Additive units of product system

0.28 MB·English

by  B. V. Rajarama Bhat

ADDITIVE UNITS OF PRODUCT SYSTEM
5 B.V.RAJARAMABHAT,MARTINLINDSAY,ANDMITHUNMUKHERJEE
1
0
2
n Abstract1
a
J

0 Abstract. We introduce the notion of additive units and roots of a unit in a spatial product system.
3 Theset of allroots ofany unitformsaHilbertspace and itsdimensionis thesameas the indexof the
productsystem. WeshowthataunitandallofitsrootsgeneratethetypeIpartoftheproductsystem.

] Using properties of roots, we also provide an alternative proof of the Powers’ problem that the cocycle
A
conjugacyclassofPowerssumisindependentofthechoiceofintertwiningisometries. Inthelastsection,
F weintroducethenotionofclusterofaproductsubsystemandestablishitsconnectionwithrandomsets
. inthesenseofTsirelson([27])andLiebscher([11]).
h
t
a 1. Introduction
m
[ Afundamentalgoalofquantumdynamicsistheclassificationofsemigroupsofunital -endomorphisms
∗
of the algebra of all bounded operators on a separable Hilbert space up to cocycle conjugacy. Associated
1
v with every such ‘E0-semigroup’, is a (tensor) product system of Hilbert spaces ([1]). This translates the
5 problem of classification of E0-semigroups up to cocycle conjugacy into the problem of classification of
7 the product systems up to isomorphism. A product system is a measurable family of separable Hilbert
6
spaces ( ) with associative identification through unitaries. A unit is a measurable
s s>0 s+t s t
7 E E ≃ E ⊗E
section of non-zero vectors (u ) , u which factorises: u = u u , s,t >0. Depending on the
0 s s>0 s ∈Es s+t s⊗ t
. existence of units, product systems are classified into three categories. A product system is said to be
1
of type I if units exist and they ‘generate’ the product system. A product system is said to be of type
0
II if it has a unit but they fail to ‘generate’ the product system. Product systems having units are also
5
1 known as spatial product systems. A product system is said to be of type III or non-spatial if it does
: not have any unit. Spatial product systems have an index. The index is a complete invariant for type I
v
product systems and each is cocycle conjugate to a CCR flow ([2]). There is an operation of tensoring
i
X on the category of product systems. The index is additive under the tensor product of spatial product
r systems. Product systems of type II and type III exist in abundance but their classification theory is far
a
from complete. It was shown that there are uncountably many cocycle conjugacy classes of type II and
type III product systems ([17],[18],[29],[28]) but we still lack good invariants to distinguish them.
Tsirelson ([27],[26]) established interesting new examples of type II product systems coming from
measure types of randomsets or generalizedrandom(Gaussian) processes. Liebscher,([11]) then made a
systematic study of measure types ofrandomsets. Givena pair of product systems, one containedin the
other, one associates a measure type of random (closed) sets of the interval [0,1]. These measure types
are stationary and factorizing over disjoint intervals. The corresponding measure type is an invariant of
the product system. See [11] for more details.
Contractive semigroups of completely positive maps are known as quantum dynamical semigroups.
The dilation theory of quantum dynamical semigroups ([4]) reveals a new approach to understand E -
0
semigroups. Every unital quantum dynamical semigroup dilates to an E -semigroup and the minimal
0
dilation is unique up to conjugacy.
Similarly, E semigroups on general C∗ algebras or von Neumann algebras correspond to product
0
systems of Hilbert modules, ([14],[20],[21]). Much of the theory of product system of Hilbert spaces
1AMS Subject Classification: 46L55, 46C05. Keywords: Product Systems, Completely Positive Semigroups, Inclusion
Systems.
1
2 BHAT,B.V.R.,LINDSAY,J.M.,ANDMUKHERJEE,M.
and the theory of E -semigroups acting on (H) can be carried through also for the product systems
0
B
of Hilbert modules and E semigroups acting on a(E), the algebra of all adjointable operators on a
0
B
Hilbertmodule. Howeverthereisnonaturaltensorproductoperationonthecategoryofproductsystems
of Hilbert modules. Skeide ([23]) overcame this by introducing the spatial product of spatial product
systems of Hilbert modules in which the reference units (normalized) are identified and under which the
indexofthespatialproductsystemofHilbertmoduleisadditive. Restrictingtothecaseofspatialproduct
systemsofHilbertspaces,wehaveanotheroperationsonthecategoryofspatialproductsystems. Suppose
and aretwospatialproductsystemswithnormalizedunits uandv respectively. The spatialproduct
E F
canbeidentifiedwiththeproductsubsystemofthetensorproduct,generatedbythetwosubsystems v
E⊗
andu .Thisraisesthequestionwhetherthespatialproductisthetensorproductornot. Powers([19])
⊗F
answered this in the negative sense by solving the seemingly different but equivalent following problem:
Suppose φ= φ :t 0 and ψ = ψ :t 0 are two E semigroups on (H) and (K) respectively
t t 0
{ ≥ } { ≥ } B B
and U = U : t 0 and V = V : t 0 are two strongly continuous semigroups of isometries which
t t
{ ≥ } { ≥ }
intertwine φ (φ (A)U = U A, A (H),t 0) and ψ respectively. Note that the intertwining
t t t t t
∀ ∈ B ≥
isometries of E -semigroups correspond bijectively to the normalized units of the associated product
0
systems. Consider the CP semigroup (Powers sum) τ on (H K) defined by
t
B ⊕
X Y φ (X) U YV∗
τ = t t t .
