arXiv:0704.1989v1  [math.OA]  16 Apr 2007
Jones index of a quantum dynamical semigroup
Anilesh Mohari
S.N.Bose Center for Basic Sciences,
JD Block, Sector-3, Calcutta-98
E-mail:anilesh@boson.bose.res.in
Abstract
In this paper we consider a completely positive map τ = (τt, t ≥ 0) with a faith-
ful normal invariant state φ on a type-II1 factor A0 and propose an index theory.
We achieve this via a more general Kolmogorov’s type of construction for station-
ary Markov processes which naturally associate a nested isomorphic von-Neumann
algebras. In particular this construction generalizes well known Jones construction
associated with a sub-factor of type-II1 factor.
http://arxiv.org/abs/0704.1989v1
1 Introduction:
Let τ = (τt, t ≥ 0) be a semigroup of identity preserving completely positive normal
maps [Da,BR] on a von-Neumann algebra A0 acting on a separable Hilbert space
H0, where either the parameter t ∈R+, the set of positive real numbers orZ
+, the
set of positive integers. In case t ∈ R+, i.e. continuous, we assume that for each
x ∈ A0 the map t→ τt(x) is continuous in the weak
∗ topology. Thus variable t ∈ IT+
where IT is either IR or IN . We assume further that (τt) admits a normal invariant
state φ0, i.e. φ0τt = φ0∀t ≥ 0.
As a first step following well known Kolmogorov’s construction of stationary
Markov processes, we employ GNS method to construct a Hilbert space H and an
increasing tower of isomorphic von-Neumann type−II factors {A[t : t ∈ R or Z}
generated by the weak Markov process (H, jt, Ft], t ∈ R or Z,Ω) [BP,AM] where
jt : A0 → A[t is an injective homomorphism from A0 into A[0 so that the projection
Ft] = jt(I) is the cyclic space of Ω generated by {js(x) : −∞ < s ≤ t, x ∈ A0}.
The tower of increasing isomorphic von-Neumann algebras {A[t, t ∈ R or Z} are
indeed a type-II∞ factor if and only if τ is not an endomorphism. In any case the
projection j0(I) is a finite projection in A[−t for all t ≤ 0. In particular we also find
an increasing tower of type-II1 factors {Mt : t ≥ 0} defined by Mt = j0(I)A[−tj0(I).
Thus Jones in-dices {[Mt : Ms] : 0 ≤ s ≤ t} are invariance for the Markov semigroup
(A0, τt, t ≥ 0, φ0) and further the map (t, s) → [Mt : Ms] is not continuous if the
variable (t, s) are continuous i.e. if τ = (τt : t ∈ IR+). In discrete time dynamical
system we find an invariance sequence {[Mn+1 : Mn] : n ≥ 0} canonically associated
with the canonical conditional expectation on a sub-factor B0 of a type-II1 factor A0
where φ0 is the unique normal trace on A0. However unlike Jones construction we
have [Mn+1 : Mn] = d
2 where d = [A0 : B0]. This shows that our construction in
a sense generalizes two step Jones construction in discrete time. A detailed study,
needs to be done to explore this new invariance, which seems to be an interesting
problem!
Acknowledgment: The author takes the opportunity to acknowledge Prof. Luigi
Accardi for an invitation to visit Centro Vito Volterra, University of Rome, Tor
Vergata during the summer 2005. The author further gratefully acknowledge Prof.
Roberto Longo and Prof. Francesco Fidaleo for valuable discussion which helped
the author to realize that the tower of type-II1 sub-factors indeed generalizing well
known Jones construction.
2 Stationary Markov Processes and Markov shift:
A family (τt, t ≥ 0) of one parameter completely positive maps on a C
∗ algebra or
a von-Neumann sub-algebra A0 is called a quantum dynamical semigroup if
τ0 = I, τs ◦ τt = τs+t, s, t ≥ 0
Moreover if τt(I) = I, t ≥ 0 it is called a Markov semigroup. We say a state φ0
on A0 is invariant for (τt) if φ0(τt(x)) = φ0(x) ∀t ≥ 0. We fix a Markov semigroup
(A0, τt, t ≥ 0) and also a (τt)−invariant state φ0.
In the following we briefly recall [AM] the basic construction of the minimal
forward weak Markov processes associated with (A0, τt, t ≥ 0, φ0). The construction
goes along the line of Kolmogorov’s construction of stationary Markov processes or
Markov shift with a modification [Sa,BP] which takes care of the fact that A0 need
not be a commutative algebra. Here we review the construction given in [AM] in
order to fix the notations and important properties.
We consider the class M of A0 valued functions x : IT → A0 so that xr 6= I for
finitely many points and equip with the point-wise multiplication (xy)r = xryr. We
define the map L : (M,M) → IC by
L(x, y) = φ0(x
τrn−1−rn(x
(.....x∗r2τr1−r2(x
yr1)yr2)...yrn−1)yrn) (2.1)
where r = (r1, r2, ..rn) r1 ≤ r2 ≤ .. ≤ rn is the collection of points in IT when either
x or y are not equal to I. That this kernel is well defined follows from our hypothesis
that τt(I) = I, t ≥ 0 and the invariance of the state φ0 for (τt). The complete
positiveness of (τt) implies that the map L is a non-negative definite form on M.
Thus there exists a Hilbert space H and a map λ : M → H such that
< λ(x), λ(y) >= L(x, y).
Often we will omit the symbol λ to simplify our notations unless more then one such
maps are involved.
We use the symbol Ω for the unique element in H associated with x = (xr =
I, r ∈ IR) and φ for the associated vector state φ on B(H) defined by φ(X) =<
Ω,XΩ >.
For each t ∈ IR we define shift operator St : H → H by the following prescription:
(Stx)r = xr+t (2.2)
It is simple to note that S = ((St, t ∈ IR)) is a unitary group of operators on H with
Ω as an invariant element.
For any t ∈ IR we set
Mt] = {x ∈ M, xr = I ∀r > t}
and Ft] for the projection onto Ht], the closed linear span of {λ(Mt])}. For any
x ∈ A0 and t ∈ IT we also set elements it(x),∈ M defined by
it(x)r =
x, if r = t
I, otherwise
So the map V+ : H0 → H defined by
V+x = i0(x)
is an isometry of the GNS space {x :< x, y >φ0= φ0(x
∗y)} into H and a simple
computation shows that < y, V ∗+StV+x >φ0=< y, τt(x) >φ0 . Hence
P 0t = V
+StV+, t ≥ 0
where P 0t x = τt(x) is a contractive semigroup of operators on the GNS space asso-
ciated with φ0.
We also note that it(x) ∈ Mt] and set ⋆-homomorphisms j
0 : A0 → B(H0])
defined by
j00(x)y = i0(x)y
for all y ∈ M0]. That it is well defined follows from (2.1) once we verify that it
preserves the inner product whenever x is an isometry. For any arbitrary element
we extend by linearity. Now we define j
0 : A → B(H) by
0 (x) = j
0(x)F0]. (2.3)
Thus j
0 (x) is a realization of A0 at time t = 0 with j
0 (I) = F0]. Now we use the
shift (St) to obtain the process j
f = (j
t : A0 → B(H), t ∈ IR) and forward filtration
F = (Ft], t ∈ IR) defined by the following prescription:
t (x) = Stj
0 (x)S
t Ft] = StF0]S
t , t ∈ IR. (2.4)
So it follows by our construction that jfr1(y1)j
(y2)...j
(yn)Ω = y where yr = yri ,
if r = ri otherwise I, (r1 ≤ r2 ≤ .. ≤ rn). Thus Ω is a cyclic vector for the
von-Neumann algebra A generated by {jfr (x), r ∈ IR, x ∈ A0}.
From (2.4) we also conclude that StXS
t ∈ A whenever X ∈ A and thus we can
set a family of automorphism (αt) on A defined by
αt(X) = StXS
Since Ω is an invariant element for (St), φ is an invariant state for (αt). Now our
aim is to show that the reversible system (A, αt, φ) satisfies (1.1) with j0 as defined
in (2.4), for a suitable choice of IE0]. To that end, for any element x ∈ M, we verify
by the relation < y,Ft]x =< y, x > for all y ∈ Mt] that
(Ft]x)r =
xr, if r < t;
τrk−t(...τrn−1−rn−2(τrn−rn−1(xrn)xrn−1)...xt), if r = t
I, if r > t
where r1 ≤ .. ≤ rk ≤ t ≤ .. ≤ rn is the support of x. We also claim that
t (x)Fs] = j
s (τt−s(x)) ∀s ≤ t. (2.5)
For that purpose we choose any two elements y, y′ ∈ λ(Ms]) and check the following
steps with the aid of (2.2):
< y,Fs]j
t (x)Fs]y
′ >=< y, it(x)y
=< y, is(τt−s(x))y
′) > .
Since λ(Ms]) spans Hs] it complete the proof of our claim.
We also verify that < z, V ∗+j
t (x)V+y >φ0= φ0(z
∗τt(x)y), hence
V ∗+j
t (x)V+ = τt(x), ∀t ≥ 0. (2.6)
For any fix t ∈ IT let A[t 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] (2.7)
for all x ∈ A0. We consider the GNS Hilbert space (Hπφ0 , πφ0(A0), ω0) associ-
ated with (A0, φ0) and define a Markov semigroup (τ
t ) on π(A0) by τ
t (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
τπt (PXP ) = Pαt(X)P (2.8)
for all X ∈ A[0 where P is the projection from [A[0Ω] onto the GNS space [j
0 (A0)Ω]
which identified with the GNS space associated with (A0, φ0). It is enough if we verify
for typical elements X = js1(x1)...jsn(xn) for any s1, s2, ..., sn ≥ 0 and xi ∈ A0 for
1 ≤ i ≤ n and n ≥ 1. We use induction on n ≥ 1. If X = js(x) for some s ≥ 0,
(2.8) follows from (2.5). Now we assume that (2.8) 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 (2.5) to conclude (2.8) by our
induction hypothesis. Now suppose sn+1 ≤ sn. In such a case if sn−1 ≤ sn we appeal
once more to (2.5) and induction hypothesis to verify (2.8) 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 (2.5) at the end of the string and conclude that our claim is true for
any typical element X and hence 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 is a sub-harmonic projection [Mo1] for (αt : t ≥ 0)
i.e. αt(P ) ≥ P for all t ≥ 0 and αt(P ) ↑ [A[0Ω] as t ↑ ∞.
THEOREM 2.1: Let (A0, τt, φ0) be a Markov semigroup and φ0 be (τt)-invariant
state on a C∗ algebra A0. Then the GNS space [π(A0)Ω] associated with φ0 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 vector state φ(X) =< Ω,XΩ >, Ω ∈ H[0
invariant for (αt : t ≥ 0).
