International Journal of Analysis and Applications
ISSN 2291-8639
Volume 10, Number 2 (2016), 71-76
http://www.etamaths.com

ADDITIVE UNITS OF PRODUCT SYSTEM OF HILBERT MODULES

BILJANA VUJOŠEVIĆ∗

Abstract. In this paper we consider the notion of additive units and roots of a central unital unit
in a spatial product system of two-sided Hilbert C∗-modules. This is a generalization of the notion of
additive units and roots of a unit in a spatial product system of Hilbert spaces introduced in [B. V.

R. Bhat, M. Lindsay, M. Mukherjee, Additive units of product system, arXiv:1501.07675v1 [math.FA]
30 Jan 2015]. We introduce the notion of continuous additive unit and continuous root of a central

unital unit ω in a spatial product system over C∗-algebra B and prove that the set of all continuous
additive units of ω can be endowed with a structure of two-sided Hilbert B−B module wherein the
set of all continuous roots of ω is a Hilbert B−B submodule.

1. Introduction

The notion of additive units and roots of a unit in a spatial product system of Hilbert spaces is
introduced and studied in [1, Section 3]. In more details, an additive unit of a unit u = (ut)t>0 in a
spatial product system E is a measurable section a = (at)t>0, at ∈ Et, that satisfies

as+t = asut + usat

for all s,t > 0, i.e. a is ”additive with respect to the given unit u”. An additive unit a = (at)t>0 of a
unit u = (ut)t>0 is a root if for all t > 0

〈at,ut〉 = 0.
In the same paper it is, also, proved that the set of all additive units of a unit u is a Hilbert space
wherein the set of all roots of u is a Hilbert subspace.

The goal of this paper is to generalize the notion of additive units and roots of a unit in a spatial
product system of Hilbert spaces (from [1, Section 3]) and to obtain some similar results as therein but
in a more general context. To this purpose, we observe a spatial product system of two-sided Hilbert
modules over unital C∗-algebra B (it presents a product system that contains a central unital unit).
We introduce the notion of continuous additive unit and continuous root of a central unital unit. Also,
we show that the set of all continuous additive units of a central unital unit is continuous in a certain
sense. Finally, we prove that the set of all continuous additive units of a central unital unit ω can be
provided with a structure of two-sided Hilbert B−B module wherein the set of all continuous roots of
ω is a Hilbert B−B submodule.

Throughout the whole paper, B denotes a unital C∗-algebra and 1 denotes its unit. Also, we use ⊗
for tensor product, although � is in common use.

The rest of this section is devoted to basic definitions.

Definition 1.1. a) A Hilbert B-module F is a right B-module with a map 〈 , 〉 : F ×F →B which
satisfies the following properties:

• 〈x,λy + µz〉 = λ〈x,y〉 + µ〈x,z〉 for x,y,z ∈ F and λ,µ ∈ C;
• 〈x,yβ〉 = 〈x,y〉β for x,y ∈ F and β ∈B;
• 〈x,y〉 = 〈y,x〉∗ for x,y ∈ F ;
• 〈x,x〉≥ 0 and 〈x,x〉 = 0 ⇔ x = 0 for x ∈ F ;

2010 Mathematics Subject Classification. 46H25.
Key words and phrases. additive unit; Hilbert module; C∗-algebra.

c©2016 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

71



72 VUJOŠEVIĆ

and F is complete with respect to the norm ‖ ·‖ = ‖〈·, ·〉‖
1
2 .

b) A Hilbert B−B module is a Hilbert B-module with a non-degenerate ∗−representation of B by
elements in the C∗-algebra Ba(F) of adjointable (and, therefore, bounded and right linear) mappings
on F . The homomorphism j : B → Ba(F) is contractive. In particular, since C∗-algebra B is unital,
the unit of B acts as the unit of Ba(F). Also, for x,y ∈ F and β ∈ B there holds 〈x,βy〉 = 〈β∗x,y〉
where βy = j(β)(y).

For basic facts about Hilbert C∗-modules we refer the reader to [5] and [6].