t(cid:18) Z W (cid:19) (cid:18) VtZUt∗ ψt(W) (cid:19)
How is the product system of the minimal dilation (in the sense of [9],[4]) of τ related to the product
systems of φ and ψ? Skeide ([22]) identified the product system as a spatial product through normalized
units. The definition of Powers’sum easily extends to CP semigroupsand the product system of Powers’
sum in that case also is the spatial product of the product systems of its summands ([7],[24]). Motivated
by this problem and its straightforward generalization to more general ‘corner’, amalgamated product
(see Section 2) through general contractive morphism of two product systems (not necessarily spatial)
was introduced in [8] which generalizes the spatial product. The spatial product may be viewed as an
amalgamatedproductthroughthe contractivemorphismdefinedthroughnormalizedunits. This answers
Powers’problem for the Powers’sum obtained from not necessarily isometric intertwining semigroups.
The structure of the spatial product, a priori depends on the choice of the reference units in their
respective factors. In fact, Tsirelson ([30]) showed that the group of all automorphisms of a product
systemmaynotacttransitivelyonthesetofallunits. Itraisesanotherquestionwhethertheisomorphism
classofthespatialproductdependsonthechoiceofthereferenceunits. Equivalently,whetherthecocycle
conjugacyclassoftheminimaldilationofPowerssumdependsonthechoiceoftheintertwiningisometries.
This was answered in the negative sense in [5]. See also [6].
In this paper, we start with a brief overview of the theory of inclusion systems and amalgamated
products to make the readers familiar with these notions which we use repeatedly. Readers are referred
to [8], [16] for more details. In Section 3, we introduce the notion of additive units and roots of a
unit in a spatial product system. Additive units are measurable sections of product system which are
‘additive with respect to a given unit’. Roots are the special additive units such that for each t> 0, the
sections are orthogonal to the unit. The set of all additive units forms a Hilbert space and the set of
all roots is a subspace of co-dimension one. We compute all the roots of the vacuum unit in CCR flows
(Γ (L2[0,t],K)). They are given by the set of all cχ , c K almost surely. From this, we establish
sym t]
| ∈
that a unit and all of its roots ‘generate’ the type I part of the product system and the dimension of the
Hilbert space of the set of all roots of a unit is the same for every unit and coincides with the index of
the product system. We also generalize the notion of additive units and roots of a unit on the level of
inclusionsystems(seeSection2). Weshowthatthesetofalladditiveunitsofaunitinaninclusionsystem
are in a bijective correspondence with the set of all additive units of the ‘lifted’ unit in the generated
algebraic product system. The behaviour of the roots under amalgamatedproduct is also studied. Using
the properties of roots, we have an alternating proof of the fact that the Powers sum is independent of
the choice of the intertwining isometries or equivalently that the isomorphism class of the amalgamated
product through normalized units is independent of the choice of the units (see Section 4). In fact, we
have an improvement of this result which says that the isomorphism class of the amalgamated product
ADDITIVE UNITS OF PRODUCT SYSTEM 3
throughstrictlycontractiveunitsisalsoindependentofthechoiceoftheunits. Thisfactwillbeexplained
elsewhere ([15]).
In Section 5, given any product subsystem of a product system , we construct an intermediate
F E
subsystem called the cluster subsystem of . A product subsystem correspondsto an ‘adapted’ family of
F
commutative projections satisfying some relation. The commutative von Neumann algebra generated by
themisuniquelydeterminedbyameasuretypeofrandomclosedsetsoftheinterval[0,1].Thedistribution
oftherandommappingwhichsendsaclosedsettoitslimitpointsisthemeasuretypeoftheclustersystem
of the originalproduct subsystem. In a special case, the measure type correspondingto a single unit and
the measure type corresponding to the type I part, both share the same relation. See Proposition 3.33,
Chapter3,[11]. Liebscher’sproofsofthosefactsuseheavymachineryfrommeasuretheoryofrandomsets
andthe directintegralconstruction. Hereweexplicitlyconstructthe clustersubsystemwithoutinvolving
any heavy machinery. We show that the measure type corresponding to the subsystem and the measure
type of its cluster are related by the above random mapping. Without using any random sets theory, we
also compute that the cluster of the subsystem generated by a single unit in a spatial product system is
the type I part of the product system.
2. Inclusion system and amalgamation
An inclusion system is a parametrized family of Hilbert spaces exactly like product system but the
connecting maps are now only isometries. These objects seem to be ubiquitous in the field of product
system. They are the recurrenttheme of studying quantum dynamics, inparticular CP semigroups. (See
[10],[14],[12],[20],[8]). Even while associating product systems to CP semigroups what one gets first are
inclusion systems, and then an inductive limit procedure gives product systems ([10],[8]). The notion of
inclusion systems is introduced in [8]. It was also introduced by Shalit and Sholel ([20]) under the name
subproduct system. The following definition is taken from [8].