(b) PA[0P is isomorphic with π(A0)
′′ where P is the projection from [A[0Ω] onto
[jf (A0)Ω];
(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 property
Following the literature [Vi,Sa,BhP,Bh] on dilation we say (A[0, αt, φ) is the min-
imal E0semigroup associated with (A0, τt, φ0). We have studied extensively asymp-
totic behavior of the dynamics (A0, τt, φ0) in [AM] and Kolmogorov’s property of
the Markov semigroup introduced in [Mo1] was explored to asymptotic behavior of
the dynamics (A[0, αt, φ). In particular this yields a criteria for the inductive limit
state canonically associated with (A[0, αt, φ) to be pure. The notion is intimately
connected with the notion of a pure E0-semigroup introduced in [Po,Ar]. For more
details we refer to [Mo2].
3 Dual Markov semigroup and Time Reverse Markov
processes:
Now we are more specific and assume that A0 is a von-Neumann algebra and each
Markov map (τt) is normal and for each x ∈ A0 the map t → τt(x) is continuous in
the weak∗ topology. We assume further that φ0 is also faithful. Following [AM2],
in the following we briefly recall the time reverse process associated with the KMS-
adjoint ( or Petz-adjoint ) quantum dynamical semigroup (A, τ̃t, φ0).
Let φ0 be a faithful state and without loss of generality let also (A0, φ0) be
in the standard form (A0, J,P, ω0) [BR] where ω0 ∈ H0, a cyclic and separating
vector for A0, so that φ0(x) =< ω0, xω0 > and the closer of the close-able operator
S0 : xω0 → x
∗ω0, S possesses a polar decomposition S = J∆
1/2 with the self-dual
positive cone P as the closure of {JxJxω0 : x ∈ A0} in H0. Tomita’s [BR] theorem
says that ∆itA0∆
−it = A0, t ∈ IR and JA0J = A
0, where A
0 is the commutant of
A0. We define the modular automorphism group σ = (σt, t ∈ IR) on A0 by
σt(x) = ∆
itx∆−it.
Furthermore for any normal state ψ on A0 there exists a unique vector ζ ∈ P
so that ψ(x) =< ζ, xζ >. Note that J π(x)J π(y)Ω = J π(x)∆
2π(y∗)Ω =
2π(x)∆
2π(y∗)Ω = π(y)∆
2π(x∗)∆−
2Ω. Thus the Tomita’s map x →
J π(x)J is an anti-linear ∗-homomorphism representation of A0. This observation
leads to a notion called backward weak Markov processes [AM].
To that end we consider the unique Markov semigroup (τ ′t) on the commutant
A′0 of A0 so that φ(τt(x)y) = φ(xτ
t(y)) for all x ∈ A0 and y ∈ A
0. We define weak
continuous Markov semigroup (τ̃t) on A0 by τ̃t(x) = Jτ
t(JxJ)J. Thus we have the
following adjoint relation
φ0(σ1/2(x)τt(y)) = φ0(τ̃t(x)σ−1/2(y)) (3.1)
for all x, y ∈ A0, analytic elements for (σt). One can as well describe the adjoint
semigroup as Hilbert space adjoint of a one parameter contractive semigroup (Pt)
on a Hilbert space defined by Pt : ∆
1/4xω0 = ∆
1/4τt(x)ω0. For more details we refer
to [Ci].
We also note that it(x) ∈ M[t and set ⋆ anti-homomorphisms j
0 : A0 → B(H[0)
defined by
jb0(x)y = yi0(σ− i
(x∗))
for all y ∈ M[0. That it is well defined follows from (2.1) once we verify by KMS
relation that it preserves the inner product whenever x is an isometry. For any
arbitrary element we extend by linearity. Now we define jb0 : A → B(H) by
jb0(x) = j
0(x)F[0. (3.2)
Thus jb0(x) is a realization of A0 at time t = 0 with j
0(I) = F[0. Now we use the
shift (St) to obtain the process j
b = (jbt : A0 → B(H), t ∈ IR) and forward filtration
F = (F[t, t ∈ IR) defined by the following prescription:
jbt (x) = Stj
0(x)S
t F[t = StF[0S
t , t ∈ IR. (3.3)
A simple computation shows for −∞ < s ≤ t <∞ that
s(x)F[t = j
t (τ̃t−s(x)) (3.4)
for all x ∈ A0. It also follows by our construction that j
(y1)j
(y2)...j
(yn)Ω =
(y) where yr = yri , if r = ri otherwise I, (r1 ≥ r2 ≥ .. ≥ rn). Thus Ω is a cyclic
vector for the von-Neumann algebra Ab generated by {jbr(x), r ∈ IR, x ∈ A0}
′′. We
also von-Neumann algebra Ab
generated by {jbr(x), r ≤ t, x ∈ A0}
′′. The following
theorems say that there is a duality between the forward and backward weak Markov
processes.
THEOREM 3.1: [AM] We consider the weak Markov
processes (A,H, Ft], F[t, St, j
t , j
t t ∈ IR, Ω) associated with (A0, τt, t ≥ 0, φ0)
and the weak Markov processes (Ã, H̃, F̃t], F̃[t, S̃t, j̃
t , j̃
t , t ∈ IR, Ω̃) associated with
(A0, τ̃t, t ≥ 0, φ0). There exists an unique anti-unitary operator U0 : H → H̃ so
(a) U0Ω = Ω̃;
(b) U0StU
0 = S̃−t for all t ∈ IR;
(c) U0j
t (x)U
0 = j̃
−t(x), U0J
t (x)U0 = j̃
−t(x) for all t ∈ IR;
(d) U0Ft]U
0 = F̃[−t, U0F[tU
0 = F̃−t] for all t ∈ IR;
THEOREM 3.2: Let (A0, τt, φ0) be as in Theorem 3.1 with φ0 as faithful. Then
the commutant of A[t is A
for each t ∈ IR.
PROOF: It is obvious that A[0 is a subset of the commutant of A
. Note also
that F[0 is an element in A
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 ∈ (Ab
)′ 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 ∈ A
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
F[0 ⊆ F[0A
F[0. Let X ∈ F[0A
F[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 F[0A
F[0 = j
b(A0)
Thus there exists an element Y ∈ Ab
so that XΩ = YΩ. 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
we conclude that F[0A
F[0 = F[0A
F[0 being a projection in A
we verify that F[0(A
)′F[0 ⊆ (F[0A
′ and so
we also have F[0(A
)′F[0 ⊆ (F[0A
′ as Ab
. Thus it is enough if we prove
[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
F[0 so that XΩ = Y Ω. Thus for any Z ∈ A[0 we have
XZΩ = Y ZΩ and thus XF[0 = Y F[0. Hence X = Y ∈ F[0A
F[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)
′ = αt(A
αt being an automorphism. This completes the proof as αt(A
) = Ab
by our
construction.
4 Subfactors:
In this section we will investigate further the sequence of von-Neumann algebra
{A[t : t ∈ IR} defined in the last section with an additional assumption that φ0
is also faithful and thus we also have in our hand backward von-Neumann algebras
: t ∈ IR}.
PROPOSITION 4.1: Let (A0, τt, φ0) be a Markov semigroup with a faithful
normal invariant state φ0. If A0 is a factor then A[0 is a factor. In such a case the
following also hold:
(a) A0 is type-I (type-II, type-III) if and only if A[0 is type-I (type -II , type-III)
respectively;
(b) H is separable if and only if H0 is separable;
(c) If H0 is separable then A0 is hyper-finite if and only if A[0 is hyper-finite.
PROOF: We first show factor property of A[0. Note that the von-Neumann algebra
generated by the backward process {jbs(x) : s ≤ 0, x ∈ A0} is a sub-algebra
of A′
, the commutant of A[0. We fix any X ∈ A[0
in the center. Then for
any y ∈ A0 we verify that Xj0(y)Ω = XF0]j0(y)Ω = F0]XF0]j0(y)Ω = j0(xy)Ω
for some x ∈ A0. Since Xj0(y) = j0(y)X we also have j0(xy)Ω = j0(yx)Ω. By
faithfulness of the state φ0 we conclude xy = yx thus x must be a scaler. Thus we
have Xj0(y)Ω = cj0(y)Ω for some scaler c ∈ IC. Now we use the property that X
commutes with forward process jt(x) : x ∈ A0, t ≥ 0 and as well as the backward
processes {jbt (x), t ≤ 0} to conclude that Xλ(t, x) = cλ(t, x). Hence X = c. Thus
A[0 is a factor.
Now if A0 is a type-I factor, then there exists a non-zero minimal projection
p ∈ A0. In such a case we claim that j0(p) is also a minimal projection in A[0. To
that end let X be any projection in A[0 so that X ≤ j0(p). Since F0]A[0F0] = j0(A0)
we conclude that F0]XF0] = j0(x) for some x ∈ A0. Hence X = j0(p)Xj0(p) =
F0]Xj0(p) = j0(xp) = j0(px) Thus by faithfulness of the state φ0 we conclude that
px = xp. Hence X = j0(q) where q is a projection smaller then equal to p. Since p
is a minimal projection in A0, q = p or q = 0 i.e. X = j0(p) or 0. So j0(p) is also
a minimal projection. Hence A[0 is a type-I factor. For the converse statement we
trace the argument in the reverse direction. Let p be a non-zero projection in A0 and
claim that there exists a minimal projection q ∈ A0 so that 0 < q ≤ p. Now since
j0(p) is a non-zero projection in a type-I factor A[0 there exists a non-zero projection
X which is minimal in A[0 so that 0 < X ≤ j0(p). Now we repeat the argument to
conclude that X = j0(q) for some projection q. Since X 6= 0 and minimal, q 6= 0
and minimal in A0. This completes the proof for type-I case. We will prove now the
case for Type-II.
Let A[0 be type-II then there exists a finite projection X ≤ F0]. Once more
X = F0]XF0] = j0(x) for some projection x ∈ A0. We claim that x is finite. To
that end let q be another projection so that q ≤ x and q = uu∗ and u∗u = x. Then
j0(q) ≤ j0(x) = X and j0(q) = j0(u)j0(u)
∗ and j0(x) = j0(u)
∗j0(u). Since X is
finite in A[0 we conclude that j0(q) = j0(x). By faithfulness of φ0 we conclude that
q = x, hence x is a finite projection. Since A0 is not type-I, it is type-II. For the
converse let A0 be type-II. So A[0 is either type-II or type-III. We will rule out that
the possibility for type-III. Suppose not, i.e. if A[0 is type-III, for every projection
p 6= 0, there exists u ∈ A[0 so that j0(p) = uu
∗ and F0] = u
∗u. In such a case
j0(p)u = uF0]. Set j0(v) = F0]uF0] for some v ∈ A0. Thus j0(pv) = j0(v). Once
more by faithfulness of the normal state φ0, we conclude pv = v. So j0(v) = uF0].
