For reader convenience we collect below some
preliminary definitions and propositions that
will be used in what follows. A quasi *-
algebra is a couple (A,A0), where Ais a vec-
tor space with involution ∗,A0is a *-algebra
and a vector subspace of Aand Ais an A0-
bimodule whose module operations and invo-
lution extend those of A0. The unit of (A,A0)
is an element e∈ A0such that xe =ex =x,
for every x∈ A. A quasi *-algebra (A,A0)
is said to be locally convex if Ais endowed
with a topology τwhich makes of Aa lo-
cally convex space and such that the involu-
tion a7→ a∗and the multiplications a7→ ab,
a7→ ba,b∈ A0, are continuous. If τis a norm
topology and the involution is isometric with
respect to the norm, we say that (A,A0) is a
normed quasi *-algebra and, if it is complete,
we say it is a Banach quasi*-algebra.
Let A#be a C*-algebra, with involution
#and norm k k#, and X[k k] a left Banach
A#-module. This means, in particular, that
there is a bounded bilinear map
(a, x)→ax
from A#× X into Xsuch that
(a1a2)x=a1(a2x),∀a1, a2∈ A#, x ∈ X .
We will always assume that the following in-
equality holds:
kaxk ≤ kak#kxk,∀x∈ X , a ∈ A#.
This implies, as shown in [7], that
kak#= sup
x∈X ;∥x∥≤1
kaxk, a ∈ A#.
A left Banach A#-module Xis called a
CQ*-algebra if
(i) in Xan involution ∗is defined and
kx∗k=kxkfor every x∈ X ;
Some classes of quasi *-algebras
S. TRIOLO
Dipartimento di Ingegneria, Universit`a di Palermo,
I-90123 Palermo, ITALY
1. Introduction
Abstract: In this paper we will continue the analysis undertaken in [1] and in [2] [20] our
investigation on the structure of Quasi-local quasi *-algebras possessing a sufficient family of
bounded positive tracial sesquilinear forms. In this paper it is shown that any Quasi-local quasi
*-algebras (A, A0), possessing a ”sufficient state” can be represented as non-commutative L2-
spaces.
Keywords: Quasi *-algebras, Non-commutative L2-spaces.
Received: October 19, 2021. Revised: June 13, 2022. Accepted: July 7, 2022. Published: August 1, 2022.
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(ii) A#is a k k-dense subspace of Xand the
module left-multiplication extends the
multiplication of A#;
(iii) A#∩ A∗
#is a *-algebra and it is dense in
A#with respect to k k#;
A CQ*-algebra is denoted with (X,∗,A#,#)
to underline the different involutions.
If ∗=#on A#, the CQ*-algebra is called
proper.
The following basic definitions and results
on non-commutative measure theory and in-
tegration are also needed in what follows.
Let Mbe a von Neumann algebra and φ
a normal faithful semifinite trace defined on
M+.
Put
J={X∈ M :φ(|X|)<∞}.
Jis a *-ideal of M.
Let P∈Proj(M), the lattice of projec-
tions of M. We say that Pis φ-finite if
P∈ J . Any φ-finite projection is finite.
Definition 1 A vector subspace Dof His
said to be strongly dense ( resp., strongly φ-
dense) if
•U′D ⊂ D for any unitary U′in M′
•there exists a sequence Pn∈Proj(M):
PnH ⊂ D ,P⊥
n↓0and (P⊥
n)is a finite
projection (resp., φ(P⊥
n)<∞).
Clearly, every strongly φ-dense domain is
strongly dense.
Throughout this paper, when we say that
an operator Tis affiliated with a von Neu-
mann algebra, written T η M, we always
mean that Tis closed, densely defined and
T U ⊇U T for every unitary operator U∈
M′.
Definition 2 An operator T η Mis called
•measurable (with respect to M) if its do-
main D(T)is strongly dense;
•φ-measurable if its domain D(T)is
strongly φ-dense.
From the definition itself it follows that, if
Tis φ-measurable, then there exists P∈
Proj(M) such that T P is bounded and
φ(P⊥)<∞.
We remind that any operator affiliated
with a finite von Neumann algebra is mea-
surable [12, Cor. 4.1] but it is not necessarily
φ-measurable.
The following statements will be used
later.
(i) Let T η Mand Q∈ M. If D(T Q) =
{ξ∈ H :Qξ ∈D(T)}is dense in H, then
T Q η M.
(ii) If Q∈Proj(M), then QMQ=
{QXQ QH;X∈ M} is a von Neumann
algebra on the Hilbert space QH; more-
over (QMQ)′=QM′Q. If T η Mand
Q∈ M and D(T Q) = {ξ∈ H :Qξ ∈
D(T)}is dense in H, then QT Q η QMQ.
Let Mbe a von Neumann algebra and φ
a normal faithful semifinite trace defined on
M+. For each p≥1, let
Jp={X∈ M :φ(|X|p)<∞}.
Then Jpis a *-ideal of M. Following [13], we
denote with Lp(φ) the Banach space comple-
tion of Jpwith respect to the norm
kXkp:= φ(|X|p)1/p, X ∈ Jp.