Definition 1. An inclusion System (E,β) is a family of Hilbert spaces E = E ,t (0, ) together
t
{ ∈ ∞ }
with isometries β :E E E , for s,t (0, ), such that r,s,t (0, ), (β 1 )β =
s,t s+t → s ⊗ t ∈ ∞ ∀ ∈ ∞ r,s ⊗ Et r+s,t
(1 β )β . It is said to be an algebraic product system if further every β is a unitary.
Er ⊗ s,t r,s+t s,t
Definition 2. Suppose (E,β) is an inclusion system. Then a family F = (F ) of closed subspaces,
t t>0
F E is said to be an inclusion subsystem of (E,β) if β (F ) F F for every s,t>0.
t ⊂ t s,t|Fs+t s+t ⊂ s⊗ t
For each t R , we set
+
∈
n
J = (t ,t ,...,t ):t >0, t =t,n 1 .
t 1 2 n i i
{ ≥ }
Xi=1
For s = (s ,s ,...,s ) J , and t = (t ,t ,...,t ) J we define s ⌣ t :=
1 2 m s 1 2 n t
∈ ∈
(s ,s ,...,s ,t ,t ,...,t ) J . Nowfixt R . OnJ ,defineapartialordert s=(s ,s ,...,s )
1 2 m 1 2 n s+t + t 1 2 m
∈ ∈ ≥
if for each i, (1 i m) there exists (unique) s J such that t = s ⌣ s ⌣ ⌣ s . The order
≤ ≤ i ∈ si 1 2 ··· m
relation makes J a directed set.
t
≥
Suppose (E,β) is an inclusion system. For s = (s1,··· ,sn) ∈ Jt, we set Es = Es1 ⊗···⊗Esn. For
s = (s1,··· ,sn) ≤ t = s1 ⌣ ··· ⌣ sn ∈ Jt, define βs,t : Es → Et by βs,t = βs1,s1 ⊗···⊗βsn,sn, where
β =I and for s=(s , ,s ) J , inductively define
s,s Es 1 ··· n ∈ s
βs,s =(I ⊗βsn−1,sn)···(I ⊗βs2,s3+···+sn)βs1,s2+s3+···+sn.
Proof of the following theorem can be found in Theorem 5, [8].
Theorem 3. Suppose (E,β) is an inclusion system. Let Et = indlimJtEs be the inductive limit of Es
over J for t>0. Then = :t>0 has the structure of an algebraic product system.
t t
E {E }
Let ( ,B) be the generated algebraic product system of the inclusion system (E,β). Note that the
E
unitary map B goes from to for every s,t > 0. In other words, algebraic product systems
s,t s+t s t
E E ⊗E
are inclusion systems with all the linking maps are unitaries. Observe that any product system is an
algebraic product system but the converse may not be true. The multiplication operation of a product
4 BHAT,B.V.R.,LINDSAY,J.M.,ANDMUKHERJEE,M.
system gives rise to the unitary maps which goes from to for every s,t > 0. Ad-joints of
s t s+t
E E ⊗E E
theseunitarymapsobviouslyassociativeandmakesitintoanalgebraicproductsystem. Thereforewecan
assumethat aproductsystemis a specialalgebraicproductsystem. Thoughthe linking mapsimplement
‘co-product’ rather than ‘product’ but abusing of terminology, we call it an algebraic product system.
Nevertheless, we can talk about an inclusion subsystem of a product system. The following important
fact that an inclusion subsystem in a product system generates a product subsystem is used throughout
without reference. For the proof, see Lemma 33, Appendix A. The following definition is taken from [8].
Definition 4. Let (E,β) be an inclusion system. Let u = u : t > 0 be a family of vectors such that
t
{ }
(1) for all t>0, u E (2) there is a k R, such that u exp(tk), for all t>0. and (3) u =0 for
t t t t
∈ ∈ k k≤ 6
some t>0. Then u is said to be a unit if
u =β∗ (u u ) s,t>0.
s+t s,t s⊗ t ∀
Let i :E be the canonical embedding.
t t t
→E
Theorem 5. Let (E,β) be an inclusion system and let ( ,B) be the algebraic product system generated
E
by it. Then the map i∗ provides a bijection between the set of all units of ( ,B) and the set of all units
E
of (E,β) by letting it acts point-wise on units.
For the proof, readers are referred to Theorem 10, [8].
Fixaunituof(E,β). Thenbythe abovetheoremthereisauniqueunituˆin( ,B)suchthatforevery
E
t>0, i∗(uˆ )=u . We say uˆ as the ‘lift’ of u. Note that if u is normalized, then uˆ is also normalized.
t t t
Amalgamation
The amalgamated product of two product systems over a contractive morphism is introduced in [8].
The index of the amalgamated product over general contractive morphism is computed in [16]. The
following theorem characterizes the amalgamated product. See Theorem 2.7, [16].
Theorem 6. Suppose ( ,WE) and ( ,WF) are two product systems and let C : ( ,WF) ( ,WE)
E F F → E
be a contractive morphism. Then there exist an algebraic product system ( ,WG) and isometric product
G
system morphisms I : and J : such that the following holds:
E →G F →G
(i) I (x),J (y) = x,C y for all x and y .
s s s s s
h i h i ∈E ∈F
(ii) =I( ) J( ).