Definition 1.2. a) A product system over C∗-algebra B is a family (Et)t≥0 of Hilbert B−B modules,
with E0 ∼= B, and a family of (unitary) isomorphisms

ϕt,s : Et ⊗Es → Et+s,
where ⊗ stands for the so-called inner tensor product obtained by identifications ub ⊗ v ∼ u ⊗ bv,
u⊗vb ∼ (u⊗v)b, bu⊗v ∼ b(u⊗v), (u ∈ Et, v ∈ Es, b ∈B) and then completing in the inner product
〈u⊗v,u1 ⊗v1〉 = 〈v,〈u,u1〉v1〉;

b) Unit on E is a family u = (ut)t≥0, ut ∈ Et, so that u0 = 1 and ϕt,s(ut ⊗us) = ut+s, which we
shall abbreviate to ut⊗us = ut+s. A unit u = (ut) is unital if 〈ut,ut〉 = 1. It is central if for all β ∈B
and all t ≥ 0 there holds βut = utβ.

Definition 1.3. The spatial product system is a product system that contains a central unital unit.

For a more detailed approach to this topic, we refer the reader to [2], [8], [9], [4].

2. Additive units

In this section we define all notions and prove auxiliary statements that are necessary for the proof
of main result that we present in Section 3.

Throughout the whole paper, ω = (ωt)t≥0 is a central unital unit in a spatial product system
E = (Et)t≥0 over unital C

∗-algebra B.

Definition 2.1. A family a = (at), at ∈ Et, is said to be an additive unit of ω if a0 = 0 and
as+t = as ⊗ωt + ωs ⊗at, s,t ≥ 0.

Definition 2.2. An additive unit a = (at) of a unit ω = (ωt) is said to be a root if 〈at,ωt〉 = 0 for
all t ≥ 0.

The previous definitions do not include any technical condition, such as measurability or continuity.
It occurs that it is sometimes more convenient to pose the continuity condition directly on units.

Definition 2.3. For β ∈B, let Fa,bβ : [0,∞) →B be the map defined by

(1) F
a,b
β (s) = 〈as,βbs〉, s ≥ 0,

where a,b are additive units of ω in E.

We say that the set of additive units of ω S is continuous if the map Fa,bβ is continuous for all
a,b ∈ S, β ∈ B. We say that a is a continuous additive unit of ω if the set {a} is continuous, i.e. if
the map F

a,a
β is continuous for each β ∈B. Denote the set of all continuous additive units of ω by Aω

and the set of all continuous roots of ω by Rω.

Remark 2.4. We should tell the difference between the continuous set of additive units of ω and
the set of continuous additive units of ω. In the second case only F

a,a
β should be continuous for all

a ∈S, β ∈B, whereas in the first case all Fa,bβ should be continuous.

The following example assures us that the set of all continuous additive units of a central unital
unit ω in a spatial product system is not empty.



ADDITIVE UNITS OF PRODUCT SYSTEM OF HILBERT MODULES 73

Example 2.5. For γ ∈ B, the family (as)s≥0, where as = sγωs = sωsγ, is an additive unit of ω
since for s,t ≥ 0 there holds

as+t = (s + t)γωs ⊗ωt = sγωs ⊗ωt + tωsγ ⊗ωt = sγωs ⊗ωt + tωs ⊗γωt =
= sγωs ⊗ωt + ωs ⊗ tγωt = as ⊗ωt + ωs ⊗at

and a0 = 0. Since F
a,a
β : s 7→ 〈sωsγ,β(sωsγ)〉 = s

2γ∗βγ is a continuous mapping for all β ∈ B, the
additive unit a belongs to Aω.

The properties of additive units of ω are given in the following lemma:

Lemma 2.6. 1. If a is a continuous additive unit of ω, then

(2) 〈ωs,as〉 = s〈ω1,a1〉, s ≥ 0.
2. If a,b are continuous roots of ω and β ∈B, then

(3) F
a,b
β (s) = sF

a,b
β (1), s ≥ 0.