Hence j0(v
∗v) = F0]. Hence v
∗v = 1 by faithfulness of φ0. Since this is true for any
non-zero projection p in A0, A0 is type-III, which is a contradiction. Now we are left
to show the statement for type-III, which is true since any factor needs to be either
of these three types. This completes the proof for (a).
(b) is obvious if IT is ZZ. In case IT = IR, we use our hypothesis that the map
(t, x) → τt(x) is sequentially jointly continuous with respect to weak
∗ topology.
For (c) we first recall from [Co] that hyper-finiteness property, being equivalent to
injective property of von-Neumann algebra, is stable under commutant and countable
intersection operation when they are acting on a separable Hilbert space. Let A0
be hyper-finite and H0 be separable. We will first prove A[0 is hyper-finite when
IT = ZZ, i.e. time variable are integers. In such a case for each n ≥ 0, jn being
injective, jn(A0)
′′ = {j0(x) : x ∈ A0}
′′ is a hyper-finite von-Neumann algebra. Thus
A[0 = {jn(A0)
′′ : n ≥ 0}′′ is also hyper-finite as they are acting on a separable
Hilbert space. In case IT = IR, for each n ≥ 1 we set von-Neumann sub-algebras
⊆ A[0 generated by the elements {jt(A0)
′′ : t = r
, 0 ≤ r ≤ n2n}. Thus each An
is hyper-finite. Since A′
n≥0(A
)′ by weak∗ continuity of the map t → τt(x),
we conclude that A[0 is also hyper-finite being generated by a countable family of
increasing hyper-finite von-Neumann algebras.
For the converse we recall for a factorM acting on a Hilbert spaceH, Tomiyama’s
property ( i.e. there exists a norm one projection E : B(H) → M, see [BR1] page-
151 for details ) is equivalent to hyper-finite property. For a hyper-finite factor A[0,
j0(A0) is a factor in the GNS space identified with the subspace F0]. Let E be the
norm one projection from B(H[0) on A[0 and verify that the completely positive map
E0 : B(H0) → A0 defined by E0(X) = F0]E(F0]XF0])F0] is a norm one projection
from B(F0]) to A0. This completes the proof for (b).
PROPOSITION 4.2: Let (A0, τt, φ0) be a dynamical system as in Proposition
4.1. If A[0 is a type-II1 factor which admits a unique normalize faithful normal
tracial state then the following hold:
(a) Ft] = I for all t ∈ IR;
(b) τ = (τt) is a semigroup of ∗−endomorphisms.
(c) A[0 = j0(A0).
PROOF: Let tr0 be the unique normalize faithful normal trace on A[0. For any fix
t ≥ 0 we set a normal state φt on A[0 by φt(x) = tr0(αt(x)). It is simple to check
that it is also a faithful normal trace. Since αt(I) = I, by uniqueness φt = tr0.
In particular tr0(F0]) = tr0(αt(F0]) = tr0(Ft]), by faithful property Ft] = F0] for all
t ≥ 0. Since Ft] ↑ 1 as t→ ∞ we have F0] = I. Hence Ft] = αt(F0]) = I for all t ∈ IR.
This proves (a). For (b) and (c) we recall that F0]jt(x)F0] = j0(τt(x)) for all t ≥ 0
and jt : A0 → A[t is an injective ∗− homomorphism. Since Ft] = F0] = I we have
jt(x) = F0]jt(x)F0] = j0(τt(x)). Hence A[0 = j0(A0) and j0(τt(x)τt(y)) = j0(τt(xy))
for all x, y ∈ A0. Now by injective property of j0, we verify (b). This completes the
proof.
We fix a type-II1 factor A0 which admits a unique normalize faithful normal
tracial state. Since A[0 is a type-II factor whenever A0 is so, we conclude that A[0 is
a type-II∞ factor whenever τt is not an endomorphism on a such a type-II1 factor.
The following proposition says much more.
PROPOSITION 4.3: Let A0 be a type-II1 factor with a unique normalize normal
trace and (A0, τt, φ0) be a dynamical system as in Proposition 4.1. Then the following
hold:
(a) j0(I) is a finite projection in A[−t for all t ≥ 0.
(b) For each t ≥ 0 Mt = j0(I)A[−tj0(I) is a type-II1 factor and M0 ⊆ Ms... ⊆
Mt ⊆ .., t ≥ s ≥ 0 are acting on Hilbert space F0] where M0 = j0(A0).
PROOF: By Proposition 4.1 A[0 is a type-II factor. Thus A[0 is either type-II1 or
type-II∞. In case it is type-II1, Proposition 4.2 says that A[−t is j0(A0), hence the
statements (a) and (b) are true with Mt = j0(A0). Thus it is good enough if we
prove (a) and (b) when A[0 is indeed a type-II∞ factor. To that end for any fix t ≥ 0
we fix a normal faithful trace tr on A[−t and consider the normal map x→ j0(x) and
thus a normal trace trace on A0 defined by x→ tr(j0(x)) for x ∈ A0. It is a normal
faithful trace on A0 and hence it is a scaler multiple of the unique trace on A0. A0
being a type-II1 factor, j0(I) is a finite projection in A[−t. Now the general theory
on von-Neumann algebra [Sa] guarantees that Mt is type-II1 factor and inclusion
follows as A−s] ⊆ A−t] whenever t ≥ s. That j0(A0) = j0(I)A[0j0(I) follows from
Proposition 4.1.
We have now one simple but useful result.
COROLLARY 4.4: Let (A0, τt, φ0) be as in Proposition 4.1. Then one of the
following statements are false:
(a) A = B(H)
(b) A0 is a type-II1 factor.
PROOF : Suppose both (a) and (b) are true. Let φt be the unique normalized trace
on Mt. As they are acting on the same Hilbert space, we note by uniqueness that φt
is an extension of φs for t ≥ s. Thus there exists a normal extension of (φt) to weak
completion M of
t≥0 Mt ( here we can use Lemma 13 page 131 [Sc] ). However if
A = B(H), M is equal to B(H0]). H0] being an infinite dimensional Hilbert space
we arrive at a contradiction.
In case M in Corollary 4.4 is a type-II1 factor, by uniqueness of the tracial
state we claim that λ(t)λ(s) = λ(t + s) where λ(t) = tr(F−t]) for all t ≥ 0. The
claim follows as von-Neumann algebra M is isomorphic with α−t(M) which is equal
to F−t]MF−t]. The map t → λ(t) being continuous we get λ(t) = exp(λt) for some
λ ≤ 0. If λ = 0 then λ(t) = 1 so by faithful property of the trace we get F−t] = F0] for
all t ≥ 0. Hence we conclude that (τt) is a family of endomorphisms by Proposition
4.2. Now for λ < 0 we have tr(F−t]) → 0 as t → ∞. As F−t] ≥ |Ω >< Ω| for all
t ≥ 0, we draw a contradiction. Thus the weak∗ completion of
t≥0 Mt is a type-II1
factor if and only if (τt) is a family of endomorphism. In otherwords if (τt) is not
a family of endomorphism then the weak∗ completion of
t≥0 Mt is not a type-II1
factor and the tracial state though exists on M is not unique.
5 Jones index of a quantum dynamical semigroup on
II1 factor:
We first recall Jones’s index of a sub-factor originated to understand the structure of
inclusions of von Neumann factors of type II1. Let N be a sub-factor of a finite factor
M . M acts naturally as left multiplication on L2(M, tr), where tr be the normalize
normal trace. The projection E0 = [Nω] ∈ N
′, where ω is the unit trace vector i.e.
tr(x) =< ω, xω > for x ∈ M , determines a conditional expectation E(x) = E0xE0
on N . If the commutant N ′ is not a finite factor, we define the index [M : N ] to
be infinite. In case N ′ is also a finite factor, acting on L2(M, tr), then the index
[M : N ] of sub-factors is defined as tr(E0)
−1, which is the Murray-von Neumann
coupling constant [MuN] of N in the standard representation L2(M, tr). Clearly
index is an invariance for the sub-factors. Jones proved [M : N ] ∈ {4 cos2(π/n) :
n = 3, 4, · · ·} ∪ [4,∞] with all values being realized for some inclusion N ⊆M .
In this section we continue our investigation in the general framework of section
4 and study the case when A0 is type-II1 which admits a unique normalize faithful
normal tracial state and (τt) is not an endomorphism on such a type-II1 factor. By
Proposition 4.3 A[0 is a type-II∞ factor and (Mt : t ≥ 0) is a family of increasing
type-II1 factor where Mt = j0(I)A[−tj0(I) for all t ≥ 0. Before we prove to discrete
time dynamics we here briefly discuss continuous case. Thus the map I : (t, s) →
[Mt : Ms], 0 ≤ s ≤ t is an invariance for the Markov semigroup (A0, τt, φ0). By our
definition I(t, t) = 1 for all t ≥ 0 and range of values Jones’s index also says that the
map (s, t) → I(s, t) is not continuous at (s, s) for all s ≥ 0. Being a discontinuous
map we also claim the map (s, t) → I(s, t) is not time homogeneous, i.e. I(s, t) 6=
I(0, t − s) for some 0 ≤ s ≤ t. If not we could have I(0, s + t) = I(0, s)I(s, s + t) =
I(0, s)I(0, t), i.e. I(0, t) = exp(λt) for some λ, this leads to a contradiction. The
non-homogenity suggest that I is far from being simple. We devote rest of the section
discussing a much simple example in discrete time dynamics.
To that end we review now Jones’s construction [Jo, OhP]. Let A0 be a type-
II1 factor and φ0 be the unique normalize normal trace. The algebra A0 acts on
L2(A0, φ0) by left multiplication π0(y)x = yx for x ∈ L
2(A0, φ0). Let ω be the
cyclic and separating trace vector in L2(A0, φ0). The projection E0 = [B0ω] induces
a trace preserving conditional expectation τ : a → E0aE0 of A0 onto B0. Thus
E0π0(y)E0 = E0π0(E(y))E0 for all y ∈ A0. Let A1 be the von-Neumann algebra
{π0(A0), E0}
′′. A1 is also a type-II1 factor and A0 ⊆ A1, where we have identified
π0(A0) with A0. Jones proved that [A1 : A0] = [A0 : B0]. Now by repeating this
canonical method we get an increasing tower of type-II1 factors A1 ⊆ A2... so that
[Ak+1 : Ak] = [A0 : B0] for all k ≥ 0. Thus the natural question: Is Jones tower
A0 ⊆ A1 ⊆ ... ⊆ Ak... related with the tower M0 ⊆ M1...Mk ⊆ Mk+1 defined in
Proposition 4.3 associated with the dynamics (A0, τn, φ0)?