G E F
W
is saidto be the amalgamatedproductof and overthe contractivemorphismC and denotedby
G E F
= . For the details of construction, we refer to Section 3, [8].
C
G E ⊗ F
3. Additive units
Suppose is a product system. The multiplication operation in is as follows: For s,t > 0, a ,
s
E E ∈ E
b , we have a b and =span . Also for a,a′ , b,b′ , we have
t s+t s+t s t s t
∈E · ∈E E E ·E ∈E ∈E
a a′,b b′ = a,b b,b′ .
h · · iEs+t h iEsh iEt
In this section, we abbreviate the multiplication a b as ab.
·
Definition 7. Let be a spatial product system and let u=(u ) be a unit of . A measurable section
t t>0
E E
(a ) of is said to be an additive unit of u if for all s,t>0,
t t>0
E
a =a u +u a .
s+t s t s t
Definition 8. An additive unit a = (a ) of a unit u = (u ) is said to be a root if a ,u = 0 for
t t>0 t t>0 t t
h i
all t>0.
Remark9. Itisclearthatthesetofalladditiveunitsofagivenunitformsavectorspaceunderpointwise
addition and point wise scalar multiplication. The set of all roots forms a vector subspace of it. Indeed
if a = (a ) and b = (b ) are two additive units(roots) of a unit u, then clearly λa := (λa )
s s>0 s s>0 s s>0
and (a+b) := (a +b ) are additive units(roots) of u = (u ) . Also note that, if a is an additive
s s s>0 s s>0
ADDITIVE UNITS OF PRODUCT SYSTEM 5
unit(roots) of u, then (a′) which is defined by a′ = exp(λs)a , is an additive unit(roots) of (u′) ,
s>0 s s s s>0
where u′ = exp(λs)u . In other words, the additive units of a unit are completely determined by the
s s
additive units of the normalized unit.
Example 10. Let u=(u ) be a unit in a product system . Then the measurable section b=(b )
s s>0 s s>0
E
given by b = λsu , for some λ C, s > 0, is an additive unit of the unit u. We call them as the trivial
s s
∈
additive units of the unit u.
Let a be an additive unit of a unit u. For s>0, consider the measurable function
f :R C
+
→
given by
f(s)= u ,a u −2.
s s s
h ik k
Then a simple computation shows that f(s + t) = f(s) + f(t), s,t > 0. This implies f(s) = sf(1).
Decomposing a =b +b′, where
s s s
b = u ,a u −2u
s s s s s
h ik k
and
b′ =a ( u ,a u −2u ),
s s− h s sik sk s
we find that b = (λsu ) for some λ C and b′ is a root of u. In other words, every additive unit
s s s>0
∈
decomposes uniquely as a trivialadditive unit anda root. Fromthe remark,we mayassume without loss
of generality that our unit u is normalized, i.e. u =1, for every s>0. Let a and b be two roots of the
s
k k
normalized unit u. Then a similar computation shows that
a ,b =s a ,b , s>0.
s s 1 1
h i h i
Now consider a,b two additive units of u. Then we can decompose
a =c +c′ , b =d +d′ , s>0,
s s s s s s
where
c =s u ,a u , d =s u ,b u ,s>0,
s 1 1 s s 1 1 s
h i h i
and c′,d′ are roots of u with
c′,d′ =s c′,d′ .
h s si h 1 1i
Now c′,d′ = (a u ,a u ),(b u ,b u ) = a ,b a ,u u ,b . From this, a simple
h 1 1i h 1 − h 1 1i 1 1 − h 1 1i 1 i h 1 1i − h 1 1ih 1 1i
computation shows a ,b =s2 a ,u u ,b +s a ,b s a ,u u ,b . In other words,
s s 1 1 1 1 1 1 1 1 1 1
h i h ih i h i− h ih i
(3.1) a ,b = θ a ,θ b ,
s s s 1 s 1
h i h i
where θs : 1 1 is given by θs =[sI+(s2 s)u1 ><u1 ]12.
E →E − | |
Proposition 11. Let u be a normalized unit of a product system . Then the set of all additive units of
E
u forms a Hilbert space under the inner product a,b =: a ,b and the set of all roots of u is a closed
h i h 1 1iE1
subspace of co-dimension one.
Proof: LetusdenotebyAE andRE bethevectorspacesofalladditiveunitsandrootsofurespectively.
u u
For a,b AE, define an inner product on AE by a,b = a ,b . Let an be a Cauchy sequence. i.e.
∈ u u h i h 1 1i { }n≥1
an am 0asm,n .NowfromEquation 3.1,wegetforeachs>0 an am = θ (an am)
k − k→ →∞ k s− s k k s 1− 1 k≤
θ an am = θ αn am 0. Let for each s>0, a =Lim an. The section (a ) is clearly
k skk 1 − 1 k k sk − k→ s n→∞ s s s>0
measurableasbeingpoint-wiselimitofmeasurablesections. Nowwewillshowthat(a ) isanadditive
s s>0
unit of u. Let ǫ>0 be given. For s,t>0 choose N such that for n>N, an a 1ǫ, an a 1ǫ
k s − sk≤ 3 k t − tk≤ 3
and an a 1ǫ. Then
k s+t− s+tk≤ 3
a a u u a
s+t s t s t
k − − k
a an + anu a u + u an u a ǫ.