3. If a is a continuous additive unit of ω, then a family (a′s)s≥0, where

(4) a′s = as −〈ωs,as〉ωs,
is a continuous root of ω.

Proof. 1. Let Ga : [0,∞) →B be the map defined by Ga(s) = 〈ωs,as〉, s ≥ 0. For s,t ≥ 0 we obtain
Ga(s + t) = 〈ωs+t,as ⊗ωt + ωs ⊗at〉 = 〈ωs ⊗ωt,as ⊗ωt〉 + 〈ωs ⊗ωt,ωs ⊗at〉 =

= 〈ωt,〈ωs,as〉ωt〉 + 〈ωt,〈ωs,ωs〉at〉 = 〈ωs,as〉 + 〈ωt,at〉 = Ga(s) + Ga(t)
and

‖Ga(s) −Ga(0)‖2 = ‖〈ωs,as〉‖2 ≤‖ωs‖2‖as‖2 =
= ‖〈as,as〉‖ = ‖F

a,a
1 (s)‖→‖F

a,a
1 (0)‖ = 0, s → 0.

Hence, the map Ga is continuous. Therefore, Ga(s) = sGa(1), i.e.

〈ωs,as〉 = s〈ω1,a1〉.
2. Let s,t ≥ 0. Since a,b ∈Rω, we see that

F
a,b
β (s + t) = 〈as ⊗ωt + ωs ⊗at,β(bs ⊗ωt + ωs ⊗ bt)〉 =

= 〈ωt,〈as,βbs〉ωt〉 + 〈at,〈ωs,βωs〉bt〉 = 〈as,βbs〉 + 〈at,βbt〉 = F
a,b
β (s) + F

a,b
β (t)

and
‖Fa,bβ (s) −F

a,b
β (0)‖

2 = ‖〈as,βbs〉‖2 ≤‖〈as,as〉‖‖β‖2‖〈bs,bs〉‖→ 0, s → 0.

Hence, the map F
a,b
β is continuous and, therefore, F

a,b
β (s) = sF

a,b
β (1).

3. For s,t ≥ 0, we obtain that
a′s+t = as ⊗ωt + ωs ⊗at −〈ωs ⊗ωt,as ⊗ωt + ωs ⊗at〉ωs ⊗ωt =

= as ⊗ωt + ωs ⊗at − (〈ωt,〈ωs,as〉ωt〉 + 〈ωt,〈ωs,ωs〉at〉)ωs ⊗ωt =
= as ⊗ωt + ωs ⊗at −〈ωs,as〉ωs ⊗ωt −〈ωt,at〉ωs ⊗ωt =

= (as −〈ωs,as〉ωs) ⊗ωt + ωs ⊗ (at −〈ωt,at〉ωt) = a′s ⊗ωt + ωs ⊗a
′
t

and
〈a′s,ωs〉 = 0.

Therefore, a′ is a root of ω.
Let β ∈B. By (4) and (2), it follows that

F
a′,a′

β (s) = F
a,a
β (s) −s

2〈a1,ω1〉β〈ω1,a1〉, s ≥ 0.

Hence, the map F
a′,a′

β is continuous which implies that a
′ ∈Rω. �

Remark 2.7. Let a be a continuous additive unit of ω. By (2) and (4), it can be decomposed as
as = s〈ω1,a1〉ωs + a′s, s ≥ 0, where a′ is a continuous root of ω.



74 VUJOŠEVIĆ

Let a,b be two continuous additive units of ω. By Remark 2.7, we can decompose them as

(5) as = s〈ω1,a1〉ωs + a′s, bs = s〈ω1,b1〉ωs + b
′
s, s ≥ 0,

where a′,b′ ∈Rω. Therefore,

F
a′,b′

β (1) = 〈a1 −〈ω1,a1〉ω1,β(b1 −〈ω1,b1〉ω1)〉 =

= F
a,b
β (1) −〈a1,ω1〉β〈ω1,b1〉, β ∈B.

