0 be a faithful normal invariant state for
τ . Then the following conditions are equivalent for x ∈ A0:
(a) τ(x∗x) = τ(x∗)τ(x) and σs(τ(x)) = τ(σs(x)), ∀ s ∈R;
(b) τ̃ τ(x) = x.
Moreover τ restricted to the sub-algebra {x : τ̃ τ(x) = x} is an isomorphism
onto the sub-algebra {x ∈ A0 : τ τ̃ (x) = x} where (σs) be the modular auto-
morphism on A0 associated with φ0.
In the following we investigate the situation further.
PROPOSITION 2.2: Let (A0, τt, φ0) be a quantum dynamical system and
φ0 be faithful invariant normal state for (τt). Then the following hold:
(a) G = {x ∈ A0 : τt(x
∗x) = τt(x
∗)τt(x), τt(xx
∗) = τt(x)τt(x
∗), σs(τt(x)) =
τt(σs(x)), ∀ s ∈R, t ≥ 0} and G is σ = (σs : s ∈R) invariant and commuting
with τ = (τt : t ≥ 0) on G. Moreover for all t ≥ 0, τ̃t(G) = G and the
conditional expectationEG : A0 → A0 onto G0 commutes with (τt).
(b) There exists a unique maximal von-Neumann algebra G0 ⊆ G
G̃ so that
σt(G0) = G0 for all t ∈R and (G0, τt, φ0) is an automorphism where for any t ≥ 0,
τ̃tτt = τtτ̃t = 1 on G0. Moreover the conditional expectation EG0 : A0 → A0
onto G0 commutes with (τt) and (τ̃t).
PROOF: The first part of (a) is a trivial consequence of Proposition 2.1 once
we note that G is closed under the action x → x∗. For the second part we recall
[Mo1] that φ0(x
∗JxJ) − φ0(τt(x
∗)Jτt(x)J) is monotonically increasing with t
and thus for any fix t ≥ 0 if τ̃tτt(x) = x then τ̃sτs(x) = x for all 0 ≤ s ≤ t.
So the sequence Gt = {x ∈ A0 : τ̃tτt(x) = x} of von-Neumann sub-algebras
decreases to G as t increases to ∞ i.e. G =
t≥0 Gt. Similarly we also have
t≥0 G̃t.
Since G̃t monotonically decreases to G̃ as t increases to infinity for any
s ≥ 0 we claim that τs(G̃
t≥0 τs(G̃
t ), where we have used the symbol
A1 = {x ∈ A : ||x|| = 1} for a von-Neumann algebra A. We will prove the
non-trivial inclusion. To that end let x ∈
t≥0 τs(G̃
t ) i.e. for each t ≥ 0 there
exists yt ∈ G̃
t so that τs(yt) = x. By weak
∗ compactness of the unit ball of A0,
we extract a subsequence tn → ∞ so that ytn → y as tn → ∞ for some y ∈ A0.
The von-Neumann algebras G̃t being monotonically decreasing, for each m ≥ 1,
ytn ∈ G̃tm for all n ≥ m. G̃tm being a von-Neumann algebra, we get y ∈ G̃tm .
As this holds for each m ≥ 1, we get y ∈ G̃. However by normality of the map
τs, we also have x = τs(y). Hence x ∈ τs(G̃
Now we verify that
s≥r τs(G̃
t≥0 τs+t(G
t ) =
s≥r τs+t(G
t ) =
0≤s≤t τt(G
s ), where we have used τt(G
t ) = G̃
t being isomorphic. Since Gt
are monotonically decreasing with t we also note that
0≤s≤t τt(G
s ) = τt(G
Hence for any r ≥ 0
τs(G̃
1) = G̃1 (2.2)
From (2.2) with r = 0 we get G̃1 ⊆ τt(G̃
1) for all t ≥ 0. For any t ≥ 0 we also
have τt(G̃
s≥t τs(G̃
1) = G̃1. Hence we conclude τt(G̃
1) = G̃1 for any t ≥ 0.
Now we can easily remove the restriction to show that τt(G̃) = G̃ for any t ≥ 0
by linearity. By symmetry τ̃t(G) = G for any t ≥ 0.
Since G is invariant under the modular automorphism (σs) by a theorem
of Takesaki [AC] there exists a norm one projection EG : A → A with range
equal to G. We claim thatEG commutes with (τt). To that end we verify for
any x ∈ A0 and y ∈ G the following equalities:
< JGyJGω0,EG(τt(x))ω0 >=< J0yJ0ω0, τt(x)ω0 >
=< J0τ̃t(y)J0ω0, xω0 >=< JG τ̃t(y)JGω0,EG(x)ω0 >
=< JGyJGω0, τt(EG(x))ω0) >
where we used the fact that τ̃ (G) = G for the third equality and range of IEG
is indeed G is used for the last equality. This completes the proof of (a).
Now for any s ≥ 0, it is obvious that τ̃s(G̃) ⊆
t≥s τ̃s(G̃t). In the following
we prove equality in the above relation. Let x ∈
t≥s τ̃s(G̃t) i.e. there exists
elements yt ∈ G̃t so that x = τ̃s(yt) for all t ≥ s. If so then we have τs(x) = yt
for all t ≥ s as G̃t ⊆ G̃s. Thus for any t ≥ s, yt = ys ∈ G̃ and x ∈ τ̃s(G̃).
Now we verify the following elementary relations: τ̃s(G̃) =
t≥s τ̃sτt(Gt) =
t≥s τ̃sτs(τt−s(Gt))) =
t≥s τt−s(Gt) =
t≥0 τt(Gs+t) where we have used the
fact that τt−s(Gt) ⊆ Gs. Thus we have
s≥0 τs(G) ⊆
s≥0 τ̃s(G̃). By the dual
symmetry, we conclude the reverse inclusion and hence
τs(G) =
τ̃s(G̃) (2.3)
We set von-Neumann algebra G0 =
s≥0 τs(G). Thus G0 ⊆ G and also
G0 ⊆ G̃ by (2.3) and for each t ≥ 0 we have τtτ̃t = τ̃tτt = 1 on G0. Since τs(G)
is monotonically decreasing, we also note that τt(G0) =
s≥0 τs+t(G) = G0.
Similarly τ̃t(G0) = G0 by (2.3). That G0 is invariant by the modular group σ
follows since G is invariant by σ = (σt) which is commuting with τ = (τt) on G.
Same is also true for (τ̃t) by (2.3). By Takesaki’s theorem [AC] once more we
guarantee that there exists a conditional expectationEG0 : A0 → A0 with range
equal to G0. Since τ̃t(G0) = G0, once more by repeating the above argument we
conclude that EG0τt = τtEG0 on A0. By symmetry of the argument,EG0 is also
commuting with τ̃ = (τ̃t)
We have the following reduction theorem.
THEOREM 2.3: Let (A0, τt, φ0) be as in Proposition 2.2. Then the follow-
ing statements are equivalent:
(a) (A0, τt, φ0) is mixing ( ergodic );
(b) (G, τt, φ0) is mixing ( ergodic );
(c) (G0, τt, φ0) is mixing ( ergodic ).