≤k s+t− s+tk k s t− s tk k s t − s tk≤
So a AE and an a 0. This proves that AE is complete with respect to the inner product. Other
part i∈s truivial. k − k→ u (cid:3)
6 BHAT,B.V.R.,LINDSAY,J.M.,ANDMUKHERJEE,M.
Proposition 12. The set of all roots of the vacuum unit in CCR flow of index k is given by the set
cχ :c K almost everywhere.
t]
{ ∈ }
Proof: It is easyto see that cχ ,c K are the roots of the vacuum unit. To provethe converse,if a is
t]
∈
a root of the vacuum unit, then in Guichardet’s picture described in Appendix B , the following identity
is valid almost everywhere,
a (σ [t,s+t] t) if σ [0,t]=
s
∩ − ∩ ∅
a (σ)= a (σ [0,t]) if σ [t,s+t]= , σ ∆(s+t)
s+t t
 ∩0 o∩therwise. ∅ ∈
Fix s R . Let k be anynatural number. Denote by [a,b]′ the complement of [a,b] in [0,s]. Then we
+
∈
have the identity, almost everywhere ,
as(σ [(k−1)s,s] (k−1)s) if σ [(k−1)s,s]′ =
k ∩ k − k ∩ k ∅
 as(σ [(k−2)s,(k−1)s] (k−2)s) if σ [(k−2)s,(k−1)s]′ =
as(σ)= k ∩ k .. k − k ∩ k .. k ∅ , σ ∈∆(s)
as(σ [0, s]) if σ [0, s]′ =
 k ∩0 k ot∩herwkise. ∅
Suppose that # σ =n, then the subsetof ∆ (s), where a is nonzero exceptona setof measurezero,
n s
is contained in
k−1[∆ (s/k)+is/k], for all k =1,2, .
∪i=0 n ···
The Lebesgue measure of the set k−1[∆ (s/k)+is/k] is sn/n!kn−1. So the Lebesgue measure of the set
∪i=0 n
k−1[∆ (s/k)+is/k] is zero for n 2. It follows that a vanishes on ∆ (s), for n 2. As it is a
∩k≥1∪i=0 n ≥ s n ≥
root, it is orthogonal to the vacuum unit, we conclude that, a is a measurable function in L2([0,s],K)
s
with the property, a.e.
(3.2) a =a +S a , r, 0<r<s.
s r r s−r
∀
where S on L2(R ,K) defined by
t +
f(s t) if s t
(3.3) (Stf)(s) = (cid:26) 0− otherw≥ise.
Foreveryx K,definethemeasurablefunctionA :R C,byA (s)= a ,xχ .Aneasycalculation
x + x s s]
∈ → h | i
shows that A (s+t)=A (s)+A (t). Its measurable solution is given by A (s)=sA (1). Let us define
x x x x x
the linear functional f : C by f(x) = a ,xχ , for x K. It is bounded as f a . So by
1 1] 1 L2
K → h | i ∈ k k ≤ k k
Riesz representation theorem there is a unique y K such that f(x)= y,x . Now for r s, z K,
∈ h i ≤ ∈
a yχ ,zχ = a ,zχ r y,z
s s] r] r r]
h − | | i h | i− h i
=rA (1) r y,z
z
− h i
=0.
As the set zχ :z K,0 r s is total in L2([0,s],K), we have a =yχ . (cid:3)
r] s s]
{ | ∈ ≤ ≤ } |
Let us denote by RE, the Hilbert space of roots of the unit u in .
u E
Theorem 13. Suppose ( ,W) is a product system and u is a normalized unit of . Then dim RE =
E E u
index .
E
Proof: First we claim that roots of u are in I, the type I part of . Given a root a of u, a = 1,
E E k k
set E = span u ,a . Then it is easy to see that (E,W ) is an inclusion system. Let Γ (L2[0,t])
s s s E sym
{ } |
be the symmetric Fock product system. Define φ :E Γ (L2[0,t] by φ (αu +βa )=αΩ +βχ .
s s sym s s s s s]
→ |
Then φ = (φ ) is an isometric morphism of inclusion system. So the product system generated by
s s>0
u and a is isomorphic to a type I product system. This proves the claim. Any isomorphism of I to
E
Γ (L2[0,t],K) (dim K =index ) sending u to vacuum unit, sends roots to roots. This implies every
sym
root of u under this will be mappeEd to a (cχ ) and vice versa. The result now follows. (cid:3)
s] s>0
|
ADDITIVE UNITS OF PRODUCT SYSTEM 7
Corollary 14. Let a be a root of a unit u in a spatial product system ( ,B). Then a I.
E ∈E
Corollary 15. Suppose ( ,B) is a spatial product system and u is a unit. Then the product system
E
generated by the unit u and all roots of it, is the type I part of ( ,B).
E
Weshallnowdefineallthesenotionsonthelevelofinclusionsystem. Wequotethefollowingdefinition
from [8].