Let s ≥ 0 and β ∈B. Since, by (3), there holds

F
a′,b′

β (s) = sF
a′,b′

β (1),

it follows that

(6) F
a′,b′

β (s) = sF
a,b
β (1) −s〈a1,ω1〉β〈ω1,b1〉.

Now, by (5) and (6), we obtain that

(7) F
a,b
β (s) = sF

a,b
β (1) + (s

2 −s)〈a1,ω1〉β〈ω1,b1〉.

It follows that the map F
a,b
β is continuous.

Therefore, we conclude that the set of all continuous additive units of ω is continuous in the sense
of Definition 2.3.

3. The result

In this section we prove the main result.
Throughout the whole section, ω = (ωt)t≥0 is a central unital unit in a spatial product system

E = (Et)t≥0 over unital C
∗-algebra B.

Theorem 3.1. The set Aω (the set of all continuous additive units of ω) is a B−B module under
the point-wise addition and point-wise scalar multiplication. The set Rω (the set of all continuous
roots of ω) is a B−B submodule in Aω.

Proof. Let a = (as), b = (bs) ∈ Aω and β ∈ B. For s ≥ 0, (a + b)s = as + bs, (aβ)s = asβ and
(βa)s = βas.

Let s,t ≥ 0. Since (a+b)s+t = (a+b)s⊗ωt +ωs⊗(a+b)t and F
a+b,a+b
β = F

a,a
β +F

b,a
β +F

a,b
β +F

b,b
β ,

it follows that a + b ∈Aω.
Let γ ∈ B. Since the unit ω is central, we obtain that (aγ)s+t = (aγ)s ⊗ ωt + ωs ⊗ (aγ)t. Also,

F
aγ,aγ
β (s) = γ

∗F
a,a
β (s)γ which implies that the map F

aγ,aγ
β is continuous. Therefore, aγ ∈Aω.

Similarly, (γa)s+t = (γa)s ⊗ωt + ωs ⊗ (γa)t. By Remark 2.7, as = s〈ω1,a1〉ωs + a′s, a′ ∈Rω, and
we obtain that F

γa,γa
β (s) = s

2〈a1,ω1〉γ∗βγ〈ω1,a1〉 + F
a′,a′

γ∗βγ(s). By (3), the map F
γa,γa
β is continuous.

Therefore, γa ∈Aω.
The associativity and the commutativity follow directly. The neutral element is 0 = (0s) and the

inverse of a is −a = (−as). The other axioms of two-sided B−B module (aβ)γ = a(βγ), β(γa) = (βγ)a,
β(a + b) = βa + βb, (a + b)β = aβ + bβ, (β + γ)a = βa + γa, a(β + γ) = aβ + aγ, 1a = a1 = a follow
directly.

If a,b ∈ Rω, then 〈as + bs,ωs〉 = 0, 〈asβ,ωs〉 = β∗〈as,ωs〉 = 0 and 〈βas,ωs〉 = 〈as,β∗ωs〉 =
〈as,ωsβ∗〉 = 〈as,ωs〉β∗ = 0. Hence, a + b,aβ,βa ∈Rω. Since also 0 = (0s) and −a = (−as) ∈Rω, we
see that Rω is a B−B submodule in Aω. �

For every B 3 β ≥ 0 there is a map 〈 , 〉β : Aω ×Aω →B given by
(8) 〈a,b〉β = 〈a1,βb1〉.

Proposition 3.2. The pairing (8) satisfies the following properties:

1. 〈a,λb + µc〉β = λ〈a,b〉β + µ〈a,c〉β for all a,b,c ∈Aω and λ,µ ∈ C;
2. 〈a,bγ〉β = 〈a,b〉βγ for all a,b ∈Aω and γ ∈B;
3. 〈a,b〉β = 〈b,a〉∗β for all a,b ∈Aω;
4. 〈a,a〉β ≥ 0 for all a ∈Aω;



ADDITIVE UNITS OF PRODUCT SYSTEM OF HILBERT MODULES 75

5. 〈a,a〉1 = 0 ⇔ a = 0 for all a ∈Aω;
6. 〈a,γb〉1 = 〈γ∗a,b〉1 for all a,b ∈Aω and γ ∈B.