PROOF: That (a) implies (b) is obvious. By Proposition 2.2. we have
EGτt(x) = τtEG(x) for any x ∈ A0 and t ≥ 0. Fix any x ∈ A0. Let x∞ be any
weak∗ limit point of the net τt(x) as t→ ∞ which is an element in G [Mo1]. In
case (b) is true, we find that x∞ =EG(x∞) = φ0(EG(x)) = φ0(x)1. Thus φ0(x)1
is the unique limit point, hence weak∗ limit of τt(x) as t → ∞ is φ0(x)1. The
equivalence statement for ergodicity also follows along the same line since the
conditional expectationEI on the the von-Neumann algebra I = {x : τt(x) =
x, t ≥ 0} commutes with (τt) and thus satisfies EIEG = EGEI = EI . This
completes the proof that (a) and (b) are equivalent. That (b) and (c) are
equivalent follows essentially along the same line since once more there exists a
conditional expectation from G to G0 commuting with (τt) and any weak
∗ limit
point of the net τt(x) as t diverges to infinity belongs to τs(G) for each s ≥ 0,
thus in G0. We omit the details.
Now we investigate asymptotic behavior for quantum dynamical system
dropping the assumption that φ0 is faithful. Let p be the support projection
of the normal state φ0 in A0. Thus we have φ0(pτt(1− p)p) = 0 for all t ≥ 0, p
being the support projection we have pτt(1− p)p = 0 i.e. p is a sub-harmonic
projection in A0 for (τt) i.e. τt(p) ≥ p for all t ≥ 0. Then it is simple to
check that (A
t , φ
0) is a quantum dynamical semigroup where A
0 = pA0p
and τ
t (x) = pτt(pxp)p for x ∈ A
0 and φ
0(x) = φ0(pxp) is faithful on A
0. In
Theorem 3.6 and Theorem 3.12 in [Mo1] we have explored how ergodicity and
strong mixing of the original dynamics (A0, τt, φ0) can be determined by that
of the reduced dynamics (A
t , φ
0). Here we add one more result in that line
of investigation.
THEOREM 2.4: Let (A0, τt, φ0) be a quantum dynamical systems with a
normal invariant state φ0 and p be a sub-harmonic projection for (τt). If
s-limitt→∞τt(p) = 1 then the following statements are equivalent:
(a) ||φτt − φ0|| → 0 as t→ ∞ for any normal state on φ on A0.
(b) ||φpτ
t − φ
0|| → 0 as t→ ∞ for any normal state φ
p on A
PROOF: That (a) implies (b) is trivial. For the converse we write
||φτt − φ0|| = supx:||x||≤1|φτt(x) − φ0(x)| ≤ sup{x:||x||≤1}|φτt(pxp) − φ0(pxp)| +
sup{x:||x||≤1}|φτt(pxp
⊥)| + sup{x:||x||≤1}|φτt(p
⊥xp)| + sup{x:||x||≤1}|φτt(p
⊥xp⊥)|.
Since τt((1 − p)x) → 0 in the weak
∗ topology and |φτt(xp
⊥)|2 ≤
|φτt(xx
∗)|φ(τt(p
⊥))| ≤ ||x||2φ(τt(p
⊥) it is good enough if we verify that (a)
is equivalent to sup{x:||x||≤1}|φτt(pxp) − φ0(pxp)| → 0 as t → ∞. To that end
we first note that limsupt→∞supx:||x||≤1|ψ(τs+t(pxp))− φ0(pxp)| is independent
of s ≥ 0 we choose. On the other hand we write τs+t(pxp) = τs(pτt(pxp)p) +
τs(pτt(pxp)p
⊥)+τs(p
⊥τt(pxp)p)+τs(p
⊥τt(pxp)p
⊥) and use the fact for any nor-
mal state φ we have limsupt→∞supx:||x||≤1|ψ(τs(zτt(pxp)p
⊥)| ≤ ||z|| |ψ(τs(p
for all z ∈ A0. Thus by our hypothesis on the support projection we conclude
that (a) hold whenever (b) is true.
In case the time variable is continuous and the von-Neumann algebra is the
set of bounded linear operators on a finite dimensional Hilbert space H0, by
exploring Lindblad’s representation [Li], Arveson [Ar] shows that a quantum
dynamical semigroup with a faithful normal invariant state is ergodic if and
only if the dynamics is mixing. In the following we prove a more general result
exploring the criteria that we have obtained in Theorem 2.3. Note at this
point that we don’t even need the generator of the Markov semigroup to be
a bounded operator for which Lindblad’s representation is not yet understood
with full generality [CE].
THEOREM 2.5: Let A0 be type-I with center completely atomic and (τt :
t ∈R) admits a normal invariant state φ0. Then (A0, τt, φ0) is strong mixing if
and only if (A0, τt, φ0) is ergodic.
PROOF: We first assume that φ0 is also faithful. We will verify now the
criteria that G0 is trivial when (τt) is ergodic. Since G0 is invariant by the
modular auto-morphism group associated with the faithful normal state φ0, by
a theorem of Takesaki [Ta] there exists a faithful normal norm one projection
from A0 onto G0. Now since A0 is a von-Neumann algebra of type-I with center
completely atomic, a result of E. Stormer [So] says that G0 is also type-I with
center completely atomic.
Let Q be a central projection in G0. Since τt(Q) is also a central projection
and τt(Q) → Q as t → 0 we conclude that τt(Q) = Q for all t ≥ 0 (center
of G being completely atomic and time variable t is continuous ). Hence by
ergodicity we conclude that Q = 0 or 1. Hence G0 can be identified with B(K)
for a separable Hilbert space K. Since (τt) on B(K) is an automorphism we find
a self-adjoint operator H in K so that τt(x) = e
itHxe−itH for any x ∈ B(K).
Since it admits an ergodic faithful normal state, by [Fr, Mo1] we conclude that
{x ∈ B(K) : xeitH = eitHx, t ∈R} = IC, which holds if and only if K is one
dimensional. Hence G0 = IC.
Now we deal with the general situation. Let p be the support projection of
φ0 in A0 and A0 being a type-I von-Neumann algebra with centre completely
atomic, the center of A
0 = pA0p being equal to the corner of the center of A0
i.e. pA0
A′0p, is also a type-I von-Neumann algebra with completely atomic
centre. (A0, τt, φ0) being ergodic, we have τt(p) ↑ 1 as t ↑ ∞ in the weak
topology and (A
t , φ
0) is ergodic. Thus by the first part of the argument,
t , φ
0) is strongly mixing. Hence by Theorem 3.12 in [Mo1] we conclude
that (A0, τt, φ0) is also strong mixing. This completes the proof.
We end this section with another simple application of Theorem 2.3 by
proving a result originated in [FNW1,FNW2,BJKW].
THEOREM 2.6: Let A0 be a type-I von-Neumann algebra with center
completely atomic and τ be a completely positive map with a faithful normal
invariant state φ0. Then the following are equivalent:
(a) (A0, τn, φ0) is strong mixing.
(b) (A0, τn, φ0) is ergodic and {w ∈ S
1, τ(x) = wx, for some non zerox ∈
A0} = {1}, where S
1 = {w ∈ IC : |w| = 1}.
PROOF: That ‘(a) implies (b)’ is rather simple and true in general for any
von-Neumann algebra. To that end let τ(x) = wx for some x 6= 0 and |w| = 1.