One usually defines L∞(φ) = M. Thus, if
φis a finite trace, then L∞(φ)⊂Lp(φ) for
every p≥1. As shown in [13], if X∈Lp(φ),
then Xis a measurable operator.
We consider now the case where Ahas a lo-
cal structure. Following [10] we construct the
local net of C*-algebras as follows.
2. Quasi Local Structure
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Let Fbe a set of indexes directed upward
and with an orthonormality relation ⊥such
that
(i.) ∀α∈ F there exists β∈ F such that
α⊥β;
(ii.) if α≤βand β⊥γ,α, β, γ ∈ F, then
α⊥γ;
(iii.) if, for α, β, γ ∈ F,α⊥βand α⊥γ,
there exists δ∈ F such that α⊥δand
δ≥β, γ.
Definition 3 Let (A,A0)be a quasi *-
algebra with unit e. We say that (A,A0)
has a local structure if there exists a net
{Aα(k.kα), α ∈ F} of subspaces of A0, in-
dexed by F, such that every Aαis a C*-
algebra (with norm k.kαand unit e) with the
properties
(a.) A0=Sα∈F Aα
(b.) if α≥βthen Aα⊃ Aβ;
(c.) if α⊥β, then xy =yx for every x∈ Aα,
y∈ Aβ.
(d.) if x∈ Aα∩ Aβ, then kxkα=kxkβ.
A quasi *-algebra (A,A0)with a local struc-
ture will be shortly called a quasi-local quasi*-
algebra.
If (A,A0) is a quasi-local quasi-*algebra,
and x∈ A0, there will be some β∈ F such
that x∈ Aβ. We put Jx={α∈ F :x∈ Aα}
and define
kxk= inf
α∈Jx
kxkα.
Then A0is a C*-normed algebra with norm
k·k.
For instance if we consider the quasi *-
algebra (Lp(I), L∞(I)), where Lp(I) (p≥1)
and L∞(I) are the usual Lebesgue spaces on
I:= [0,1]. Put ω(f) = R1
0f(x)dx,f∈Lp(I).
If 1 ≤p < 2, ωis not representable. Indeed,
if f∈Lp(I)\L2(I), there cannot exist any
γf>0 such that
|ω(fφ)|=Z1
0
f(x)φ(x)dx≤γfω(φ∗φ)1/2=γfkφk2,∀φ∈L∞(I),
since this would imply that f∈L2(I) (see
[1]).
The following proposition extends the
GNS construction to quasi *-algebras. The
prof hers in [9].
Proposition 4 Let ωbe a linear functional
on Aτ−continuous satisfying the following
requirements:
(L1) ω(a∗a) = ω(aa∗)≥0for all a∈ A0;
(L2) ω(b∗x∗a) = ω(a∗xb),∀a, b ∈ A0,x∈
A;
(L3) ∀x∈ A there exists γx>0such that
|ω(x∗a)| ≤ γxω(a∗a)1/2.
Then there exists a triple (πω, λω,Hω)
such that
•πωis a ultra-cyclic *-representation of A
with ultra-cyclic vector ξω;
•λωis a linear map of Ainto Hω
with λω(A0) = Dπω,ξω=λω(e)and
πω(x)λω(a) = λω(xa), for every x∈
A, a ∈ A0.
•ω(x) = hπω(x)ξω|ξωi, for every x∈ A.
•π∗
ω(a)λω(x) = λω(ax), for every x∈ A,
a∈ Ao.
Definition 5 A linear functional ω τ −con-
tinuous satisfying (L1),(L2),(L3) is called
representable. We denote by T(A)the set
of representable linear functionals on A.
Once we have constructed in the previous sec-
tion some Quasi-local quasi *-algebras of op-
erators affiliated to a given von Neumann al-
gebra, it is natural to pose the question under
3. A Representation Theorem
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which conditions can an abstract Quasi-local
quasi *-algebras be realized as a Quasi-local
quasi *-algebras of this type.
We denotes by [M] the closed subspace of
Hspanned by [M] for any subset Mof H.
Thus for every ω∈ T (A) we put Hα:=
[πω(Aα)ξω] then {πωAα,Hα, ξω}} it is a rep-
resentation of Aα.
Thus, let πωbe the ultra-cyclic *-
representation of Awith ultra-cyclic vector
ξωand πω(Aα)′′ the von Neumann algebra
generated by πω(Aα).
For every ω∈ T (A) and a∈ Aα, we put
φω,α(πω(a)) = ω(a) = hπω(a)ξω|ξωi.
Then, for each ω∈ T (A), φω,α is a posi-
tive bounded linear functional on the opera-
tor algebra πω(Aα).Clearly, for every a∈ Aα
|φω(πω(a)) |=| hπω(a)ξω|ξωi |≤ kπω(a)kkξωk2
Thus φω,α is continuous on πω(Aα).