To that end recall the von-Neumann sub-factors M0 ⊆ M1 and the induced
representation of M1 on Hilbert subspace H[−1,0] generated by {j0(x0)j−1(x−1)Ω :
x0, x−1 ∈ A0}. Ω is the trace vector for M0 i.e. φ0(x) =< Ω, j0(x)Ω >. However
the vector state given by Ω is not the trace vector for M1 as M1 6= M0 ( If so we
check by trace property that φ0(τ(x)yτ(z)) = φ(j0(x)j−1(y)j0(z)) = φ0(τ(zx)y) for
any x, y, z ∈ A0 and so τ(zx) = τ(z)τ(x) for all z, x ∈ A0. Hence by Proposition 4.2
we have M1 = M0). Nevertheless M1 being a type-II1 factor there exists a unique
normalize trace on M1.
PROPOSITION 5.1: M1 ≡ A2 and [M1 : M0] = d
2 where d = [A0 : B0].
PROOF: Let φ1 be the unique normalize normal trace on A1 and H1 = L
2(A1, φ1).
We consider the left action π1(x) : y → xy of A1 on H1. Thus π0(A0) is also acting
on H1. Since E0π0(x)E0 = E0π0(τ(x))E0 = E0π0(τ(x)), for any element X ∈ A1,
E0X = E0π0(x) for some x ∈ A0. Thus π1(E0) is the projection on the subspace
{E0π0(x) : x ∈ A0}.
For any y ∈ A0 we set
(a) k−1(y) on the subspace π1(E0) by k−1(y)E0π0(x) = E0π0(yx) for x ∈ A0 and
extend it to H1 trivially. That k−1(y) is well defined and an isometry for an isometry
y follows from the following identities:
φ1((E0π0(yz))
∗E0π0(yx)) = φ1(π0(z
∗y∗)E0π0(yx))
= φ1(E0π0(yx)π0(z
∗y∗)) by trace property
= φ1(E0)φ0(π0(yx)π0(z
∗y∗)) being a trace and φ1(E0π0(x)) = φ1(E0)φ0(π0(x))
= φ1(E0)φ0(π0(z
∗y∗)π0(yx)) = φ1(E0)φ0(π0(z
= φ1((E0π0(z))
∗E0π0(x))
(b) k0(y)x = π0(y)x for x ∈ A1. Thus y → k0(y) is an injective ∗-representation of
A0 in L
2(A1, φ1).
For y, z ∈ A0 we verify that
< E0π0(y), k−1(1)k0(x)k−1(1)E0π0(z) >1=< E0π0(y), E0π0(x)E0π0(z) >1
=< E0π0(y), E0π0(τ(x))E0π0(z) >1
=< E0π0(y), E0π0(τ(x))π0(z) >1
Thus k−1(1)k0(x)k−1(1) = k−1(τ(x)) for all x ∈ A0. Note that k−1(1) = π1(E0)
and the identity operator in H1 is a cyclic vector for the von-Neumann algebra
{k0(x), k−1(x), x ∈ A0}
′′. We have noted before that the vector Ω need not be
the tracial vector for M1 and also verify by a direct computation that the space
{k0(y)k−1(x)1 : x, y ∈ A0} is equal to {yE0x : y, x ∈ A0} which is a proper subspace
of L2(A1, φ1).
Now we claim that the type-II1 factor M1 is isomorphic to the von-Neumann
algebra {k−1(x), k0(x), x ∈ A0)}
′′. To that end we define an unitary operator from
L2(A1, φ1) to L
2(M1, tr1) by taking an element kt1(x1)..ktn(xn) to jt1(x1)..jtn(xn),
where tk are either 0 or −1. That it is an unitary operator follows by the tracial
property of the respective states and weak Markov property of the homomorphisms.
We leave the details and without lose of generality we identify these two weak Markov
processes. Since k−1(1) = π1(E0), we conclude that π1(A1) ⊆ M1. In fact strict
inclusion hold unless B0 = A0.
However by our construction A1 = π0(A0) ∨ E0 is acting on L
2(A0, φ0) and
A2 = π1(A1) ∨ E1 is acting on L
2(A1, φ1) where E1 is the cyclic subspace of 1
generated by π1(π0(A0)) i.e. E1 = [π1(π0(A0))1]. From (a) we also have
k−1(y)π1(E0)E1 = π1(E0)k0(y)E1 (5.1)
for all y ∈ A0.
By Temperley-Lieb relation [Jo] we have π1(E0)E1π1(E0) =
π1(E0) and thus
post multiplying (5.1) by π1(E0) we have
k−1(y)π1(E0) = π1(E0)k0(y)E1π1(E0) (5.2)
So it is clear now that k−1(y) ∈ π1(A1) ∨ E1 for all y ∈ A0. Thus π1(A1) ⊆ M1 ⊆
π1(A1) ∨E1 = A2.
We claim also that E1 ∈ M1. We will show that any unitary ele-
ment u commuting with M1 is also commuting with E1. By (5.2) we have
π1(E0)k0(y)(uE1u
∗ − E1)π1(E0) = 0 for all y ∈ A0. By taking adjoint we have
π1(E0)(uE1u
∗ − E1)k0(y)π1(E0) = 0 for all y ∈ A0. Since A1 = π0(A0) ∨
E0 we conclude by cylicity of the trace vector that π1(E0)(uE1u
∗ − E1) =
0. So we have E1π1(E0)uE1u
∗ = E1π1(E0)E1 = E1 by Temperley-Lieb rela-
tion. So u∗E1uπ1(E0) = u
∗E1π1(E0)uE1u
∗u = u∗E1u By taking adjoint we get
π1(E0)u
∗E1u = u
∗E1u. Since same is true for u
∗, we conclude that u∗E1u = E1 for
any unitary u ∈ M′1. Hence E1 ∈ M1.
Hence M1 = A2. Since [M1 : M0] = [M1 : A1][A1 : M0] and [A1 : A0] = [A0 :
B0] = d, we conclude the result.
THEOREM 5.2: Mm ≡ A2m for all m ≥ 1.
PROOF: Proposition 5.1 gives a proof for m = 1. The proof essentially follows the
same steps as in Proposition 5.1. We use induction method for m ≥ 1. Assume it is
true for 1, 2, ..m. Now consider the Hilbert space L2(A2m+1, tr2m+1), m ≥ 1 and we
set homomorphism k−1, k0 from A2m into B(L
2(A2m+1, tr2m+1)) in the following:
(a) k0(x)y = xy for all y ∈ A2m+1 and x ∈ A2m
(b) π2m+1(E2m) is the projection on the subspace {E2my : y ∈ A2m} and k−1(x)
defined on the subspace E2m by
k−1(x)E2my = E2mxy
for all x ∈ A2m and y ∈ A2m. That k−1 is an homomorphism follows as in Proposition
5.1. Thus an easy adaptation of Proposition 5.1 says that M = {k0(x), k−1(x) :
x ∈ A2m}
′′ is a type-II1 factor and proof will be complete once we show that it is
isomorphic to Mm+1.
To that end we check as in Proposition 5.1 that
k−1(E2m) = k−1(I), k−1(I)k0(x)k−1(I) = k−1(E2mxE2m)
for all x ∈ A2m and k−1(x)E2m = E2mk0(x)E2m+1 where E2m+1 is the cyclic space
of the trace vector generated by A2m. Thus following Proposition 5.1 we verify
now that type-II1 factor M = {k0(x), k−1(I), E2m+1, x ∈ A2m}
′′ is isomorphic to
Mm+1 = {J−1(x), J0(x) x ∈ Am}
′′, where we used notation J−1(x) = x for all
x ∈ Am, J0(x) = SJ−1(x)S
∗ for all x ∈ A2m where we have identified A2m ≡ Mm
with {jk(x),−m − 1 ≤ k ≤ −1, x ∈ A0}
′′ and S is the (right) Markov shift. This
completes the proof.
REFERENCES
• [AM] Accardi, L., Mohari, A.: Time reflected Markov processes. Infin. Dimens.
Anal. Quantum Probab. Relat. Top., vol-2, no-3, 397-425 (1999).
• [Ar] Arveson, W.: Pure E0-semigroups and absorbing states, Comm. Math.
Phys. 187 , no.1, 19-43, (1997)
• [Bh] Bhat, B.V.R.: An index theory for quantum dynamical semigroups, Trans.
Amer. Maths. Soc. vol-348, no-2 561-583 (1996).
• [BP] Bhat, B.V.R., Parthasarathy, K.R.: Kolmogorov’s existence theorem for
Markov processes on C∗-algebras, Proc. Indian Acad. Sci. 104,1994, p-253-
• [BR] Bratelli, Ola., Robinson, D.W. : Operator algebras and quantum statis-
tical mechanics, I,II, Springer 1981.
• [Da] Davies, E.B.: Quantum Theory of open systems, Academic press, 1976.
• [El] Elliot. G. A.: On approximately finite dimensional von-Neumann algebras
I and II, Math. Scand. 39 (1976), 91-101; Canad. Math. Bull. 21 (1978), no.
4, 415–418.
• [Jo] Jones, V. F. R.: Index for subfactors. Invent. Math. 72 (1983), no. 1,
1–25.
• [Mo1] Mohari, A.: Markov shift in non-commutative probability, Jour. Func.
Anal. 199 (2003) 189-209.
• [Mo2] Mohari, A.: Pure inductive limit state and Kolmogorov’s property, ....
• [Mo3] Mohari, A.: SU(2) symmetry breaking in quantum spin chain, ....
• [MuN] Murray, F. J.; von Neumann, J., On rings of operators. (English)[J]
Ann. Math., Princeton, (2)37, 116-229.
• [OP] Ohya, M., Petz, D.: Quantum entropy and its use, Text and monograph
in physics, Springer-Verlag 1995.
• [Po] Powers, Robert T.: An index theory for semigroups of ∗-endomorphisms
of B(H) and type II1 factors. Canad. J. Math. 40 (1988), no. 1, 86–114.
• [Sak] Sakai, S.: C∗-algebras and W∗-algebras, Springer 1971.
• [Sa] Sauvageot, Jean-Luc: Markov quantum semigroups admit covariant
Markov C∗-dilations. Comm. Math. Phys. 106 (1986), no. 1, 91103.
