ISSN 2291-8639

Volume 10, Number 2 (2016), 71-76 http://www.etamaths.com

ADDITIVE UNITS OF PRODUCT SYSTEM OF HILBERT MODULES

BILJANA VUJOˇSEVI ´C∗

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 HilbertC∗_{-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 unitωin a spatial product system overC∗-algebraBand prove that the set of all continuous additive units ofωcan be endowed with a structure of two-sided HilbertB − Bmodule wherein the set of all continuous roots ofωis a HilbertB − Bsubmodule.

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 unitu= (ut)t>0 in a

spatial product systemE is a measurable sectiona= (at)t>0, at∈Et,that satisfies

as+t=asut+usat

for alls, t >0, i.e. ais ”additive with respect to the given unit u”. An additive unita= (at)t>0 of a

unitu= (ut)t>0 is a root if for allt >0

hat, uti= 0.

In the same paper it is, also, proved that the set of all additive units of a unitu is a Hilbert space wherein the set of all roots ofuis 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 unitalC∗-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 unitω can be provided with a structure of two-sided HilbertB − Bmodule wherein the set of all continuous roots of

ωis a HilbertB − B submodule.

Throughout the whole paper, Bdenotes a unitalC∗-algebra and 1 denotes its unit. Also, we use⊗ for tensor product, althoughis in common use.

The rest of this section is devoted to basic definitions.

Definition 1.1. a) A HilbertB-moduleF is a rightB-module with a map h, i:F×F → B which satisfies the following properties:

• hx, λy+µzi=λhx, yi+µhx, ziforx, y, z∈F andλ, µ∈C;

• hx, yβi=hx, yiβ forx, y∈F andβ∈ B;

• hx, yi=hy, xi∗ _{for}_{x, y}_{∈}_{F}_{;}

• hx, xi ≥0 andhx, xi= 0⇔x= 0 forx∈F;

2010Mathematics 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.

andF is complete with respect to the normk · k=kh·,·ik12.

b) A Hilbert B − B module is a Hilbert B-module with a non-degenerate ∗−representation of B by elements in theC∗-algebraBa(F) of adjointable (and, therefore, bounded and right linear) mappings onF. The homomorphism j :B →Ba(F) is contractive. In particular, sinceC∗-algebraB is unital, the unit of B acts as the unit of Ba(F). Also, for x, y∈F and β ∈ B there holdshx, βyi=hβ∗x, yi

whereβy=j(β)(y).

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

Definition 1.2. a) A product system overC∗-algebraBis a family(Et)t≥0of HilbertB−Bmodules,

withE0∼=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

hu⊗v, u1⊗v1i=hv,hu, u1iv1i;

b) Unit on E is a familyu= (ut)t≥0,ut∈Et, so thatu0= 1 andϕt,s(ut⊗us) =ut+s, which we shall abbreviate tout⊗us=ut+s. A unitu= (ut)is unital ifhut, uti= 1. It is central if for allβ∈ B and allt≥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 unitalC∗-algebraB.

Definition 2.1. A familya= (at),at∈Et, is said to be an additive unit ofω ifa0= 0and

as+t=as⊗ωt+ωs⊗at, s, t≥0.

Definition 2.2. An additive unita= (at)of a unit ω= (ωt)is said to be a root ifhat, ωti= 0 for allt≥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. Forβ∈ B, letF_{β}a,b : [0,∞)→ B be the map defined by

(1) F_{β}a,b(s) =has, βbsi, s≥0,

wherea, bare additive units of ω inE.

We say that the set of additive units of ω S is continuous if the map F_{β}a,b is continuous for all
a, b∈ S, β∈ B. We say thata is a continuous additive unit of ω if the set{a} is continuous, i.e. if
the mapF_{β}a,ais continuous for each β∈ B. Denote the set of all continuous additive units ofω byAω
and the set of all continuous roots ofω 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_{β}a,a should be continuous for all

a∈ S, β∈ B, whereas in the first case allF_{β}a,b should be continuous.

Example 2.5. For γ ∈ B, the family (as)s≥0, where as=sγωs =sωsγ, is an additive unit of ω since fors, 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 anda0 = 0. Since F

a,a

β :s7→ hsωsγ, β(sωsγ)i=s2γ∗βγ is a continuous mapping for all β ∈ B, the additive unitabelongs toAω.

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

Lemma 2.6. 1. If ais a continuous additive unit ofω, then

(2) hωs, asi=shω1, a1i, s≥0.

2. Ifa, b are continuous roots ofω andβ ∈ B, then

(3) F_{β}a,b(s) =sF_{β}a,b(1), s≥0.

3. Ifais a continuous additive unit of ω, then a family(a0s)s≥0, where

(4) a0_{s}=as− hωs, asiωs,

is a continuous root ofω.