By [17, Theorem 10.1.2], φω,α is weakly
continuous and so it extends uniquely to
πω(Aα)′′ . Moreover, since φω,α is a trace on
πω(Aα):
φω,α(a∗a) = ω(a∗a) = ω(aa∗) = φω,α(aa∗)
the extension gφω,α is a trace on Mω:=
πω(Aα)′′ too.
Theorem 6 If (A,A0)has a local structure,
ωa state over Asatisfying (L1, L2, L3) such
that πω,the canonical cyclic representation
of Aassociated with ω([9]), is continuous,
therefore (πω(A), πω(A0)) has a local struc-
ture
Proof: It is easy to verify the following prop-
erty
πω(A0) = ∪α{πωAα(Aα)}=∪α{πω(Aα)} ⊆ B(Hω)
with πω(Aα)⊆B(Hα) is a C* algebra
which norm k·kB(Hα)≤ k · k.
The family of C*-algebras {πω(Aα), α ∈
F} with C*-norm k.kB(Hα), indexed by F,
satisfies the following property (a.) if α≥β
then πω(Aα)⊃πω(Aβ); (b.) there exists a
unique identity πω(e) for all πω(Aα) and the
C*-norm k.kB(Hα)are equals a k.kB(H); (c.) if
α⊥βthen for all X∈πω(Aα), Y∈πω(Aβ)
there exist x∈ Aα,y∈ Aβsuch that πω(x) =
Xand πω(y) = Ybut xy =yx for all x∈
Aα,y∈ Aβtherefore XY =πω(x)πω(y) =
πω(xy) = πω(yx) = πω(y)πω(x) = Y X.
Thus πω(A0) is, a quasi-local C*-algebra
of operator.
2
Let Pαthe operator of projection of Hin
HαPut Mω
α:= MωPα, where, as before, Pα
denotes the support of gφω,α.
Each Mω
αis a von Neumann algebra and
gφω,α is faithful in MPα[6, Proposition V.
2.10].
More precisely,
Mω
α:= MωPα={Z=XPα,for some X∈ Mω}.
In this case, putting Hα=PαH, we have
H=M
α∈I
Hα={(fα) : fα∈ Hα,X
α∈I
kfαk2<∞}.
Each vector X={fα}α∈I ∈ H is de-
noted by X=P⊕
α∈Ifα(Definition 3.4, [6]).
For each bounded sequencese {Aα}α∈I ∈
Qα∈I Mω
α,we define an operator A(follow-
ing [6]) on Hby
AX := A
⊕
X
α∈I
fα=
⊕
X
α∈I
Aαfα.
Clearly Ais a bounded operator on Hwe
denote it by A=P⊕
α∈IAα.
Let P⊕
α∈IMω
αthe set of all such A, by Propo-
sition 3.3 [6], P⊕
α∈IMω
αis a von Neumann al-
gebra on H.The algebra P⊕
α∈IMω
αis called
the diret sum of {Mα}.Of course for ev-
ery ωwe have πω(A0) = P⊕
α∈IMω
αand
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fφω:= Pα∈Igφω,α is a faithful semifinite nor-
mal trace on Mω.
The previous discussion can be summer-
ized in the following
Theorem 7 If (A,A0)has a local structure,
ωa state over Asatisfying (L1, L2, L3) such
that πω,the canonical cyclic representation
of Aassociated with ω([9]), is continuous,
therefore ( L2(fφω), πω(A0)′′)has a local struc-
ture.
Theorem 8 If (A,A0)has a local structure
the CQ*-Algebra ( L2(fφω), πω(Ao)′′)) consists
of operators affiliated with πω(Ao)′′.
Proposition 9 If (A,A0)is a quasi-local
quasi-*algebra, ω∈ T (A)and (Hω, πω, ξω)is
a the canonical cyclic representation of Aas-
sociated with ω, let τthe weakest locally topol-
ogy on Asuch that πωis continuous from
Ao(τ)into πω(Ao).
then there exist a quasi-local quasi *-
algebra ( L2(fφω), πω(A0)) and a onomorphism
Φ : x∈ A → Φ(x) := e
X∈ L2(fφω)
with the following properties:
(i) Φextends the representation πωof A0;
(ii) Φ(x∗) = Φ(x)∗,∀x∈ A;
(iii) Φ(xy) = Φ(x)Φ(y)for every x, y ∈ A
such that x∈ A0or y∈ A0.
Proof:
For every element x∈ A, there exists a
sequence {an}of elements of A0converging
to xwith respect τ. Put Xn=πω(an). Then,
φω(|Xn−Xm|2)→0.Let e
Xbe the k·k2-limit
of the sequence (Xn).We define Φ(x) := e
X.
2
Concluding remark – We have discussed
the possibility of constructing an rappp-
resentation of Quasi-local quasi *-algebras
(A,A0), possessing a ”sufficient state” on
a non-commutative L2-spaces. A more re-
stricted choice could only be obtained by re-
quiring that the Radon-Nikodym theorem in
quasi * -algebras is satisfied [11]. We hope to
discuss this aspect in a further paper.
Acknowledgment The author thank the
referees for they useful comments and sug-
gestions.
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