Then τn(x) = wnx and since the sequence wn has a limit point say z, |z| = 1
we conclude by strong mixing that zx = φ0(x)I. Hence x is a scaler and thus
x = τ(x), x 6= 0. So w = 1 and x = φ0(x)I. Ergodic property also follows by
strong mixing as x = φ0(x)I for any x for which τ(x) = x.
Now for the converse we will use our hypothesis that φ0 is faithful and A0
is a type-I von-Neumann algebra with completely atomic. To that end we plan
to verify that G0 consists of scalers only and appeal to Theorem 2.3 for strong
mixing. Since there exists a conditional expectation fromA0 onto G0, by a The-
orem of Stormer [So] G0 is once more a type-I von-Neumann algebra with center
completely atomic. Let E be a non-zero atomic projection in the center of G0.
τ being an automorphism on G0, each element in the sequence {τk(E) : k ≥ 0}
is an atomic projection in the center of G0. If τn(E)
τm(E) 6= 0 and n ≥ m
we find that τm(τn−m(E)
E) 6= 0 and thus by faithful and invariance prop-
erty of φ0, we get φ(τn−m(E)
E) > 0. Once more by faithfulness we find
that τn−m(E)
E 6= 0. So by atomic property of E and τn−m(E) we con-
clude that τn−m(E) = E. Thus either the elements in the infinite sequence
E, τ(E), ...., τn(E).... are all mutually orthogonal or there exists a least pos-
itive integer n ≥ 1 so that the projections E, τ(E), .., τn−1(E) are mutually
orthogonal and τn(E) = E. However for such an infinite sequence with mutu-
ally orthogonal projection we have 1 = φ0(I) ≤ φ0(
0≤n≤m−1 τn(E)) = mφ0(E)
for all m ≥ 1. Hence φ0(E) = 0 contradicting that E is non-zero and φ0 is
faithful.
Thus for any w ∈ S1 with wn = 1, we have τ(x) = wx, where x =
0≤k≤n−1w
kτk(E) 6= 0. By (b) we have w = 1. Hence n = 1. In other
words we have τ(E) = E for any atomic projection in the center of G0. Now by
ergodicity we have E = I. Thus G0 is a type-I factor say isomorphic to B(K)
for some Hilbert space K and τ(x) = uxu∗ for some unitary element u in G0.
Since (G0, τn, φ0) is ergodic we have {u, u∗}
′′ = B(K), which holds if and only
if K is one dimensional ( check for an alternative proof that τ(u) = u, thus
u = I by ergodicity and thus τ(x) = x for all x ∈ G0 ). Hence G0 = IC. This
complete the proof that (b) implies (a).
3 Minimal endomorphisms and Markov semi-
groups :
An E0-semigroup (αt) is a weak
∗-continuous one-parameter semigroup of unital
∗-endomorphisms on a von-Neumann algebra A acting on a Hilbert space H.
Following [Po1,Po2,Ar] we say (αt) is a shift if
t≥0 αt(A) = IC. For each
t ≥ 0, αt being an endomorphism, αt(A) is itself a von-Neumann algebra and
t≥0 αt(A) is a limit of a sequence of decreasing von-Neumann algebras.
Exploring this property Arveson proved that (αt) is pure if and only if ||ψ1αt−
ψ2αt|| → 0 as t→ ∞ for any two normal states ψ1, ψ2 on A. These criteria gets
further simplified in case (αt) admits a normal invariant state ψ0 for which we
have (αt) is a shift (in his terminology it is called pure, here we prefer Powers’s
terminology as the last section will illustrate a shift need not be pure in its
inductive limit ) if and only if ||ψαt−ψ0|| → 0 as t→ ∞ for any normal state
ψ. In such a case ψ0 is the unique normal invariant state. However a shift (αt)
in general may not admit a normal invariant state [Po2,BJP] and this issue is
itself an interesting problem.
One natural question that we wish to address here whether similar result
is also true for a Markov semigroup (τt) defined on an arbitrary von-Neumann
algebra A0. This issue is already investigated in [Ar] where A0 = B(H) and
the semigroup (τt) is assumed to be continuous in the strong operator topology.
He explored associated minimal dilation to an E0-semigroups and thus make
possible to prove that the associated E0-semigroup is a shift if and only if
||φ1τt − φ2τt|| → 0 as t → ∞ for any two normal states φ1, φ2 on A0. In case
(τt) admits a normal invariant state the criteria gets simplified once more. In
this section we will investigate this issue further for an arbitrary von-Neumann
algebra assuming that (τt) is admits a normal invariant state φ0.
To that end, we consider [Mo1] the minimal stationary weak Markov for-
ward process (H, Ft], jt,Ω, t ∈ R) and Markov shift (St) associated with
(A0, τt, φ0) and set A[t to be the von-Neumann algebra generated by the
family of operators {js(x) : t ≤ s < ∞, x ∈ A0}. We recall that
js+t(x) = S
t js(x)St, t, s ∈R and thus αt(A[0) ⊆ A[0 whenever t ≥ 0. Hence
(αt, t ≥ 0) is a E0-semigroup on A[0 with a invariant normal state Ω and
js(τt−s(x)) = Fs]αt(jt−s(x))Fs] (3.1)
for all x ∈ A0. We consider the GNS Hilbert space (Hπφ0 , πφ0(A0), ω0) as-
sociated with (A0, φ0) and define a Markov semigroup (τ
t ) on π(A0) by
τπt (π(x)) = π(τt(x). Furthermore we now identify Hφ0 as the subspace of
H by the prescription πφ0(x)ω0 → j0(x)Ω. In such a case π(x) is identified as
j0(x) and aim to verify for any t ≥ 0 that
t (PXP ) = Pαt(X)P (3.2)
for all X ∈ A[0 where P is the projection from H on the GNS space. We
use induction on n ≥ 1. If X = js(x) for some s ≥ 0, (4.2) follows
from (4.1). Now we assume that (3.2) is true for any element of the form
js1(x1)...jsn(xn) for any s1, s2, ..., sn ≥ 0 and xi ∈ A0 for 1 ≤ i ≤ n.
Fix any s1, s2, , sn, sn+1 ≥ 0 and consider X = js1(x1)...jsn+1(xn+1). Thus
Pαt(X)P = j0(1)js1+t(x1)...jsn+t(xn+1)j0(1). If sn+1 ≥ sn we use (3.1) to
conclude (3.2) by our induction hypothesis. Now suppose sn+1 ≤ sn. In
that case if sn−1 ≤ sn we appeal to (3.1) and induction hypothesis to verify
(3.2) for X . Thus we are left to consider the case where sn+1 ≤ sn ≤ sn−1
and by repeating this argument we are left to check only the case where
sn+1 ≤ sn ≤ sn−1 ≤ .. ≤ s1. But s1 ≥ 0 = s0 thus we can appeal to (3.1)
at the end of the string and conclude that our claim is true for all elements
in the ∗− algebra generated by these elements of all order. Thus the result
follows by von-Neumann density theorem. We also note that P = τπt (1) is a
sub-harmonic projection [Mo1] for (αt : t ≥ 0) i.e. αt(P ) ≥ P for all t ≥ 0.