Proof. 1. LetGa _{: [0}_{,}_{∞}_{)}_{→ B}_{be the map defined by}_{G}a_{(}_{s}_{) =}_{h}_{ω}

s, asi, s≥0.Fors, t≥0 we obtain

Ga(s+t) =hωs+t, as⊗ωt+ωs⊗ati=hωs⊗ωt, as⊗ωti+hωs⊗ωt, ωs⊗ati=

=hωt,hωs, asiωti+hωt,hωs, ωsiati=hωs, asi+hωt, ati=Ga(s) +Ga(t)

and

kGa(s)−Ga(0)k2_{=}_{kh}_{ω}

s, asik2≤ kωsk2kask2=

=khas, asik=kF1a,a(s)k → kF

a,a

1 (0)k= 0, s→0.

Hence, the mapGa _{is continuous. Therefore,}_{G}a_{(}_{s}_{) =}_{sG}a_{(1), i.e.}

hωs, asi=shω1, a1i.

2. Let s, t≥0. Sincea, b∈ Rω, we see that

F_{β}a,b(s+t) =has⊗ωt+ωs⊗at, β(bs⊗ωt+ωs⊗bt)i=

=hωt,has, βbsiωti+hat,hωs, βωsibti=has, βbsi+hat, βbti=F a,b β (s) +F

a,b β (t) and

kF_{β}a,b(s)−F_{β}a,b(0)k2_{=}_{kh}_{a}

s, βbsik2≤ khas, asikkβk2khbs, bsik →0, s→0.

Hence, the mapF_{β}a,b is continuous and, therefore,F_{β}a,b(s) =sF_{β}a,b(1).

3. For s, t≥0, we obtain that

a0s+t=as⊗ωt+ωs⊗at− hωs⊗ωt, as⊗ωt+ωs⊗atiωs⊗ωt=

=as⊗ωt+ωs⊗at−(hωt,hωs, asiωti+hωt,hωs, ωsiati)ωs⊗ωt=

=as⊗ωt+ωs⊗at− hωs, asiωs⊗ωt− hωt, atiωs⊗ωt=

= (as− hωs, asiωs)⊗ωt+ωs⊗(at− hωt, atiωt) =as0 ⊗ωt+ωs⊗a0t

and

ha0_{s}, ωsi= 0.
Therefore,a0 is a root ofω.

Letβ ∈ B. By (4) and (2), it follows that

F_{β}a0,a0(s) =F_{β}a,a(s)−s2ha1, ω1iβhω1, a1i, s≥0.

Hence, the mapF_{β}a0,a0 is continuous which implies thata0_{∈ R}

ω.

Leta, bbe two continuous additive units ofω. By Remark 2.7, we can decompose them as

(5) as=shω1, a1iωs+a0s, bs=shω1, b1iωs+b0s, s≥0,

wherea0, b0∈ Rω. Therefore,

F_{β}a0,b0(1) =ha1− hω1, a1iω1, β(b1− hω1, b1iω1)i=

=F_{β}a,b(1)− ha1, ω1iβhω1, b1i, β∈ B.

Lets≥0 andβ∈ B. Since, by (3), there holds

F_{β}a0,b0(s) =sF_{β}a0,b0(1),

it follows that

(6) F_{β}a0,b0(s) =sF_{β}a,b(1)−sha1, ω1iβhω1, b1i.

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

(7) F_{β}a,b(s) =sF_{β}a,b(1) + (s2−s)ha1, ω1iβhω1, b1i.

It follows that the mapF_{β}a,b is continuous.

Therefore, we conclude that the set of all continuous additive units ofω 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 unitalC∗-algebraB.

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

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.

Lets, t≥0. Since (a+b)s+t= (a+b)s⊗ωt+ωs⊗(a+b)tandF_{β}a+b,a+b=F_{β}a,a+F_{β}b,a+F_{β}a,b+F_{β}b,b,
it follows thata+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_{β}aγ,aγ(s) =γ∗F_{β}a,a(s)γwhich implies that the mapF_{β}aγ,aγ is continuous. Therefore,aγ∈ Aω.

Similarly, (γa)s+t= (γa)s⊗ωt+ωs⊗(γa)t. By Remark 2.7,as=shω1, a1iωs+a0s, a0∈ Rω, and
we obtain thatF_{β}γa,γa(s) =s2ha1, ω1iγ∗βγhω1, a1i+Fa

0_{,a}0

γ∗_{βγ}(s). By (3), the mapF
γa,γa

β is continuous.

Therefore,γa∈ Aω.

The associativity and the commutativity follow directly. The neutral element is 0 = (0s) and the inverse ofais−a= (−as). The other axioms of two-sidedB−Bmodule (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 =afollow directly.

If a, b ∈ Rω, then has+bs, ωsi = 0, hasβ, ωsi = β∗has, ωsi = 0 and hβas, ωsi = has, β∗ωsi =

has, ωsβ∗i=has, ωsiβ∗= 0. Hence,a+b, aβ, βa∈ Rω. Since also 0 = (0s) and−a= (−as)∈ Rω, we

see thatRωis a B − B submodule inAω.