Proof. 1, 2, 3 - Straightforward calculation.
4 - Since β ≥ 0, it follows that β = γ∗γ for some γ ∈B. Thus, 〈a,a〉β = 〈a1,γ∗γa1〉 = 〈γa1,γa1〉≥ 0.
5 - If 〈a,a〉1 = 0, then a1 = 0 by (8). By Remark 2.7, as = s〈ω1,a1〉ωs + a′s, a′ ∈ Rω and s ≥ 0,

implying that as = a
′
s. Therefore, 〈as,as〉 = s〈a′1,a′1〉 by (3). Now, it follows that 〈as,as〉 = 0, i.e.

as = 0 for all s ≥ 0.
6 - Straightforward calculation. �

Theorem 3.3. The set Aω (the set of all continuous additive units of ω) is a Hilbert B−B module
under the inner product 〈 , 〉 : Aω ×Aω →B defined by
(9) 〈a,b〉 = 〈a1,b1〉, a,b ∈Aω.
The set Rω (the set of all continuous roots of ω) is a Hilbert B−B submodule in Aω.

Proof. We notice that the mapping 〈 , 〉 in (9) is equal to the mapping 〈 , 〉1 in (8). Therefore, by
Theorem 3.1 and Proposition 3.2, we obtain that 〈 , 〉 is a B-valued inner product on B−B module
Aω. Therefore, Aω is a pre-Hilbert B−B module. Now, we need to prove that Aω is complete with
respect to the inner product (9).

Let (an) be a Cauchy sequence in Aω and s ≥ 0. If β = 1 and a = b = am −an in (7), it follows
that

‖ams −a
n
s‖

2 ≤ (s2 + 2s)‖am1 −a
n
1‖

2 = (s2 + 2s)‖am −an‖2.
(The last equality follows by (9).) Thus, (ans ) is a Cauchy sequence in Es and denote

(10) as = lim
n→∞

ans .

Let ε > 0 and s,t ≥ 0. There is n0 ∈ N so that ‖ans −as‖≤
ε
3
, ‖ant −at‖≤

ε
3

and ‖ans+t−as+t‖≤
ε
3

for n > n0. Then,

‖as+t −as ⊗ωt −ωs ⊗at‖≤‖as+t −ans+t‖ + ‖a
n
s+t −as ⊗ωt −ωs ⊗at‖≤

≤‖as+t −ans+t‖ + ‖(a
n
s −as) ⊗ωt‖ + ‖ωs ⊗ (a

n
t −at)‖≤ ε.

Hence, a is an additive unit of ω. Let β ∈B. By (1), (10) and (7),

F
a,a
β (s) = limn→∞

F
an,an

β (s) = limn→∞
[sF

an,an

β (1) + (s
2 −s)〈an1 ,ω1〉β〈ω1,a

n
1〉] =

= sF
a,a
β (1) + (s

2 −s)〈a1,ω1〉β〈ω1,a1〉.
Hence, the map F

a,a
β is continuous, i.e. a ∈Aω. By (9) and (10), ‖a

n −a‖ = ‖an1 −a1‖→ 0, n →∞.
Therefore, Aω is complete with respect to the inner product (9).

Let (an) be a sequence in Rω satisfying lim
n→∞

an = a. The only question is whether the contin-

uous additive unit a belongs to Rω. However, this immediately follows from (10) since 〈as,ωs〉 =
lim
n→∞

〈ans ,ωs〉 = 0 for all s ≥ 0. �

References

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[math.FA] 30 Jan 2015.

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[4] D. J. Kečkić, B. Vujošević, On the index of product systems of Hilbert modules, Filomat, 29 (2015), 1093-1111.
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76 VUJOŠEVIĆ

Faculty of Mathematics, University of Belgrade, Studentski trg 16-18, 11000 Beograd, Serbia

∗Corresponding author: bvujosevic@matf.bg.ac.rs