PROPOSITION 3.1: Let (A0, τt, φ0) be a quantum dynamical semigroup
with a normal invariant state for (τt). Then the GNS space Hπφ0 associated
with the normal state φ0 on A0 can be realized as a closed subspace of a unique
Hilbert space H[0 up to isomorphism so that the following hold:
(a) There exists a von-Neumann algebra A[0 acting on H[0 and a unital ∗-
endomorphism (αt, t ≥ 0) on A[0 with a pure vector state φ(X) =< Ω, XΩ >,
Ω ∈ H[0 invariant for (αt : t ≥ 0).
(b) PAP is isomorphic with π(A0) where P is the projection onto Hπφ0 ;
(c) Pαt(X)P = τ
t (PXP ) for all t ≥ 0 and X ∈ A[0;
(d) The closed span generated by the vectors {αtn(PXnP )....αt1(PX1P )Ω :
0 ≤ t1 ≤ t2 ≤ .. ≤ tk ≤ ....tn, X1, .., Xn ∈ A[0, n ≥ 1} is H[0.
PROOF: The uniqueness up to isomorphism follows from the minimality prop-
erty (d).
Following the literature [Vi,Sa,BhP] on dilation we say (A[0, αt, φ) is the
minimal E0-semigroup associated with (A0, τt, φ0). By a theorem [Ar, Proposi-
tion 1.1 ] we conclude that
t≥0 αt(A[0) = IC if and only if for any normal state
ψ on A[0, ||ψαt − ψ0|| → 0 as t→ ∞, where ψ0(X) =< Ω, XΩ > for X ∈ A0].
In the following proposition we explore that fact that P is a sub-harmonic
projection for (αt) and by our construction αt(P ) = Ft] ↑ 1 as t→ ∞.
PROPOSITION 3.2: ||ψαt − ψ0|| → 0 as t → ∞ for all normal state ψ on
A[0 if and only if ||φτt − φ0|| → 0 as t → ∞ for all normal state φ on π(A0)
where π is the GNS space associated with (A0, φ0).
PROOF: Since Fs] ↑ 1 in strong operator topology by our construction and
π(A0) is isomorphic to F0]A[0F0], we get the result by a simple application of
Theorem 2.4.
THEOREM 3.3: Let τ = (τt, t ≥ 0) be a weak
∗ continuous Markov semi-
group on A0 with an invariant normal state φ0. Then there exists a weak
continuous E0-semigroup α = (αt, t ≥ 0) on a von-Neumann algebra A[0
acting on a Hilbert space H so that
Pαt(X)P = τ
t (PXP ), t ≥ 0
for all X ∈ A[0, where P is a sub-harmonic projection for (αt) such that
αt(P ) ↑ I.
Moreover the following statements are equivalent:
t≥0 αt(A[0) =C
(b) ||φτπt − φ0|| → 0 as t→ ∞ for any normal state φ on π(A0)
PROOF: For convenience of notation we denote π(A0)
′′ as A0 in the following
proof. That (a) and (b) are equivalent follows by a Theorem of Arveson [Ar]
and Proposition 3.2.
Following [AM,Mo1] we say (H, St, Ft],Ω) is a Kolmogorov’s shift if strong
limt→−∞Ft] = |Ω >< Ω|. We also recall here that Kolmogorov’s shift property
holds if and only if φ0(τt(x)τt(y)) → φ0(x)φ0(y) as t→ ∞ for all x, y ∈ A0. In
such a case A = B(H) ( see the paragraph before Theorem 3.9 in [Mo1] ). If φ0
is faithful then A0 and π(A0) are isomorphic, thus
t≥0 αt(A[0) =C if and only
if ||φτt − φ0|| → 0 as t→ ∞ for any normal state φ on A0. Such a property is
often called strong ergodic property. Our next result says that there is a duality
between strong ergodicity and Kolmogorov’s shift property. To that end we
recall the backward process (H, jbt , F[t,Ω) as defined in [AcM,Mo1] where Ft]
be the projection on the subspace generated by the vectors {λ : IR → A0 :
support of λ ⊆ (−∞, t]} and for any x ∈ A0, j
t (x) is the trivial extension of
it’s action on Ft] which takes an typical vector λ to λ
′ where λ′(s) = λ(s) for any
s < t and λ′(t) = λ(t)σ i
(x). For any analytic element x for the automorphism
group, we check first that jbt is indeed an isometry if x is so. Now we extend as
analytic elements are weak∗ dense to all isometrics and extend by linearity to all
elements of A0. We recall here that we have backward Markov property for the
process (jbs) as F[tj
s(x)F[t = j
t (τ̃t−s(x)) for all t ≥ s where (A0, τ̃t, t ≥ 0, φ0 is
the dual Markov semigroup defined in (3.1). As in the forward process we have
now F[tA
t]F[t = j
t (A0) where for each t ∈ IR we set A
t] for the von-Neumann
algebra {jbs(x) : s ≤ t, x ∈ A0}
THEOREM 3.4: Let (A0, τt, φ0) be a Markov semigroup with a faithful
normal invariant state φ0. Then the following are equivalent:
(a) φ0(τ̃t(x)τ̃t(y)) → φ0(x)φ0(y) as t→ ∞ for any x, y ∈ A0.
(b) ||φτt − φ0|| → 0 as t→ ∞ for any normal state φ on A0.
PROOF: For each t ∈R let Abt] be the von-Neumann algebra generated by the
backward processes {jbs(x) : −∞ < s ≤ t} [Mo1]. Assume (a). By Theorem
3.9 and Theorem 4.1 in [Mo1] we verify that weak∗ closure of
t] is B(H).
Since for each t ∈ R the commutant of Abt] contains A[t we conclude that
t∈RA[t is trivial. Hence (b) follows once we appeal to Theorem 3.3. For
the converse, it is enough if we verify that φ0(τ̃t(x)Jτ̃t(y)J) → φ0(x)φ0(y) as
t→ ∞ for any x, y ∈ A0 with y ≥ 0 and φ0(y) = 1. To that end we check the
following easy steps φ0(τ̃t(x)Jτ̃t(y)J) = φ0(τt(τ̃t(x))JyJ) and for any normal
state φ, |φ ◦ τt(τ̃t(x))−φ0(x)| ≤ ||φ ◦ τt−φ0||||τ̃t(x)|| ≤ ||φ ◦ τt−φ0||||x||. Thus
the result follows once we note that φ defined by φ(x) = φ0(xJyJ) is a normal
state.
THEOREM 3.5: Let (A0, τt, φ0) be a Markov semigroup with a normal in-
variant state φ0. Consider the following statements:
(a) φ0(τt(x)τt(y)) → φ0(x)φ0(y) as t→ ∞ for all x, y ∈ A0.
(b) the strong limt→−∞Ft] = |Ω >< Ω|.
(c) A = B(H)
Then (a) and (b) are equivalent statements and in such a case (c) is also
true. If φ0 is also faithful (c) is also equivalent to (a) ( and hence ( b)).
PROOF: That (a) and (b) are equivalent is nothing but a restatement of
Theorem 3.9 in [Mo1]. That (b) implies (c) is obvious since the projection
[A′Ω], where A′ is the commutant of A, is the support of the vector state in A.
We will prove now (c) implies (a). In case A = B(H), we have
t] =C,
thus in particular
t≤0 αt(A
0]) = C. Hence by Theorem 3.3 applied for the
time-reverse endomorphism we verify that ||φτ̃t− φ0|| → 0 as t→ ∞. Now (a)
follows once we appeal to Theorem 3.4 for the adjoint semigroups since ˜̃τ t = τt.