• [Vi] Vincent-Smith, G. F.: Dilation of a dissipative quantum dynamical system
to a quantum Markov process. Proc. London Math. Soc. (3) 49 (1984), no.
1, 5872.

  In this paper we consider a semigroup of completely positive maps
$\tau=(\tau_t,t \ge 0)$ with a faithful normal invariant state $\phi$ on a
type-$II_1$ factor $\cla_0$ and propose an index theory. We :achieve this via a
more general Kolmogorov's type of construction for stationary Markov processes
which naturally associate a nested isomorphic von-Neumann algebras. In
particular this construction generalizes well known Jones construction
associated with a sub-factor of type-II$_1$ factor.

Introduction:
Let τ = (τt, t ≥ 0) be a semigroup of identity preserving completely positive normal
maps [Da,BR] on a von-Neumann algebra A0 acting on a separable Hilbert space
H0, where either the parameter t ∈R+, the set of positive real numbers orZ
+, the
set of positive integers. In case t ∈ R+, i.e. continuous, we assume that for each
x ∈ A0 the map t→ τt(x) is continuous in the weak
∗ topology. Thus variable t ∈ IT+
where IT is either IR or IN . We assume further that (τt) admits a normal invariant
state φ0, i.e. φ0τt = φ0∀t ≥ 0.
As a first step following well known Kolmogorov’s construction of stationary
Markov processes, we employ GNS method to construct a Hilbert space H and an
increasing tower of isomorphic von-Neumann type−II factors {A[t : t ∈ R or Z}
generated by the weak Markov process (H, jt, Ft], t ∈ R or Z,Ω) [BP,AM] where
jt : A0 → A[t is an injective homomorphism from A0 into A[0 so that the projection
Ft] = jt(I) is the cyclic space of Ω generated by {js(x) : −∞ < s ≤ t, x ∈ A0}.
The tower of increasing isomorphic von-Neumann algebras {A[t, t ∈ R or Z} are
indeed a type-II∞ factor if and only if τ is not an endomorphism. In any case the
projection j0(I) is a finite projection in A[−t for all t ≤ 0. In particular we also find
an increasing tower of type-II1 factors {Mt : t ≥ 0} defined by Mt = j0(I)A[−tj0(I).
Thus Jones in-dices {[Mt : Ms] : 0 ≤ s ≤ t} are invariance for the Markov semigroup
(A0, τt, t ≥ 0, φ0) and further the map (t, s) → [Mt : Ms] is not continuous if the
variable (t, s) are continuous i.e. if τ = (τt : t ∈ IR+). In discrete time dynamical
system we find an invariance sequence {[Mn+1 : Mn] : n ≥ 0} canonically associated
with the canonical conditional expectation on a sub-factor B0 of a type-II1 factor A0
where φ0 is the unique normal trace on A0. However unlike Jones construction we
have [Mn+1 : Mn] = d
2 where d = [A0 : B0]. This shows that our construction in
a sense generalizes two step Jones construction in discrete time. A detailed study,
needs to be done to explore this new invariance, which seems to be an interesting
problem!
Acknowledgment: The author takes the opportunity to acknowledge Prof. Luigi
Accardi for an invitation to visit Centro Vito Volterra, University of Rome, Tor
Vergata during the summer 2005. The author further gratefully acknowledge Prof.
Roberto Longo and Prof. Francesco Fidaleo for valuable discussion which helped
the author to realize that the tower of type-II1 sub-factors indeed generalizing well
known Jones construction.
2 Stationary Markov Processes and Markov shift:
A family (τt, t ≥ 0) of one parameter completely positive maps on a C
∗ algebra or
a von-Neumann sub-algebra A0 is called a quantum dynamical semigroup if
τ0 = I, τs ◦ τt = τs+t, s, t ≥ 0
Moreover if τt(I) = I, t ≥ 0 it is called a Markov semigroup. We say a state φ0
on A0 is invariant for (τt) if φ0(τt(x)) = φ0(x) ∀t ≥ 0. We fix a Markov semigroup
(A0, τt, t ≥ 0) and also a (τt)−invariant state φ0.
In the following we briefly recall [AM] the basic construction of the minimal
forward weak Markov processes associated with (A0, τt, t ≥ 0, φ0). The construction
goes along the line of Kolmogorov’s construction of stationary Markov processes or
Markov shift with a modification [Sa,BP] which takes care of the fact that A0 need
not be a commutative algebra. Here we review the construction given in [AM] in
order to fix the notations and important properties.
We consider the class M of A0 valued functions x : IT → A0 so that xr 6= I for
finitely many points and equip with the point-wise multiplication (xy)r = xryr. We
define the map L : (M,M) → IC by
L(x, y) = φ0(x
τrn−1−rn(x
(.....x∗r2τr1−r2(x
yr1)yr2)...yrn−1)yrn) (2.1)
where r = (r1, r2, ..rn) r1 ≤ r2 ≤ .. ≤ rn is the collection of points in IT when either
x or y are not equal to I. That this kernel is well defined follows from our hypothesis
that τt(I) = I, t ≥ 0 and the invariance of the state φ0 for (τt). The complete
positiveness of (τt) implies that the map L is a non-negative definite form on M.
Thus there exists a Hilbert space H and a map λ : M → H such that
< λ(x), λ(y) >= L(x, y).
Often we will omit the symbol λ to simplify our notations unless more then one such
maps are involved.
We use the symbol Ω for the unique element in H associated with x = (xr =
I, r ∈ IR) and φ for the associated vector state φ on B(H) defined by φ(X) =<
Ω,XΩ >.
For each t ∈ IR we define shift operator St : H → H by the following prescription:
(Stx)r = xr+t (2.2)
It is simple to note that S = ((St, t ∈ IR)) is a unitary group of operators on H with
Ω as an invariant element.
For any t ∈ IR we set
Mt] = {x ∈ M, xr = I ∀r > t}
and Ft] for the projection onto Ht], the closed linear span of {λ(Mt])}. For any
x ∈ A0 and t ∈ IT we also set elements it(x),∈ M defined by
it(x)r =
x, if r = t
I, otherwise
So the map V+ : H0 → H defined by
V+x = i0(x)
is an isometry of the GNS space {x :< x, y >φ0= φ0(x
∗y)} into H and a simple
computation shows that < y, V ∗+StV+x >φ0=< y, τt(x) >φ0 . Hence
P 0t = V
+StV+, t ≥ 0
where P 0t x = τt(x) is a contractive semigroup of operators on the GNS space asso-
ciated with φ0.
We also note that it(x) ∈ Mt] and set ⋆-homomorphisms j
0 : A0 → B(H0])
defined by
j00(x)y = i0(x)y
for all y ∈ M0]. That it is well defined follows from (2.1) once we verify that it
preserves the inner product whenever x is an isometry. For any arbitrary element
we extend by linearity. Now we define j
0 : A → B(H) by
0 (x) = j
0(x)F0]. (2.3)
Thus j
0 (x) is a realization of A0 at time t = 0 with j
0 (I) = F0]. Now we use the
shift (St) to obtain the process j
f = (j
t : A0 → B(H), t ∈ IR) and forward filtration
F = (Ft], t ∈ IR) defined by the following prescription:
t (x) = Stj
0 (x)S
t Ft] = StF0]S
t , t ∈ IR. (2.4)
So it follows by our construction that jfr1(y1)j
(y2)...j
(yn)Ω = y where yr = yri ,
if r = ri otherwise I, (r1 ≤ r2 ≤ .. ≤ rn). Thus Ω is a cyclic vector for the
von-Neumann algebra A generated by {jfr (x), r ∈ IR, x ∈ A0}.
From (2.4) we also conclude that StXS
t ∈ A whenever X ∈ A and thus we can
set a family of automorphism (αt) on A defined by
αt(X) = StXS
Since Ω is an invariant element for (St), φ is an invariant state for (αt). Now our
aim is to show that the reversible system (A, αt, φ) satisfies (1.1) with j0 as defined
in (2.4), for a suitable choice of IE0]. To that end, for any element x ∈ M, we verify
by the relation < y,Ft]x =< y, x > for all y ∈ Mt] that
(Ft]x)r =
xr, if r < t;
τrk−t(...τrn−1−rn−2(τrn−rn−1(xrn)xrn−1)...xt), if r = t
I, if r > t
where r1 ≤ .. ≤ rk ≤ t ≤ .. ≤ rn is the support of x. We also claim that
t (x)Fs] = j
s (τt−s(x)) ∀s ≤ t. (2.5)
For that purpose we choose any two elements y, y′ ∈ λ(Ms]) and check the following
steps with the aid of (2.2):
< y,Fs]j
t (x)Fs]y
′ >=< y, it(x)y
=< y, is(τt−s(x))y
′) > .
Since λ(Ms]) spans Hs] it complete the proof of our claim.
We also verify that < z, V ∗+j
t (x)V+y >φ0= φ0(z
∗τt(x)y), hence
V ∗+j
t (x)V+ = τt(x), ∀t ≥ 0. (2.6)
For any fix t ∈ IT let A[t 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] (2.7)
for all x ∈ A0. We consider the GNS Hilbert space (Hπφ0 , πφ0(A0), ω0) associ-
ated with (A0, φ0) and define a Markov semigroup (τ
t ) on π(A0) by τ
t (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
τπt (PXP ) = Pαt(X)P (2.8)
for all X ∈ A[0 where P is the projection from [A[0Ω] onto the GNS space [j
0 (A0)Ω]
which identified with the GNS space associated with (A0, φ0). It is enough if we verify
for typical elements X = js1(x1)...jsn(xn) for any s1, s2, ..., sn ≥ 0 and xi ∈ A0 for
1 ≤ i ≤ n and n ≥ 1. We use induction on n ≥ 1. If X = js(x) for some s ≥ 0,
(2.8) follows from (2.5). Now we assume that (2.8) 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 (2.5) to conclude (2.8) by our
induction hypothesis. Now suppose sn+1 ≤ sn. In such a case if sn−1 ≤ sn we appeal
once more to (2.5) and induction hypothesis to verify (2.8) 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 (2.5) at the end of the string and conclude that our claim is true for
any typical element X and hence 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 is a sub-harmonic projection [Mo1] for (αt : t ≥ 0)
i.e. αt(P ) ≥ P for all t ≥ 0 and αt(P ) ↑ [A[0Ω] as t ↑ ∞.
THEOREM 2.1: Let (A0, τt, φ0) be a Markov semigroup and φ0 be (τt)-invariant
state on a C∗ algebra A0. Then the GNS space [π(A0)Ω] associated with φ0 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 vector state φ(X) =< Ω,XΩ >, Ω ∈ H[0
invariant for (αt : t ≥ 0).