Definition 16. Let (E,β) be an inclusion system and let u be a normalized unit of (E,β). A section
(a ) of (E,β) is said to be an additive unit of the unit u if
t t>0
a =β∗ (a u +u a ) and a 2 k(s+s2), s>0, for some k 0.
s+t s,t s⊗ t s⊗ t k sk ≤ ≥
Definition 17. An additive unit a=(a ) of a unit u=(u ) is said to be a root if a ,u =0 for
t t>0 t t>0 t t
h i
all t>0.
Proposition18. Let(E,β)beaninclusionsystemandlet( ,B)bethealgebraicproductsystemgenerated
E
by it. Then i∗ provides a bijection between the set of all additive units of u in ( ,B) and the set of all
E
additive units of i∗(u) in (E,β) by letting it acts point-wise on units. More over if i∗(a) is a root of i∗(u),
then a is a root of u.
Proof: Suppose u is a unit of the algebraic product system ( ,B). Then by Theorem 5, i∗(u) is a unit
E
of the of the inclusion system and i∗(ˆu)=u. Let a be an additive unit of u. Consider i∗(a). Now
β∗ [i∗(a ) i∗(u )+i∗(u ) i∗(a )]=[(i i )β ]∗[a u +u a ]
s,t s s ⊗ t t s s ⊗ t t s⊗ t s,t s⊗ t s⊗ t
=[B i ]∗[a u +u a ]
s,t s+t s t s t
⊗ ⊗
=i∗ a .
s+t s+t
Hence i∗(a) is an additive unit of the unit i∗(u).
Nowweprovetheinjectivityofi∗.Considertwoadditiveunits aandbofthe unituin( ,B)suchthat
i∗tat = i∗tbt for all t > 0. Fix t > 0. For s = (s1,s2,...,sn) ∈Jt, Define as = nj=1us1 ⊗usE2 ⊗···usj−1 ⊗
asj ⊗usj+1 ⊗···⊗usn and bs = nj=1us1 ⊗us2 ⊗···⊗usj−1 ⊗bsj ⊗usj+1 ⊗P···⊗usn. Now for s∈Jt,
P
i∗sat =i∗sBt∗,sas
=(Bt,sis)∗as
n
=(i∗ i∗ ) u u u a u u
s1 ⊗···⊗ sn s1 ⊗ s2 ⊗···⊗ sj−1 ⊗ sj ⊗ sj+1 ⊗···⊗ sn
Xj=1
n
=(i∗ i∗ ) u u u b u u
s1 ⊗···⊗ sn s1 ⊗ s2 ⊗···⊗ sj−1 ⊗ sj ⊗ sj+1 ⊗···⊗ sn
Xj=1
=(Bt,sis)∗bs
=i∗sBt∗,sbs
=i∗b .
s t
This implies isi∗sat =isi∗sbt. The net of projection isi∗s :s Jt convergesstrongly to the identity. So
{ ∈ }
we get a =b .
t t
Conversely,letubeaunitandabeanadditiveunitofuin(E,β).Fixt>0.Fors=(s ,s ,...,s ) J ,
1 2 n t
Define as = nj=1us1 ⊗us2 ⊗···⊗usj−1 ⊗asj ⊗usj+1 ⊗···⊗usn. Now the family {isas : s ∈ J∈t} is
bounded as P
n
kisask2 ≤ k(si+s2i)
Xi=1
k(s+s2).
≤
8 BHAT,B.V.R.,LINDSAY,J.M.,ANDMUKHERJEE,M.
It follows from the hypothesis that, for s t J ,
t
≤ ∈
as =βs∗,tat.
Now for s t J ,
t
≤ ∈
isi∗sitat =isβs∗,tat =isas.
Given ǫ>0, x Et, choose s Jt such that (I isi∗s)x <ǫ. Then for t s, we have
∈ ∈ k − k ≥
itat isas,x = (I isi∗s)itat,x
h − i h − i
= itat,(I isi∗s)x
h − i
itat (I isi∗s)x
≤k kk − k
[k(s+s2)]21ǫ.
≤
So for each x t, isas,x :s Jt is a weakly Cauchy net. Set φ(x)= lim isas,x . Then φ: t C
∈E {h i ∈ } s∈Jth i E →
is a bounded linear functional with φ k(s + s2). So there is a unique vector aˆ such that
t t
k k ≤ ∈ E
φ(x)= aˆt,x . This implies for every x t, isas,x = aˆt,x . Now for s Jt,
h i ∈E h i h i ∈
isi∗saˆt = limisi∗sitat
t∈Jt
=tl∈imJtisβs∗,tat
=isas.
This shows that isas : s Jt converges to aˆt in the Hilbert space norm. Let uˆ be the lift of u in the
{ ∈ }
algebraic product system. Our claim is that aˆ=(aˆ ) is an additive unit of the unit uˆ=(uˆ ) in the
t t>0 t t>0
algebraic product system. For x , y ,
s t
∈E ∈E
aˆs uˆt+uˆs aˆt,x y = lim (is it)[as ut+us at],x y
h ⊗ ⊗ ⊗ i s∈Js,t∈Jth ⊗ ⊗ ⊗ ⊗ i
= lim (is it)as⌣t,(x y)
s∈Js,t∈Jth ⊗ ⊗ i
= lim Bs,tis⌣tas⌣t,x y
s∈Js,t∈Jth ⊗ i
=hs⌣lti∈mJs+tis⌣tas⌣t,Bs∗,t(x⊗y)i
= B aˆ ,x y .
s,t s+t
h ⊗ i
This proves the claim.