THEOREM 3.6: Let (A0, τt, φ0) be as in Theorem 3.1. Then the following
hold:
(a) If (A0, τt, φ0) is mixing then αt(X) → φ(X) as t→ ∞ for all X ∈ B, where
B is the C∗ completion of the ∗ algebra generated by {jt(x) : t ∈ IR, x ∈ A0}.
(b) If (A0, τt, φ0) is mixing and A is a type-I factor then A = B(H).
PROOF: For (a) we refer to [AM, Mo1]. By our hypothesis A is a type-I
von-Neumann factor and thus there exists an irreducible representation π of B
in a Hilbert space Hπ quasi equivalent to πφ. There exists a density matrix ρ
on Hπ such that φ(X) = tr(π(X)ρ) for all X ∈ B. Thus there exists a unitary
representation t→ Ut on Hπ so that
Utπ(X)U
t = π(αt(X))
for all t ∈ IR and X ∈ B. Since φ = φαt on B we also have U
t ρUt = ρ. We
claim that ρ is a one dimensional projection. Suppose not and then there exists
at least two characteristic unit vectors f1, f2 for ρ so that f1, f2 are character-
istic vector for unitary representation Ut. Hence we have < fi, π(X)fi >=<
fi, π(αt(X))fi > for all t ∈ IR and i = 1, 2. By taking limit we conclude by (a)
that < fi, π(X)fi >= φ(X) < fi, fi >= φ(X) for i = 1, 2 for all X ∈ B. This
violets irreducibility of representation π.
PROPOSITION 3.7: Let (A0, τt, φ0) be as in Theorem 3.5 with φ0 as
faithful. Then the commutant of A[t is A
t] for each t ∈ IR.
PROOF: It is obvious that A[0 is a subset of the commutant of A
0]. Note also
that F[0 is an element in A
0] which commutes with all the elements in A[0. As
a first step note that it is good enough if we show that F[0(A
′F[0 = F[0A[0F[0.
As for some X ∈ (Ab0])
′ and Y ∈ A[0 if we have XF[0 = F[0XF[0 = F[0Y F[0 =
Y F[0 then we verify that XZf = Y Zf where f is any vector so that F[0f = f
and Z ∈ Ab0] and thus as such vectors are total in H we get X = Y ). Thus
all that we need to show that F[0(A
′F[0 ⊆ F[0A[0F[0 as inclusion in other
direction is obvious. We will explore in following the relation that F0]F[0 =
F[0F0] = F{0} i.e. the projection on the fiber at 0 repeatedly. A simple proof
follows once we use explicit formulas for F0] and F[0 given in [Mo1].
Now we aim to prove that F[0A
[0F[0 ⊆ F[0A
0]F[0. Let X ∈ F[0A
[0F[0 and
verify that XΩ = XF0]Ω = F0]XF0]Ω = F{0}XF{0}Ω ∈ [j
0(A0)
′′Ω]. On the
other-hand we note by Markov property of the backward process (jbt ) that
0]F[0 = j
b(A0)
′′. Thus there exists an element Y ∈ Ab0] so that XΩ = Y Ω.
Hence XZΩ = Y ZΩ for all Z ∈ A[0 as Z commutes with both X and Y . Since
{ZΩ : Z ∈ A[0} spans F[0, we get the required inclusion. Since inclusion in the
other direction is trivial as F[0 ∈ A
[0 we conclude that F[0A
[0F[0 = F[0A
0]F[0.
F[0 being a projection in A
0] we verify that F[0(A
′F[0 ⊆ (F[0A
0]F[0)
′ and
so we also have F[0(A
′F[0 ⊆ (F[0A
[0F[0)
′ as Ab0] ⊆ A
[0. Thus it is enough if
we prove that
[0F[0 = (F[0A[0F[0)
We will verify the non-trivial inclusion for the above equality. Let X ∈
(F[0A[0F[0)
′ then XΩ = XF0]Ω = F0]XF0]Ω = F{0}XF{0}Ω ∈ [j
0(A0)Ω]. Hence
there exists an element Y ∈ F[0A
[0F[0 so that XΩ = Y Ω. Thus for any Z ∈ A[0
we have XZΩ = Y ZΩ and thusXF[0 = Y F[0. Hence X = Y ∈ F[0A
[0F[0. Thus
we get the required inclusion.
Now for any value of t ∈ IR we recall that αt(A[0) = A[t and αt(A[0)
[0), αt being an automorphism. This completes the proof as αt(A
0]) = A
by our construction.
One interesting problem that we raised in [Mo1] whether Kolmogorov’s
property is time reversible i.e. whether Ft] → |Ω >< Ω| as t → −∞ if and
only if F[t → |Ω >< Ω| as t → ∞. That it is true in classical case follows
by Kolmogorov-Sinai-Rohlin theory on dynamical entropy for the associated
Markov shift [Pa]. In the present general set up, it is true if A0 is a type-I von-
Neumann algebra with centre atomic [Mo1]. It is obviously true if the Markov
semigroup is KMS symmetric. But in general it is false. In the last section we
will give a class of counter example. This indicates that the quantum counter
part of Kolmogorov property is unlikely to be captured by a suitable notion of
quantum dynamical entropy with Kolmogorov-Sinai-Rohlin property.
4 Inductive limit state and purity:
Let (B0, λt, t ≥ 0, ψ) be a unital ∗− endomorphism with an invariant normal
state ψ on a von-Neumann algebra B0 acting on a Hilbert space H. Let P
be the support projection for ψ. We set A0 = PBP , a von-Neumann algebra
acting on H0, the closed subspace P , and τt(x) = Pλt(PxP )P , for any x ∈ A0
and t ≥ 0. Since λt(P ) ≥ P , it is simple to verify [Mo1] that (A0, τt, ψ0) is a
quantum dynamical semigroup with a faithful normal invariant state ψ0, where
ψ0(x) = ψ(PxP ) for x ∈ A0. Now we set j0(x) = PxP and jt(x) = λt(j0(x))
for t ≥ 0 and x ∈ A0. A routine verification says that Fs]jt(x)Fs] = js(τt−s(x))
for 0 ≤ s ≤ t, where Fs] = λs(P ), s ≥ 0. Let A[0 be the von-Neumann algebra
{jt(x) : t ≥ 0, x ∈ A0}
′′. As in Section 4 we check that Pαt(X)P = τt(PXP )
for all X ∈ A[0. However are these vectors {λtn(PXnP )....λt1(PX1P )f : f ∈
H0, 0 ≤ t1 ≤ t2 ≤ .. ≤ tk ≤ ..tn, X1, .., Xn ∈ B0, n ≥ 1} total in H? As
an counter example in discrete time we consider an endomorphism on B(H)
[BJP] with a pure mixing state and note that A0 is only scalers. Thus the
cyclic space generated by the process (jt) on the pure state is itself. Thus
the problem is rather delicate even when the von-Neumann algebra is the
algebra of all bounded operators on K. We will not address this problem
here. Since λt(P )λtn(PXnP )...λt1(PXP )H0 = λtn(PXnP )...λt1(PXP )Ω for
t ≥ tn, limt→∞λt(P ) = 1 is a necessary condition for cyclic property. The same
counter example shows that it is not sufficient. In the following we explore
the fact the support projection P is indeed an element in the von-Neumann
algebra A generated by the process (kt(x) : t ≥ 0, x ∈ A0) and asymptotic
limit of the endomorphism (B0, λt, t ≥ 0, ψ) is related with that of minimal
endomorphism (A[0, αt, t ≥ 0φ).