(b) PA[0P is isomorphic with π(A0)
′′ where P is the projection from [A[0Ω] onto
[jf (A0)Ω];
(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 property
Following the literature [Vi,Sa,BhP,Bh] on dilation we say (A[0, αt, φ) is the min-
imal E0semigroup associated with (A0, τt, φ0). We have studied extensively asymp-
totic behavior of the dynamics (A0, τt, φ0) in [AM] and Kolmogorov’s property of
the Markov semigroup introduced in [Mo1] was explored to asymptotic behavior of
the dynamics (A[0, αt, φ). In particular this yields a criteria for the inductive limit
state canonically associated with (A[0, αt, φ) to be pure. The notion is intimately
connected with the notion of a pure E0-semigroup introduced in [Po,Ar]. For more
details we refer to [Mo2].
3 Dual Markov semigroup and Time Reverse Markov
processes:
Now we are more specific and assume that A0 is a von-Neumann algebra and each
Markov map (τt) is normal and for each x ∈ A0 the map t → τt(x) is continuous in
the weak∗ topology. We assume further that φ0 is also faithful. Following [AM2],
in the following we briefly recall the time reverse process associated with the KMS-
adjoint ( or Petz-adjoint ) quantum dynamical semigroup (A, τ̃t, φ0).
Let φ0 be a faithful state and without loss of generality let also (A0, φ0) be
in the standard form (A0, J,P, ω0) [BR] where ω0 ∈ H0, a cyclic and separating
vector for A0, so that φ0(x) =< ω0, xω0 > and the closer of the close-able operator
S0 : xω0 → x
∗ω0, S possesses a polar decomposition S = J∆
1/2 with the self-dual
positive cone P as the closure of {JxJxω0 : x ∈ A0} in H0. Tomita’s [BR] theorem
says that ∆itA0∆
−it = A0, t ∈ IR and JA0J = A
0, where A
0 is the commutant of
A0. We define the modular automorphism group σ = (σt, t ∈ IR) on A0 by
σt(x) = ∆
itx∆−it.
Furthermore for any normal state ψ on A0 there exists a unique vector ζ ∈ P
so that ψ(x) =< ζ, xζ >. Note that J π(x)J π(y)Ω = J π(x)∆
2π(y∗)Ω =
2π(x)∆
2π(y∗)Ω = π(y)∆
2π(x∗)∆−
2Ω. Thus the Tomita’s map x →
J π(x)J is an anti-linear ∗-homomorphism representation of A0. This observation
leads to a notion called backward weak Markov processes [AM].
To that end we consider the unique Markov semigroup (τ ′t) on the commutant
A′0 of A0 so that φ(τt(x)y) = φ(xτ
t(y)) for all x ∈ A0 and y ∈ A
0. We define weak
continuous Markov semigroup (τ̃t) on A0 by τ̃t(x) = Jτ
t(JxJ)J. Thus we have the
following adjoint relation
φ0(σ1/2(x)τt(y)) = φ0(τ̃t(x)σ−1/2(y)) (3.1)
for all x, y ∈ A0, analytic elements for (σt). One can as well describe the adjoint
semigroup as Hilbert space adjoint of a one parameter contractive semigroup (Pt)
on a Hilbert space defined by Pt : ∆
1/4xω0 = ∆
1/4τt(x)ω0. For more details we refer
to [Ci].
We also note that it(x) ∈ M[t and set ⋆ anti-homomorphisms j
0 : A0 → B(H[0)
defined by
jb0(x)y = yi0(σ− i
(x∗))
for all y ∈ M[0. That it is well defined follows from (2.1) once we verify by KMS
relation that it preserves the inner product whenever x is an isometry. For any
arbitrary element we extend by linearity. Now we define jb0 : A → B(H) by
jb0(x) = j
0(x)F[0. (3.2)
Thus jb0(x) is a realization of A0 at time t = 0 with j
0(I) = F[0. Now we use the
shift (St) to obtain the process j
b = (jbt : A0 → B(H), t ∈ IR) and forward filtration
F = (F[t, t ∈ IR) defined by the following prescription:
jbt (x) = Stj
0(x)S
t F[t = StF[0S
t , t ∈ IR. (3.3)
A simple computation shows for −∞ < s ≤ t <∞ that
s(x)F[t = j
t (τ̃t−s(x)) (3.4)
for all x ∈ A0. It also follows by our construction that j
(y1)j
(y2)...j
(yn)Ω =
(y) where yr = yri , if r = ri otherwise I, (r1 ≥ r2 ≥ .. ≥ rn). Thus Ω is a cyclic
vector for the von-Neumann algebra Ab generated by {jbr(x), r ∈ IR, x ∈ A0}
′′. We
also von-Neumann algebra Ab
generated by {jbr(x), r ≤ t, x ∈ A0}
′′. The following
theorems say that there is a duality between the forward and backward weak Markov
processes.
THEOREM 3.1: [AM] We consider the weak Markov
processes (A,H, Ft], F[t, St, j
t , j
t t ∈ IR, Ω) associated with (A0, τt, t ≥ 0, φ0)
and the weak Markov processes (Ã, H̃, F̃t], F̃[t, S̃t, j̃
t , j̃
t , t ∈ IR, Ω̃) associated with
(A0, τ̃t, t ≥ 0, φ0). There exists an unique anti-unitary operator U0 : H → H̃ so
(a) U0Ω = Ω̃;
(b) U0StU
0 = S̃−t for all t ∈ IR;
(c) U0j
t (x)U
0 = j̃
−t(x), U0J
t (x)U0 = j̃
−t(x) for all t ∈ IR;
(d) U0Ft]U
0 = F̃[−t, U0F[tU
0 = F̃−t] for all t ∈ IR;
THEOREM 3.2: Let (A0, τt, φ0) be as in Theorem 3.1 with φ0 as faithful. Then
the commutant of A[t is A
for each t ∈ IR.
PROOF: It is obvious that A[0 is a subset of the commutant of A
. Note also
that F[0 is an element in A
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 ∈ (Ab
)′ 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 ∈ A
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
F[0 ⊆ F[0A
F[0. Let X ∈ F[0A
F[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 F[0A
F[0 = j
b(A0)
Thus there exists an element Y ∈ Ab
so that XΩ = YΩ. 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
we conclude that F[0A
F[0 = F[0A
F[0 being a projection in A
we verify that F[0(A
)′F[0 ⊆ (F[0A
′ and so
we also have F[0(A
)′F[0 ⊆ (F[0A
′ as Ab
. Thus it is enough if we prove
[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
F[0 so that XΩ = Y Ω. Thus for any Z ∈ A[0 we have
XZΩ = Y ZΩ and thus XF[0 = Y F[0. Hence X = Y ∈ F[0A
F[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)
′ = αt(A
αt being an automorphism. This completes the proof as αt(A
) = Ab
by our
construction.
4 Subfactors:
In this section we will investigate further the sequence of von-Neumann algebra
{A[t : t ∈ IR} defined in the last section with an additional assumption that φ0
is also faithful and thus we also have in our hand backward von-Neumann algebras
: t ∈ IR}.
PROPOSITION 4.1: Let (A0, τt, φ0) be a Markov semigroup with a faithful
normal invariant state φ0. If A0 is a factor then A[0 is a factor. In such a case the
following also hold:
(a) A0 is type-I (type-II, type-III) if and only if A[0 is type-I (type -II , type-III)
respectively;
(b) H is separable if and only if H0 is separable;
(c) If H0 is separable then A0 is hyper-finite if and only if A[0 is hyper-finite.
PROOF: We first show factor property of A[0. Note that the von-Neumann algebra
generated by the backward process {jbs(x) : s ≤ 0, x ∈ A0} is a sub-algebra
of A′
, the commutant of A[0. We fix any X ∈ A[0
in the center. Then for
any y ∈ A0 we verify that Xj0(y)Ω = XF0]j0(y)Ω = F0]XF0]j0(y)Ω = j0(xy)Ω
for some x ∈ A0. Since Xj0(y) = j0(y)X we also have j0(xy)Ω = j0(yx)Ω. By
faithfulness of the state φ0 we conclude xy = yx thus x must be a scaler. Thus we
have Xj0(y)Ω = cj0(y)Ω for some scaler c ∈ IC. Now we use the property that X
commutes with forward process jt(x) : x ∈ A0, t ≥ 0 and as well as the backward
processes {jbt (x), t ≤ 0} to conclude that Xλ(t, x) = cλ(t, x). Hence X = c. Thus
A[0 is a factor.
Now if A0 is a type-I factor, then there exists a non-zero minimal projection
p ∈ A0. In such a case we claim that j0(p) is also a minimal projection in A[0. To
that end let X be any projection in A[0 so that X ≤ j0(p). Since F0]A[0F0] = j0(A0)
we conclude that F0]XF0] = j0(x) for some x ∈ A0. Hence X = j0(p)Xj0(p) =
F0]Xj0(p) = j0(xp) = j0(px) Thus by faithfulness of the state φ0 we conclude that
px = xp. Hence X = j0(q) where q is a projection smaller then equal to p. Since p
is a minimal projection in A0, q = p or q = 0 i.e. X = j0(p) or 0. So j0(p) is also
a minimal projection. Hence A[0 is a type-I factor. For the converse statement we
trace the argument in the reverse direction. Let p be a non-zero projection in A0 and
claim that there exists a minimal projection q ∈ A0 so that 0 < q ≤ p. Now since
j0(p) is a non-zero projection in a type-I factor A[0 there exists a non-zero projection
X which is minimal in A[0 so that 0 < X ≤ j0(p). Now we repeat the argument to
conclude that X = j0(q) for some projection q. Since X 6= 0 and minimal, q 6= 0
and minimal in A0. This completes the proof for type-I case. We will prove now the
case for Type-II.
Let A[0 be type-II then there exists a finite projection X ≤ F0]. Once more
X = F0]XF0] = j0(x) for some projection x ∈ A0. We claim that x is finite. To
that end let q be another projection so that q ≤ x and q = uu∗ and u∗u = x. Then
j0(q) ≤ j0(x) = X and j0(q) = j0(u)j0(u)
∗ and j0(x) = j0(u)
∗j0(u). Since X is
finite in A[0 we conclude that j0(q) = j0(x). By faithfulness of φ0 we conclude that
q = x, hence x is a finite projection. Since A0 is not type-I, it is type-II. For the
converse let A0 be type-II. So A[0 is either type-II or type-III. We will rule out that
the possibility for type-III. Suppose not, i.e. if A[0 is type-III, for every projection
p 6= 0, there exists u ∈ A[0 so that j0(p) = uu
∗ and F0] = u
∗u. In such a case
j0(p)u = uF0]. Set j0(v) = F0]uF0] for some v ∈ A0. Thus j0(pv) = j0(v). Once
more by faithfulness of the normal state φ0, we conclude pv = v. So j0(v) = uF0].