For x E , we have
t
∈
i∗aˆ ,x = aˆ ,i x
h t t i h t t i
= lim irar,itx
r∈Jth i
=rl∈imJthi∗tirar,xi
=rl∈imJthβt∗,ri∗rirar,xi
=rl∈imJthβt∗,rar,xi
= a ,x .
t
h i
This implies i∗aˆ =a .
t t t
Finally, if b is a root of a unit v in the inclusion system (E,β), then
ˆbt,vˆt = lim irbr,irvr
h i r∈Jth i
= lim br,vr
r∈Jth i
n
= lim v v v b v v ,v v v
r∈JtXj h r1 ⊗ r2 ⊗··· rj−1 ⊗ rj ⊗ rj+1 ⊗···⊗ rn r1 ⊗ r2 ⊗···⊗ rni
ADDITIVE UNITS OF PRODUCT SYSTEM 9
=0.
This proves the last assertion. (cid:3)
Here we show how the root space behaves under the amalgamationthrough partial isometry. Suppose
and are two product systems and C = (C ) : is a morphism of partial isometry. Also
t t>0 t t
E F F → E
assumethat isaproductsystem. Letv =(v ) beanormalizedunitof suchthatC∗C v =v
E⊗CF t t>0 F t t t t
for all t > 0. Then Cv := (C v ) is a normalized unit of . We have for every t > 0, I C v = J v ,
t t t>0 t t t t t
E
where I : ( ) and J : ( ) are injection morphisms. Let us denote the common
t t C t t t C t
E → E ⊗ F F → E ⊗ F
unit by u in . i.e. u =I (C v )=J (v ) for all t >0. Denoting E := a : a RE and
E ⊗C F t t t t t t RCv { 1 ∈E1 ∈ Cv}
F := b :b RF . Giventwo closedsubspaces H andH′ ofa Hilbert space G, denote by H H′
Rv { 1 ∈F1 ∈ v} ∨
the smallest closed subspace of G containing H and H′.
Theorem 19. Suppose ( ,WE) and ( ,WF) are two spatial product systems and suppose C =(C ) :
t t>0
E F
is a morphism of partial isometry such that the amalgamated product is a product
t t C
F → E E ⊗ F
system. Also Suppose v = (v ) is a normalized unit of such that C∗C v = v for all t > 0. Then
t t>0 F t t t t
Cv := (C v ) and v = (v ) are identified in . Denote the common unit by u = (u ) in
t t t>0 t t>0 C t t>0
E ⊗ F
E ⊗F. Then REu⊗CF =RECv⊕C1 RFv.
Proof: We may assume from Theorem 2.7, [16], that and are subsystems of the amalgamated
E F
product . As C is a morphism of partial isometry, we get from [16], Proposition 2.10, that for
C
E ⊗ F
each t > 0, P and P commute as elements in (( ) ). So P = P P , which implies
Et Ft B E ⊗C F t Et∩Ft Et Ft
:= ( ) is a product subsystem. In this identification, we have u = v = Cv. Hence u is a
t t t>0
E ∩F E ∩F
normalizedunitofE∩F andREu⊕C1RuF coincideswithREu∨RFu insideREu⊗CF.Sotoprovethe theorem,
it is enough to show that
E F = E⊗CF.
Ru∨Ru Ru
Clearly REu ∨RFu ⊂ REu⊗CF. Now for a ∈ RuE⊗CF, consider b = (bt)t>0 where bt = PEtat, b′ = (b′t)t>0
where b′ =P a and b′′ =(b′′) where b′′ =P a . We claim that
t Ft t t t>0 t Et∩Ft t
b RE, b′ RF, b′′ RE∩F.
∈ u ∈ u ∈ u
Note that for every s>0,
u =P =P u =P u .
s Es Fs s Es∩Fs s
As P =(P ) is a projection morphism from ( ,WE⊗F) to ( ,WE), we have
E Es s>0 E ⊗C F E
(P P )W(E⊗CF) =WE P , s,t>0.
Es ⊗ Et s+t s,t Es+t
This implies
WE b =WE P a
s,t s+t s,t Es+t s+t
=(P P )W(E⊗CF)a
Es ⊗ Et s+t s+t
=(P P )(a u +u a )
Es ⊗ Et s⊗ t s⊗ t
=(b u +u b ).
s t s t
⊗ ⊗
This shows b RE. Similarly we have, b′ RF and b′′ RE∩F. Also note that b,b′,b′′ RE⊗CF. Set
∈ u ∈ u ∈ u ∈ u
c=b+b′ b′′. Then we have for all t>0,
−
P (a c )=b b =0. P (a c )=0,.
Et t− t t− t Ft t− t
Therefore
P (a c )=0.