In the following we consider a little more general situation. Let B0 be a C
algebra, (λt : t ≥ 0) be a semigroup of injective endomorphisms and ψ be an
invariant state for (λt : t ≥ 0). We extend (λt) to an automorphism on the C
algebra B−∞ of the inductive limit
λt B0 →
λt B0
and extend also the state ψ to B−∞ by requiring (λt) invariance. Thus there
exists a directed set ( i.e. indexed by IT , by inclusion B[−s ⊆ B[−t if and only
if t ≥ s ) of C∗-subalgebras B[t of B−∞ so that the uniform closure of
s∈IT B[s
is B[−∞. Moreover there exists an isomorphism
i0 : B0 → B[0
( we refer [Sa] for general facts on inductive limit of C∗-algebras). It is simple
to note that it = λt ◦ i0 is an isomorphism of B0 onto B[t and
ψ−∞it = ψ
on B0. Let (Hπ, π,Ω) be the GNS space associated with (B[−∞, ψ[−∞) and (λt)
be the unique normal extension to π(B−∞)
′′. Thus the vector state ψΩ(X) =<
Ω, XΩ > is an invariant state for automorphism (λt). As λt(B[0) ⊆ B[0 for all
t ≥ 0, (π(B[0)
′′, λt, t ≥ 0, ψΩ) is a quantum dynamics of endomorphisms. Let
Ft] be the support projection of the normal vector state Ω in the von-Neumann
sub-algebra π(B[t)
′′. Ft] ∈ π(B[t)
′′ ⊆ π(B[−∞)
′′ is a monotonically decreasing
sequence of projections as t → −∞. Let projection Q be the limit. Thus
Q ≥ [π(B[−∞)
′Ω] ≥ |Ω >< Ω|. So Q = |Ω >< Ω| ensures that ψ on B[−∞ is
pure. We aim to investigate when Q is pure i.e. Q = |Ω >< Ω|.
To that end we set von-Neumann algebra N0 = F0]π(B[0)
′′F0] and define
family {kt : N0 → π(B−∞)
′′, t ∈ IT} of ∗−homomorphisms by
kt(x) = λt(F0]xF0]), x ∈ N0
It is a routine work to check that (kt : t ∈ IT ) is the unique up to isomor-
phism ( in the cyclic space of the vector Ω generated by the von-Neumann
algebra {kt(x) : t ∈ IT, x ∈ N0} ) forward weak Markov process associ-
ated with (N0, ηt, ψ0) where ηt(x) = F0]αt(F0]xF0])F0] for all t ≥ 0. It is
minimal once restricted to the cyclic space generated by the process. Thus
Q = |Ω >< Ω| when restricted to the cyclic subspace of the process if and only
if ψ0(ηt(x)ηt(y)) → ψ0(x)ψ0(y) as t→ ∞ for all x, y ∈ N0.
PROPOSITION 4.1: Let G0] be the cyclic subspace of the vector Ω gen-
erated by π(B[0).
(a) G0] ∈ π(B[0)
′ and the map h : π(B[0)
′′ → G0]π(B[0)
′′G0] defined by X →
G0]XG0] is an homomorphism and the range is isomorphic to π0(B0)
′′, where
(Hπ0, π0) is the GNS space associated with (B0, ψ).
(b) Identifying the range of h with π0(B0)
′′ we have
h ◦ λt(X) = λt(h(X))
for all X ∈ π(B[0)
′′ and t ≥ 0.
(c) Let P be the support projection of the state ψ in von-Neumann algebra
π0(B0)
′′ and A0 = Pπ0(B0)
′′P . We set τt(x) = Pλt(PxP )P for all t ≥ 0, x ∈
A0 and ψ0(x) = ψ(PxP ) for x ∈ A0. Then
(i) h(F0]) = P and h(N0) = A0;
(ii) h(ηt(x)) = τt(h(x)) for all t ≥ 0.
PROOF: The map π(X)Ω → π0(X)Ω0 has an unitary extension which in-
tertwines the GNS representation (H0, π0) with the sub-representation of B[0
on the cyclic subspace G0]. Thus (a) follows. (b) is a simple consequence as
i0 : B0 → B[0 is a C
∗ isomorphism which covariant with respect to (λt) for all
t ≥ 0 i.e λti0(x) = i0(λt(x)) for all x ∈ B0. That h(F0]) = P is simple as h is
an isomorphism and thus also a normal map taking support projection F0] of
the state ψ in π(B[0)
′′ to support projection P of the state ψ in π0(B0)
′′. Now
by homomorphism property of the map h and commuting property with (λt)
we also check that h(N0) = h(F0]π(B[0)
′′F0]) = Pπ0(B0)
′′P = A0 and
h(ηt(x)) = h(F0])λt(h(F0])h(x)h(F0]))
= Pλt(Ph(x)P )P = τt(h(x))
for all t ≥ 0.
THEOREM 4.2: Q is pure if and only if φ0(τt(x)τt(y)) → φ0(x)ψ0(y) as
t→ ∞ for all x, y ∈ A0.
PROOF: For any fix t ∈ IT since kt(A0) = Ft]π(B[t)
′′Ft], for any X ∈ B[t
we have QXΩ = QFt]XFt]Ω = Qkt(x)Ω for some x ∈ A0. Hence Q =
|Ω >< Ω| if and only if Q = |Ω >< Ω| on the cyclic subspace generated by
{kt(x), t ∈ IT, x ∈ A0}. Theorem 3.5 says now that Q = |Ω >< Ω| if and only
if ψ0(ηt(x)ηt(y)) → ψ0(x)ψ0(y) as t→ ∞ for all x ∈ N0, Since h is an homomor-
phism and hηt(x) = τt(h(x)), we also have h(ηt(x))ηt(y)) = τt(h(x))τt(h(x)).
Since φ0 ◦ h = ψ0 we complete the proof.
COROLLARY 4.3: ψ[−∞ is a pure state if φ0(τt(x)τt(y)) → φ0(x)ψ0(y) as
t→ ∞ for all x, y ∈ A0.
PROOF: It follows by Theorem 4.2 as Q ≤ [π(B[−∞)
′Ω] ≤ |Ω >< Ω|.
Our analysis above put very little light whether the sufficient condition
given in Corollary 4.3 is also necessary for purity. We will get to this point in
next section where we will deal with a class of examples.
5 Kolmogorov’s property and pure transla-
tion invariant states:
Let ω be a translation invariant state on UHFd algebra A = ⊗ZZMd and ω
be the restriction of ω to UHFd algebra B0 = ⊗INMd. There is a one to one
correspondence between a translation invariant state ω and λ (one sided shift
) invariant state ω′ on UHFd algebra ⊗INMd. Powers’s [Po] criteria easily
yields that ω is a factor state if and only if ω′ is a factor state. A question
that comes naturally here which property of ω′ is related with the purity of
ω. A systematic account of this question was initiated in [BJKW] inspired by
initial success of [FNW1,FNW2,BJP] and a sufficient condition is obtained. In
a recent article [Mo2] this line of investigation was further explored and we
obtained a necessary and sufficient condition for a translation invariant lattice
symmetric factor state to be pure and the criteria can be described in terms
of Popescu elements canonically associated with Cuntz’s representation. That
the state is lattice symmetric played an important role in the duality argument
used in the proof.