Hence j0(v
∗v) = F0]. Hence v
∗v = 1 by faithfulness of φ0. Since this is true for any
non-zero projection p in A0, A0 is type-III, which is a contradiction. Now we are left
to show the statement for type-III, which is true since any factor needs to be either
of these three types. This completes the proof for (a).
(b) is obvious if IT is ZZ. In case IT = IR, we use our hypothesis that the map
(t, x) → τt(x) is sequentially jointly continuous with respect to weak
∗ topology.
For (c) we first recall from [Co] that hyper-finiteness property, being equivalent to
injective property of von-Neumann algebra, is stable under commutant and countable
intersection operation when they are acting on a separable Hilbert space. Let A0
be hyper-finite and H0 be separable. We will first prove A[0 is hyper-finite when
IT = ZZ, i.e. time variable are integers. In such a case for each n ≥ 0, jn being
injective, jn(A0)
′′ = {j0(x) : x ∈ A0}
′′ is a hyper-finite von-Neumann algebra. Thus
A[0 = {jn(A0)
′′ : n ≥ 0}′′ is also hyper-finite as they are acting on a separable
Hilbert space. In case IT = IR, for each n ≥ 1 we set von-Neumann sub-algebras
⊆ A[0 generated by the elements {jt(A0)
′′ : t = r
, 0 ≤ r ≤ n2n}. Thus each An
is hyper-finite. Since A′
n≥0(A
)′ by weak∗ continuity of the map t → τt(x),
we conclude that A[0 is also hyper-finite being generated by a countable family of
increasing hyper-finite von-Neumann algebras.
For the converse we recall for a factorM acting on a Hilbert spaceH, Tomiyama’s
property ( i.e. there exists a norm one projection E : B(H) → M, see [BR1] page-
151 for details ) is equivalent to hyper-finite property. For a hyper-finite factor A[0,
j0(A0) is a factor in the GNS space identified with the subspace F0]. Let E be the
norm one projection from B(H[0) on A[0 and verify that the completely positive map
E0 : B(H0) → A0 defined by E0(X) = F0]E(F0]XF0])F0] is a norm one projection
from B(F0]) to A0. This completes the proof for (b).
PROPOSITION 4.2: Let (A0, τt, φ0) be a dynamical system as in Proposition
4.1. If A[0 is a type-II1 factor which admits a unique normalize faithful normal
tracial state then the following hold:
(a) Ft] = I for all t ∈ IR;
(b) τ = (τt) is a semigroup of ∗−endomorphisms.
(c) A[0 = j0(A0).
PROOF: Let tr0 be the unique normalize faithful normal trace on A[0. For any fix
t ≥ 0 we set a normal state φt on A[0 by φt(x) = tr0(αt(x)). It is simple to check
that it is also a faithful normal trace. Since αt(I) = I, by uniqueness φt = tr0.
In particular tr0(F0]) = tr0(αt(F0]) = tr0(Ft]), by faithful property Ft] = F0] for all
t ≥ 0. Since Ft] ↑ 1 as t→ ∞ we have F0] = I. Hence Ft] = αt(F0]) = I for all t ∈ IR.
This proves (a). For (b) and (c) we recall that F0]jt(x)F0] = j0(τt(x)) for all t ≥ 0
and jt : A0 → A[t is an injective ∗− homomorphism. Since Ft] = F0] = I we have
jt(x) = F0]jt(x)F0] = j0(τt(x)). Hence A[0 = j0(A0) and j0(τt(x)τt(y)) = j0(τt(xy))
for all x, y ∈ A0. Now by injective property of j0, we verify (b). This completes the
proof.
We fix a type-II1 factor A0 which admits a unique normalize faithful normal
tracial state. Since A[0 is a type-II factor whenever A0 is so, we conclude that A[0 is
a type-II∞ factor whenever τt is not an endomorphism on a such a type-II1 factor.
The following proposition says much more.
PROPOSITION 4.3: Let A0 be a type-II1 factor with a unique normalize normal
trace and (A0, τt, φ0) be a dynamical system as in Proposition 4.1. Then the following
hold:
(a) j0(I) is a finite projection in A[−t for all t ≥ 0.
(b) For each t ≥ 0 Mt = j0(I)A[−tj0(I) is a type-II1 factor and M0 ⊆ Ms... ⊆
Mt ⊆ .., t ≥ s ≥ 0 are acting on Hilbert space F0] where M0 = j0(A0).
PROOF: By Proposition 4.1 A[0 is a type-II factor. Thus A[0 is either type-II1 or
type-II∞. In case it is type-II1, Proposition 4.2 says that A[−t is j0(A0), hence the
statements (a) and (b) are true with Mt = j0(A0). Thus it is good enough if we
prove (a) and (b) when A[0 is indeed a type-II∞ factor. To that end for any fix t ≥ 0
we fix a normal faithful trace tr on A[−t and consider the normal map x→ j0(x) and
thus a normal trace trace on A0 defined by x→ tr(j0(x)) for x ∈ A0. It is a normal
faithful trace on A0 and hence it is a scaler multiple of the unique trace on A0. A0
being a type-II1 factor, j0(I) is a finite projection in A[−t. Now the general theory
on von-Neumann algebra [Sa] guarantees that Mt is type-II1 factor and inclusion
follows as A−s] ⊆ A−t] whenever t ≥ s. That j0(A0) = j0(I)A[0j0(I) follows from
Proposition 4.1.
We have now one simple but useful result.
COROLLARY 4.4: Let (A0, τt, φ0) be as in Proposition 4.1. Then one of the
following statements are false:
(a) A = B(H)
(b) A0 is a type-II1 factor.
PROOF : Suppose both (a) and (b) are true. Let φt be the unique normalized trace
on Mt. As they are acting on the same Hilbert space, we note by uniqueness that φt
is an extension of φs for t ≥ s. Thus there exists a normal extension of (φt) to weak
completion M of
t≥0 Mt ( here we can use Lemma 13 page 131 [Sc] ). However if
A = B(H), M is equal to B(H0]). H0] being an infinite dimensional Hilbert space
we arrive at a contradiction.
In case M in Corollary 4.4 is a type-II1 factor, by uniqueness of the tracial
state we claim that λ(t)λ(s) = λ(t + s) where λ(t) = tr(F−t]) for all t ≥ 0. The
claim follows as von-Neumann algebra M is isomorphic with α−t(M) which is equal
to F−t]MF−t]. The map t → λ(t) being continuous we get λ(t) = exp(λt) for some
λ ≤ 0. If λ = 0 then λ(t) = 1 so by faithful property of the trace we get F−t] = F0] for
all t ≥ 0. Hence we conclude that (τt) is a family of endomorphisms by Proposition
4.2. Now for λ < 0 we have tr(F−t]) → 0 as t → ∞. As F−t] ≥ |Ω >< Ω| for all
t ≥ 0, we draw a contradiction. Thus the weak∗ completion of
t≥0 Mt is a type-II1
factor if and only if (τt) is a family of endomorphism. In otherwords if (τt) is not
a family of endomorphism then the weak∗ completion of
t≥0 Mt is not a type-II1
factor and the tracial state though exists on M is not unique.
5 Jones index of a quantum dynamical semigroup on
II1 factor:
We first recall Jones’s index of a sub-factor originated to understand the structure of
inclusions of von Neumann factors of type II1. Let N be a sub-factor of a finite factor
M . M acts naturally as left multiplication on L2(M, tr), where tr be the normalize
normal trace. The projection E0 = [Nω] ∈ N
′, where ω is the unit trace vector i.e.
tr(x) =< ω, xω > for x ∈ M , determines a conditional expectation E(x) = E0xE0
on N . If the commutant N ′ is not a finite factor, we define the index [M : N ] to
be infinite. In case N ′ is also a finite factor, acting on L2(M, tr), then the index
[M : N ] of sub-factors is defined as tr(E0)
−1, which is the Murray-von Neumann
coupling constant [MuN] of N in the standard representation L2(M, tr). Clearly
index is an invariance for the sub-factors. Jones proved [M : N ] ∈ {4 cos2(π/n) :
n = 3, 4, · · ·} ∪ [4,∞] with all values being realized for some inclusion N ⊆M .
In this section we continue our investigation in the general framework of section
4 and study the case when A0 is type-II1 which admits a unique normalize faithful
normal tracial state and (τt) is not an endomorphism on such a type-II1 factor. By
Proposition 4.3 A[0 is a type-II∞ factor and (Mt : t ≥ 0) is a family of increasing
type-II1 factor where Mt = j0(I)A[−tj0(I) for all t ≥ 0. Before we prove to discrete
time dynamics we here briefly discuss continuous case. Thus the map I : (t, s) →
[Mt : Ms], 0 ≤ s ≤ t is an invariance for the Markov semigroup (A0, τt, φ0). By our
definition I(t, t) = 1 for all t ≥ 0 and range of values Jones’s index also says that the
map (s, t) → I(s, t) is not continuous at (s, s) for all s ≥ 0. Being a discontinuous
map we also claim the map (s, t) → I(s, t) is not time homogeneous, i.e. I(s, t) 6=
I(0, t − s) for some 0 ≤ s ≤ t. If not we could have I(0, s + t) = I(0, s)I(s, s + t) =
I(0, s)I(0, t), i.e. I(0, t) = exp(λt) for some λ, this leads to a contradiction. The
non-homogenity suggest that I is far from being simple. We devote rest of the section
discussing a much simple example in discrete time dynamics.
To that end we review now Jones’s construction [Jo, OhP]. Let A0 be a type-
II1 factor and φ0 be the unique normalize normal trace. The algebra A0 acts on
L2(A0, φ0) by left multiplication π0(y)x = yx for x ∈ L
2(A0, φ0). Let ω be the
cyclic and separating trace vector in L2(A0, φ0). The projection E0 = [B0ω] induces
a trace preserving conditional expectation τ : a → E0aE0 of A0 onto B0. Thus
E0π0(y)E0 = E0π0(E(y))E0 for all y ∈ A0. Let A1 be the von-Neumann algebra
{π0(A0), E0}
′′. A1 is also a type-II1 factor and A0 ⊆ A1, where we have identified
π0(A0) with A0. Jones proved that [A1 : A0] = [A0 : B0]. Now by repeating this
canonical method we get an increasing tower of type-II1 factors A1 ⊆ A2... so that
[Ak+1 : Ak] = [A0 : B0] for all k ≥ 0. Thus the natural question: Is Jones tower
A0 ⊆ A1 ⊆ ... ⊆ Ak... related with the tower M0 ⊆ M1...Mk ⊆ Mk+1 defined in
Proposition 4.3 associated with the dynamics (A0, τn, φ0)?