Et∨Ft t− t
Note that( ) )isaninclusionsystemwhichgeneratesthe productsystem . Alsonotethat
t t t>0 C
E ∨F E⊗ F
(P (a c )) is a root of u in the inclusion system ( ) ) while (a c ) is a root of u
Et∨Ft t− t t>0 Et∨Ft t>0 t− t t>0
in the product system ( ). As ( ) ) generates the product system ( ), we have from
C t t t>0 C
E ⊗ F E ∨F E ⊗ F
the injectivity of the map i∗ described in Theorem 18, for all t >0, a =c . So a = b b′′+b′, where
b1−b′1′ ∈REu and b′1 ∈RFu. Hence REu∨RFu =REu⊗CF. t t 1 1− 1 1 (cid:3)
10 BHAT,B.V.R.,LINDSAY,J.M.,ANDMUKHERJEE,M.
Suppose ( ,WE) and ( ,WF) are two product systems. Let u0 and v0 be two normalized units of
E F E
and respectively. Consider , where C = u0 v0 . In the amalgamatedproduct system ,
F E⊗CF t | tih t| E⊗CF
u0 and v0 are identified. We denote the common unit by σ.
Corollary 20. Let , , u0, v0, σ be as above. Then E⊗CF = E F.
E F Rσ Ru0 ⊕Rv0
Proof: For x E , y F,
∈Ru0 ∈Rv0
x,y = x,C y
h iC1 h 1 i
= x, u0 v0 y
h | 1ih 1| i
= x,u0 v0,y
h 1ih 1 i
=0.
(cid:3)
Remark 21. It is noted that thecondition on C that it is a partial isometry in Theorem 19 is a necessary
condition. It may not be true for general contractive morphism. Let = Cu and = Cv be two type
t t t t
E F
I product systems with u v < 1 for some t > 0. Let C = u v . Then E = 0 and F = 0. On
0 k tkk tk t | tih t| Ru Rv
the other side, we have is a type I product system. Though a priori, it is not clear whether in
C 1
E ⊗ F
this case, E ⊗C F is a product system. But this is indeed true ([15]). Therefore REσ⊗CF 6= {0} for every
unit σ in E ⊗C F. Hence REu⊗CF 6=REu⊕C1 RFu.
4. Amalgamation through normalized units is independent of the choice of units
In this section, we will show that the amalgamation through normalized unit does not depend on the
choice of the units. Proof of this fact is almost visible when we use the theory of random sets ([11]). In
[5], a short and self-contained proof has been presented. Also see [3]. Here we will prove this fact using
roots.
First, we show that the amalgamation of two spatial product systems through normalized units can
be identified with the product subsystem of the tensor product of the two systems. Let and be two
E F
spatialproductsystemsanduandv betwonormalizedunitsof and respectively. Defineacontractive
E F
morphismC =(C ) : byC = u v .Denote := .Fortwoproductsubsystems
t t>0 t t t t t u,v C
F →E | ih | E⊗ F E⊗ F
and ′ of the product system , we denote by ′ the smallest product subsystem of containing
G G H G G H
and ′. W
G G
Proposition 22. Suppose and are two spatial product systems and u and v are two normalized units
E F
of and respectively. Then is isomorphic to the product system generated by v and u
u,v
E F E⊗ F E⊗ ⊗F
inside , i.e. ( v) (u ).
u,v
E ⊗F E ⊗ F ≃ E ⊗ ⊗F
W
Proof: Asuandvarenormalized,weseethatI : vandJ : u areisometricmorphisms
E →E⊗ F → ⊗F
of product system. Also note that for x and y , I(x),J(y) = x, u v y . Now from the
s s t t
∈ E ∈ F h i h | ih | i
property of amalgamation(Theorem 2.7, [16]) we conclude that ( v) (u ) as
u,v
E⊗ F ≃ E⊗ ⊗F ⊂E⊗F
algebraic product systems. Now transferringthe measurable structure of ( v) (Wu ) onto u,v
E⊗ ⊗F E⊗ F
via the isomorphism, we can make u,v into a product system and the isoWmorphism becomes the
isomorphism of product systems. E ⊗ F (cid:3)
Suppose is a productsystem and u=(u ) is a normalizedunit of . Then for every interval[s,t],
t t>0
E E
0<s<t<1, we may identify, . Let P =P =1 P 1 .
E1 ≃Es⊗Et−s⊗E1−t s,t Es⊗Cut−s⊗E1−t Es ⊗ Cut−s ⊗ E1−t
From Proposition 3.18, [11], we know that (s,t) P is jointly continuous. So in the compact simplex
s,t
→
0 s t 1 , it is uniformly continuous. i.e. P goes to identity strongly as (t s) 0. In this
s,t
{ ≤ ≤ ≤ } − →
section, we denote the multiplication operation of the product system by i.e. a , b , we have
s t
◦ ∈ E ∈ E
a b . We write P as 1 P 1 . This is to differentiate the multiplication operation of
◦ ∈Es+t s,t Es ◦ Cut−s ◦ E1−t
the product system with the tensor product operation on the category of product systems. Though note
that this is not the usual operator multiplications as they are not acting on the same space. We hope
these notations do not lead any confusion.
For n≥1, we have Pi−n1,ni =1En1 ◦···◦1En1 ◦PCun1 ◦···◦1En1, where PCun1 on the i-th place.