Here as an application of our general result, we aim now to find one more
useful criteria for a translation invariant factor state ω on a one dimensional
quantum spin chain ⊗ZZMd to be pure. We also prove that purity of a lattice
symmetric translation invariant state ω is equivalent to Kolmogorov’s property
of a Markov semigroup canonically associated with ω.
First we recall that the Cuntz algebra Od(d ∈ {2, 3, .., }) is the universal
C∗-algebra generated by the elements {s1, s2, ..., sd} subject to the relations:
s∗i sj = δ
1≤i≤d
i = 1.
There is a canonical action of the group U(d) of unitary d× d matrices on
Od given by
βg(si) =
1≤j≤d
for g = ((gij) ∈ U(d). In particular the gauge action is defined by
βz(si) = zsi, z ∈ IT = S
1 = {z ∈ IC : |z| = 1}.
If UHFd is the fixed point subalgebra under the gauge action, then UHFd is
the closure of the linear span of all wick ordered monomials of the form
si1...siks
...s∗j1
which is also isomorphic to the UHFd algebra
Md∞ = ⊗
so that the isomorphism carries the wick ordered monomial above into the
matrix element
(1)⊗ ei2j2(2)⊗ ....⊗ e
(k)⊗ 1⊗ 1....
and the restriction of βg to UHFd is then carried into action
Ad(g)⊗Ad(g)⊗ Ad(g)⊗ ....
We also define the canonical endomorphism λ on Od by
λ(x) =
1≤i≤d
and the isomorphism carries λ restricted to UHFd into the one-sided shift
y1 ⊗ y2 ⊗ ...→ 1⊗ y1 ⊗ y2....
on ⊗∞1 Md. Note that λβg = βgλ on UHFd.
Let d ∈ {2, 3, .., , ..} and ZZd be a set of d elements. I be the set of finite
sequences I = (i1, i2, ..., im) where ik ∈ ZZd and m ≥ 1. We also include empty
set ∅ ∈ I and set s∅ = 1 = s
∅, sI = si1......sim ∈ Od and s
I = s
...s∗i1 ∈ Od.
Let ω be a translation invariant state on A = ⊗ZZMd where Md is (d × d)
matrices with complex entries. Identifying ⊗INMd with UHFd we find a one to
one relation from a λ invariant state on UHFd with that of an one sided shift
invariant state on AR = ⊗INMd. Let ω
′ be an λ-invariant state on the UHFd
sub-algebra of Od. Following [BJKW, section 7], we consider the set
Kω′ = {ψ : ψ is a state on Od such that ψλ = ψ and ψ|UHFd = ω
By taking invariant mean on an extension of ω′ to Od, we verify that Kω′ is non
empty and Kω′ is clearly convex and compact in the weak topology. In case
ω′ is an ergodic state ( extremal state ) Kω′ is a face in the λ invariant states.
Before we recall Proposition 7.4 of [BJKW] in the following proposition.
PROPOSITION 5.1: Let ω′ be ergodic. Then ψ ∈ Kω′ is an extremal point
in Kω′ if and only if ω̂ is a factor state and moreover any other extremal point
in Kω′ have the form ψβz for some z ∈ IT .
We fix any ω̂ ∈ Kω′ point and consider the associated Popescu system
(K,M, vk,Ω) described as in Proposition 2.4. A simple application of Theorem
3.6 in [Mo2] says that the inductive limit state ω̂−∞ on the inductive limit
(Od, ω̂) →
λ (Od, ω̂) →
λ (Od, ω̂) is pure if φ0(τn(x)τn(y)) → φ0(x)φ0(y) for all
x, y ∈ M as n→ ∞. This criteria is of limited use in determining purity of ω
unless we have πω̂(UHFd)
′′ = πω̂(Od)
′′. We prove a more powerful criteria in
the next section, complementing a necessary and sufficient condition obtained
by [Mo2], for a translation invariant factor state ω to be pure.
To that end note that the von-Neumann algebra {SIS
J : |I| = |J | < ∞}
acts on the cyclic subspace of Hπω̂ generated by the vector Ω. This is iso-
morphic with the GNS representation associated with (B0, ω
′). The inductive
limit (B−∞, ω̂−∞) [Sa] described as in Proposition 3.6 in [Mo2] associated with
(B0, λn, n ≥ 0, ω
′) is UHFd algebra ⊗ZZMd and the inductive limit state is ω.
Let Q be the support projection of the state ω̂ in π0(B0)
′′ and A0 =
Qπ(B0)
′′Q. Since ψΩ(Λ(X)) = ψΩ(X) for all X ∈ πω̂(UHFd)
′′, Λ(Q) ∈
πω̂(UHFd)
′′ and Λ(Q) ≥ Q [Mo1]. Thus QΛ(I − Q)Q = 0 and we have
(I − Q)S∗kQ = 0 for all 1 ≤ k ≤ d. The reduced Markov map η : A0 → A0 is
defined by
η(x) = QΛ(QxQ)Q (5.1)
for all x ∈ A0 which admits a faithful normal state φ0 defined by
ψ0(x) = ψΩ(QxQ), x ∈ A0 (5.2)
In particular, Λn(Q) ↑ I as n → ∞. Hence {SIf : |I| < ∞, Qf = f, f ∈
Hπ} is total in Hπω̂ .
We set lk = QSkQ, where lk need not be an element in A0. However
J ∈ A0 provided |I| = |J | < ∞. Nevertheless we have QΩ = Ω and thus
verify that
ω̂(sIs
J) =< Ω, SIS
< Ω, QSIS
JQΩ >=< Ω, lI l
for all |I|, |J | <∞. In particular we have
ω′(sIs
J) = ψ0(lI l
for all |I| = |J | <∞.
For each n ≥ 1 we note that {SIS
J : |I| = |J | ≤ n}
Λn(πω̂(UHFd)
πω̂(UHFd)
′′ and thus πω̂(UHFd)
′′ ⊆ (
n≥1Λn(πω̂(UHFd)
′′)′.
Hence
Λn(πω̂(UHFd)
′′) ⊆ πω̂(UHFd)
πω̂(UHFd)
′. (5.3)
Now by Proposition 1.1 in [Ar, see also Mo2] ||ψΛn − ψΩ|| → 0 as n → ∞
for any normal state ψ on πω̂(UHFd)
′′ if ω′ is a factor state. Thus we have
arrived at the following well-known result of R. T. Powers [Pow1,BR].
THEOREM 5.2: Let ω′ be a λ invariant state on UHFd ⊗INMd. Then the
following statements are equivalent:
(a) ω′ is a factor state;
(b) For any normal state ψ on A0, ||ψηn − ψ0|| → 0 as n→ ∞;
(c) For any x ∈ UHFd ⊗INMd
sup||y||≤1|ω
′(xλn(y))− ω
′(x)ω′(y)| → 0
as n→ ∞;
(d) ω′(xλn(y)) → ω
′(x)ω′(y) as n→ ∞ for all x, y ∈ UHFd ⊗INMd;
PROOF: For any normal state ψ on A0 we note that ψP (X) = ψ(PXP ) is
a normal state on πω̂(UHFd)
′′ and ||ψηn − ψ0|| ≤ ||ψPΛn − ψΩ||. Thus by the
above argument (a) implies (b). That (c) implies (d) and (d) implies (a) are
obvious. We will prove that (b) implies (c). Note that for (c) it is good enough
if we verify for all non-negative x ∈ UHFd with finite support and ω
′(x) = 1.