For every B 3β ≥0 there is a map h, iβ:Aω× Aω→ B given by

(8) ha, biβ=ha1, βb1i.

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

1. ha, λb+µciβ=λha, biβ+µha, ciβ for alla, b, c∈ Aω andλ, µ∈C; 2. ha, bγiβ=ha, biβγ for alla, b∈ Aω andγ∈ B;

5. ha, ai1= 0⇔a= 0 for alla∈ Aω;

6. ha, γbi1=hγ∗a, bi1 for all a, b∈ Aω andγ∈ B.

Proof. 1, 2, 3 - Straightforward calculation.

4 - Sinceβ ≥0, it follows thatβ=γ∗γfor someγ∈ B. Thus,ha, aiβ=ha1, γ∗γa1i=hγa1, γa1i ≥0.

5 - If ha, ai1= 0, then a1 = 0 by (8). By Remark 2.7,as=shω1, a1iωs+a0s,a0 ∈ Rω and s≥0, implying that as = a0s. Therefore, has, asi =sha01, a01iby (3). Now, it follows that has, asi= 0, i.e. as= 0 for alls≥0.

6 - Straightforward calculation. _{}

Theorem 3.3. The setAω(the set of all continuous additive units ofω) is a HilbertB − B module under the inner producth, i:Aω× Aω→ Bdefined by

(9) ha, bi=ha1, b1i, a, b∈ Aω.

The setRω (the set of all continuous roots of ω) is a Hilbert B − B submodule inAω.

Proof. We notice that the mapping h , i in (9) is equal to the mappingh , i1 in (8). Therefore, by

Theorem 3.1 and Proposition 3.2, we obtain thath, iis aB-valued inner product on B − B module Aω. Therefore,Aω is a pre-HilbertB − B module. Now, we need to prove thatAω is complete with respect to the inner product (9).

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

kam_{s} −an_{s}k2_{≤}_{(}_{s}2_{+ 2}_{s}_{)}_{k}_{a}m

1 −a

n

1k

2_{= (}_{s}2_{+ 2}_{s}_{)}_{k}_{a}m_{−}_{a}n_{k}2_{.}

(The last equality follows by (9).) Thus, (an

s) is a Cauchy sequence inEsand denote

(10) as= lim

n→∞a n s.

Letε >0 ands, t≥0. There isn0∈Nso thatkans−ask ≤ ε_{3},kant−atk ≤ _{3}ε andkans+t−as+tk ≤ ε_{3}
forn > n0. Then,

kas+t−as⊗ωt−ωs⊗atk ≤ kas+t−ans+tk+ka n

s+t−as⊗ωt−ωs⊗atk ≤

≤ kas+t−ans+tk+k(a n

s −as)⊗ωtk+kωs⊗(ant −at)k ≤ε. Hence,ais an additive unit ofω. Letβ∈ B. By (1), (10) and (7),

F_{β}a,a(s) = lim
n→∞F

an_{,a}n

β (s) = lim_{n}_{→∞}[sF
an_{,a}n

β (1) + (s

2_{−}_{s}_{)}_{h}_{a}n

1, ω1iβhω1, an1i] =

=sF_{β}a,a(1) + (s2−s)ha1, ω1iβhω1, a1i.

Hence, the mapF_{β}a,ais continuous, i.e.a∈ Aω. By (9) and (10),kan−ak=kan1−a1k →0, n→ ∞.

Therefore,Aω is complete with respect to the inner product (9).

Let (an_{) be a sequence in} _{R}

ω satisfying lim n→∞a

n _{=} _{a}_{. The only question is whether the }

contin-uous additive unit a belongs to Rω. However, this immediately follows from (10) since has, ωsi = lim

n→∞ha n

s, ωsi= 0 for alls≥0.

References

[1] B. V. R. Bhat, Martin Lindsay and Mithun Mukherjee, Additive units of product system, arXiv:1501.07675v1 [math.FA] 30 Jan 2015.

[2] S. D. Barreto, B. V. R. Bhat, V. Liebscher and M. Skeide, TypeIproduct systems of Hilbert modules,J. Funct. Anal.212(2004), 121–181.

[3] B. V. R. Bhat and M. Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups,Infin. Dimens. Anal. Quantum Probab. Relat. Top.3(2000), 519–575.

[4] D. J. Keˇcki´c, B. Vujoˇsevi´c, On the index of product systems of Hilbert modules,Filomat,29(2015), 1093-1111. [5] E. C. Lance, HilbertC∗-Modules: A toolkit for operator algebraists,Cambridge University Press,(1995). [6] V. M. Manuilov and E. V. Troitsky, HilbertC∗-Modules,American Mathematical Society(2005).

[7] M. Skeide, Dilation theory and continuous tensor product systems of Hilbert modules,PQQP: Quantum Probability and White Noise Analysis XV(2003), World Scientific.

[8] M. Skeide, Hilbert modules and application in quantum probability,Habilitationsschrift, Cottbus (2001).

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