To that end recall the von-Neumann sub-factors M0 ⊆ M1 and the induced
representation of M1 on Hilbert subspace H[−1,0] generated by {j0(x0)j−1(x−1)Ω :
x0, x−1 ∈ A0}. Ω is the trace vector for M0 i.e. φ0(x) =< Ω, j0(x)Ω >. However
the vector state given by Ω is not the trace vector for M1 as M1 6= M0 ( If so we
check by trace property that φ0(τ(x)yτ(z)) = φ(j0(x)j−1(y)j0(z)) = φ0(τ(zx)y) for
any x, y, z ∈ A0 and so τ(zx) = τ(z)τ(x) for all z, x ∈ A0. Hence by Proposition 4.2
we have M1 = M0). Nevertheless M1 being a type-II1 factor there exists a unique
normalize trace on M1.
PROPOSITION 5.1: M1 ≡ A2 and [M1 : M0] = d
2 where d = [A0 : B0].
PROOF: Let φ1 be the unique normalize normal trace on A1 and H1 = L
2(A1, φ1).
We consider the left action π1(x) : y → xy of A1 on H1. Thus π0(A0) is also acting
on H1. Since E0π0(x)E0 = E0π0(τ(x))E0 = E0π0(τ(x)), for any element X ∈ A1,
E0X = E0π0(x) for some x ∈ A0. Thus π1(E0) is the projection on the subspace
{E0π0(x) : x ∈ A0}.
For any y ∈ A0 we set
(a) k−1(y) on the subspace π1(E0) by k−1(y)E0π0(x) = E0π0(yx) for x ∈ A0 and
extend it to H1 trivially. That k−1(y) is well defined and an isometry for an isometry
y follows from the following identities:
φ1((E0π0(yz))
∗E0π0(yx)) = φ1(π0(z
∗y∗)E0π0(yx))
= φ1(E0π0(yx)π0(z
∗y∗)) by trace property
= φ1(E0)φ0(π0(yx)π0(z
∗y∗)) being a trace and φ1(E0π0(x)) = φ1(E0)φ0(π0(x))
= φ1(E0)φ0(π0(z
∗y∗)π0(yx)) = φ1(E0)φ0(π0(z
= φ1((E0π0(z))
∗E0π0(x))
(b) k0(y)x = π0(y)x for x ∈ A1. Thus y → k0(y) is an injective ∗-representation of
A0 in L
2(A1, φ1).
For y, z ∈ A0 we verify that
< E0π0(y), k−1(1)k0(x)k−1(1)E0π0(z) >1=< E0π0(y), E0π0(x)E0π0(z) >1
=< E0π0(y), E0π0(τ(x))E0π0(z) >1
=< E0π0(y), E0π0(τ(x))π0(z) >1
Thus k−1(1)k0(x)k−1(1) = k−1(τ(x)) for all x ∈ A0. Note that k−1(1) = π1(E0)
and the identity operator in H1 is a cyclic vector for the von-Neumann algebra
{k0(x), k−1(x), x ∈ A0}
′′. We have noted before that the vector Ω need not be
the tracial vector for M1 and also verify by a direct computation that the space
{k0(y)k−1(x)1 : x, y ∈ A0} is equal to {yE0x : y, x ∈ A0} which is a proper subspace
of L2(A1, φ1).
Now we claim that the type-II1 factor M1 is isomorphic to the von-Neumann
algebra {k−1(x), k0(x), x ∈ A0)}
′′. To that end we define an unitary operator from
L2(A1, φ1) to L
2(M1, tr1) by taking an element kt1(x1)..ktn(xn) to jt1(x1)..jtn(xn),
where tk are either 0 or −1. That it is an unitary operator follows by the tracial
property of the respective states and weak Markov property of the homomorphisms.
We leave the details and without lose of generality we identify these two weak Markov
processes. Since k−1(1) = π1(E0), we conclude that π1(A1) ⊆ M1. In fact strict
inclusion hold unless B0 = A0.
However by our construction A1 = π0(A0) ∨ E0 is acting on L
2(A0, φ0) and
A2 = π1(A1) ∨ E1 is acting on L
2(A1, φ1) where E1 is the cyclic subspace of 1
generated by π1(π0(A0)) i.e. E1 = [π1(π0(A0))1]. From (a) we also have
k−1(y)π1(E0)E1 = π1(E0)k0(y)E1 (5.1)
for all y ∈ A0.
By Temperley-Lieb relation [Jo] we have π1(E0)E1π1(E0) =
π1(E0) and thus
post multiplying (5.1) by π1(E0) we have
k−1(y)π1(E0) = π1(E0)k0(y)E1π1(E0) (5.2)
So it is clear now that k−1(y) ∈ π1(A1) ∨ E1 for all y ∈ A0. Thus π1(A1) ⊆ M1 ⊆
π1(A1) ∨E1 = A2.
We claim also that E1 ∈ M1. We will show that any unitary ele-
ment u commuting with M1 is also commuting with E1. By (5.2) we have
π1(E0)k0(y)(uE1u
∗ − E1)π1(E0) = 0 for all y ∈ A0. By taking adjoint we have
π1(E0)(uE1u
∗ − E1)k0(y)π1(E0) = 0 for all y ∈ A0. Since A1 = π0(A0) ∨
E0 we conclude by cylicity of the trace vector that π1(E0)(uE1u
∗ − E1) =
0. So we have E1π1(E0)uE1u
∗ = E1π1(E0)E1 = E1 by Temperley-Lieb rela-
tion. So u∗E1uπ1(E0) = u
∗E1π1(E0)uE1u
∗u = u∗E1u By taking adjoint we get
π1(E0)u
∗E1u = u
∗E1u. Since same is true for u
∗, we conclude that u∗E1u = E1 for
any unitary u ∈ M′1. Hence E1 ∈ M1.
Hence M1 = A2. Since [M1 : M0] = [M1 : A1][A1 : M0] and [A1 : A0] = [A0 :
B0] = d, we conclude the result.
THEOREM 5.2: Mm ≡ A2m for all m ≥ 1.
PROOF: Proposition 5.1 gives a proof for m = 1. The proof essentially follows the
same steps as in Proposition 5.1. We use induction method for m ≥ 1. Assume it is
true for 1, 2, ..m. Now consider the Hilbert space L2(A2m+1, tr2m+1), m ≥ 1 and we
set homomorphism k−1, k0 from A2m into B(L
2(A2m+1, tr2m+1)) in the following:
(a) k0(x)y = xy for all y ∈ A2m+1 and x ∈ A2m
(b) π2m+1(E2m) is the projection on the subspace {E2my : y ∈ A2m} and k−1(x)
defined on the subspace E2m by
k−1(x)E2my = E2mxy
for all x ∈ A2m and y ∈ A2m. That k−1 is an homomorphism follows as in Proposition
5.1. Thus an easy adaptation of Proposition 5.1 says that M = {k0(x), k−1(x) :
x ∈ A2m}
′′ is a type-II1 factor and proof will be complete once we show that it is
isomorphic to Mm+1.
To that end we check as in Proposition 5.1 that
k−1(E2m) = k−1(I), k−1(I)k0(x)k−1(I) = k−1(E2mxE2m)
for all x ∈ A2m and k−1(x)E2m = E2mk0(x)E2m+1 where E2m+1 is the cyclic space
of the trace vector generated by A2m. Thus following Proposition 5.1 we verify
now that type-II1 factor M = {k0(x), k−1(I), E2m+1, x ∈ A2m}
′′ is isomorphic to
Mm+1 = {J−1(x), J0(x) x ∈ Am}
′′, where we used notation J−1(x) = x for all
x ∈ Am, J0(x) = SJ−1(x)S
∗ for all x ∈ A2m where we have identified A2m ≡ Mm
with {jk(x),−m − 1 ≤ k ≤ −1, x ∈ A0}
′′ and S is the (right) Markov shift. This
completes the proof.
REFERENCES
• [AM] Accardi, L., Mohari, A.: Time reflected Markov processes. Infin. Dimens.
Anal. Quantum Probab. Relat. Top., vol-2, no-3, 397-425 (1999).
• [Ar] Arveson, W.: Pure E0-semigroups and absorbing states, Comm. Math.
Phys. 187 , no.1, 19-43, (1997)
• [Bh] Bhat, B.V.R.: An index theory for quantum dynamical semigroups, Trans.
Amer. Maths. Soc. vol-348, no-2 561-583 (1996).
• [BP] Bhat, B.V.R., Parthasarathy, K.R.: Kolmogorov’s existence theorem for
Markov processes on C∗-algebras, Proc. Indian Acad. Sci. 104,1994, p-253-
• [BR] Bratelli, Ola., Robinson, D.W. : Operator algebras and quantum statis-
tical mechanics, I,II, Springer 1981.
• [Da] Davies, E.B.: Quantum Theory of open systems, Academic press, 1976.
• [El] Elliot. G. A.: On approximately finite dimensional von-Neumann algebras
I and II, Math. Scand. 39 (1976), 91-101; Canad. Math. Bull. 21 (1978), no.
4, 415–418.
• [Jo] Jones, V. F. R.: Index for subfactors. Invent. Math. 72 (1983), no. 1,
1–25.
• [Mo1] Mohari, A.: Markov shift in non-commutative probability, Jour. Func.
Anal. 199 (2003) 189-209.
• [Mo2] Mohari, A.: Pure inductive limit state and Kolmogorov’s property, ....
• [Mo3] Mohari, A.: SU(2) symmetry breaking in quantum spin chain, ....
• [MuN] Murray, F. J.; von Neumann, J., On rings of operators. (English)[J]
Ann. Math., Princeton, (2)37, 116-229.
• [OP] Ohya, M., Petz, D.: Quantum entropy and its use, Text and monograph
in physics, Springer-Verlag 1995.
• [Po] Powers, Robert T.: An index theory for semigroups of ∗-endomorphisms
of B(H) and type II1 factors. Canad. J. Math. 40 (1988), no. 1, 86–114.
• [Sak] Sakai, S.: C∗-algebras and W∗-algebras, Springer 1971.
• [Sa] Sauvageot, Jean-Luc: Markov quantum semigroups admit covariant
Markov C∗-dilations. Comm. Math. Phys. 106 (1986), no. 1, 91103.
• [Vi] Vincent-Smith, G. F.: Dilation of a dissipative quantum dynamical system
to a quantum Markov process. Proc. London Math. Soc. (3) 49 (1984), no.
1, 5872.