In such a case for large values of n the map πω̂(y) → ω
′(xλn(y)) determines a
normal state on πω̂(UHFd)
′′. Hence (c) follows whenever (b) hold.
COROLLARY 5.3: Let ω be a translation invariant state on UHFd ⊗ZZMd.
Then the following are equivalent:
(a) ω is a factor state;
(b) ω(xλn(y)) → ω(x)ω(y) as n→ ∞ for all x, y ∈ UHFd ⊗ZZMd;
PROOF: First we recall ω is a factor state if and only if ω is an extremal point
in the translation invariant state i.e. ω is an ergodic state for the translation
map. Since the cluster property (b) implies ergodicity, (a) follows. For the
converse note that ω is a ergodic state for the translation map if and only if
ω′ is ergodic for λ on UHFd ⊗INMd. Hence by Theorem 3.2 we conclude that
statement (b) hold for any local elements x, y ∈ UHFd ⊗ZZMd. Now we use
the fact that local elements are dense in the C∗ norm to complete the proof.
PROPOSITION 5.4: Let ω be a translation invariant extremal state on A
and ψ be an extremal point in Kω. Then following hold:
(a) H = {z ∈ S1 : ψβz = ψ} is a closed subgroup of S
1 and π(Od)
′′βH =
π(UHFd)
′′. Furthermore we have
n(π(Od)
′′) = π(Od)
′′ ⋂ π(UHFd)′;
(b) If H = S1 then π(Od)
′′ ⋂ π(UHFd)′ = IC;
(c) Let (H, π,Ω) be the GNS representation of (Od, ψ) and P be the support
projection of the state ψ in π(Od)
′′. Then P ∈ π(UHFd)
′′ is also the support
projection of the state ψ in π(UHFd)
PROOF: First part of (a) is noting but a restatement of Proposition 2.5
in [Mo2] modulo the factor property of π(UHFd)
′′. For a proof of the factor
property we refer to Lemma 7.11 in [BJKW] modulo a modification described
in Proposition 3.2 in [Mo2].
We aim now to show that
n≥1 Λ
n(π(Od)
′′) = π(Od)
′′ ⋂ π(UHFd)′. It is
obvious by Cuntz relation that
n(π(Od)
′′) ⊆ π(Od)
′′ ⋂ π(UHFd)′. For
the converse letX ∈ π(Od)
π(UHFd)
′ and fix any n ≥ 1 and set Yn = S
with |I| = n. Since X ∈ π(UHFd)
′ we verify that S∗IXSI = S
IXSIS
JSJ =
S∗ISIS
JXSJ = S
JXSJ for any |J | = n. Thus Yn is independent of the multi-
index that we choose. Once gain as X ∈ π(UHFd)
′ we also check that Λn(Yn) =
J :|J |=n SJS
IXSIS
J = X . Hence X ∈
n≥1 Λ
n(π(Od)
Now π(UHFd)
′′ being a factor, a general result in [BJKW, Lemma 7.12] says
that π(Od)
π(OHd )
′ is a commutative von-Neumann algebra generated by an
unitary operator u so that βz(u) = γ(z)u for all z ∈ H and some character γ of
H . Furthermore there exists a z0 ∈ H so that βz0(x) = uxu
∗ for all x ∈ π(Od)
Thus we also have βz0(u) = u = γ(z0)u. So we have γ(z0) = 1. H being S
1 the
character can be written as γ(z) = zk all z ∈ H and for some k ≥ 1. Hence
ukx(uk)∗ = βzk
(x) = x. π(Od)
′′ being a factor uk is a scaler. By multiplying a
proper factor we can choose an unitary u ∈ π(Od)
π(UHFd)
′ so that uk = 1.
However we also check that for all z ∈ S1 we have βz(u
k) = γ(z)kuk i.e.
γ(z)k = 1 for all z ∈ S1 as uk = 1. Hence γ(z) = 1 for all z ∈ S1. Thus
βz(u) = u for all z ∈ S
1 and u is scaler as u is also an element in π(UHFd)
′′ by
the first part. π(UHFd)
′′ being a factor we conclude that u is a scaler. Hence
πψ(Od)
′′ ⋂π(UHFd)′ is trivial. This completes the proof of (b).
It is obvious that βz(P ) = P for all z ∈ H and thus by (a) P ∈ π(UHFd)
and thus also the support projection in π(UHFd)
′′ of the state ψ. (c) is a simple
consequence of (a) and Corollary 4.3.
THEOREM 5.4: Let ω be a translation invariant state on UHFd ⊗ZZMd
and P be the support projection of ψ ∈ Kω′ in π(Od)
′′. Further let A0 be the
von-Neumann algebra Pπ(UHFd)
′′P acting on the subspace P and completely
positive map τ : A0 → A0 defined by τ(x) = PΛ(PxP )P , i.e. τ(x) =
k lkxl
be the completely positive map on A0 where lk = Pπ(sk)P for all 1 ≤ k ≤ d.
Then the following hold:
(a) If φ0(τ
n(x)τn(y)) → φ0(x)φ0(y) as n→ ∞ for all x, y ∈ A0 then ω is pure;
(b) If H = S1 then ||φτn − φ0|| → 0 as n→ ∞ for any normal state on A0;
PROOF: (a) follows by an easy application of Corollary 4.3. For a proof for
(b) we appeal to [Ar, Proposition 1.1] and the last statement in Proposition
5.3 (a).
By a duality argument, Theorem 3.4 in [Mo2], ||ψηn − ψ0|| → 0 as n→ ∞
for any normal state ψ if and only if |ψ0(η̃n(x)η̃n(y)) → ψ0(x)ψ0(y)| as n →
∞ for any x, y ∈ A0, where (A0, η̃, φ0) the KMS-adjoint Markov semigroup
[OP,AcM,Mo1] of (A0, η, φ0).
We recall the unique KMS state ψ = ψβ on Od where β = ln(d) is a
factor state and ψβ ∈ Kω where ω
′ is the unique trace on UHFd. For a proof
that H = S1 for ψβ we refer to [BR]. ω is the unique trace on A and so is
a factor state. Hence by Proposition 5.4 (d) we have πψ(Od)
′′ ⋂ πψ(UHFd)′ is
trivial. Thus
n≥1Λ(πψ(Od)
′′) = IC. In particular
n≥1 Λ
n(πψ(UHFd)
′′) = IC.
On the other hand ψβ being faithful, the support projection is the identity
operator and thus canonical Markov semigroup τ is equal to Λ. Λ being an
endomorphism and ψβ being faithful, we easily verify that τ does not admit
Kolmogorov property. On the other handH = S1 and so by Proposition 5.4 (d)
||φτn−φ0|| → 0 as n→ ∞ for any normal state φ on A0. This example unlike
in the classical case shows that Kolmogorov’s property of a non-commutative
dynamical system in general is not time reversible.
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