Closed subspaces and some basic topological properties of noncommutative Orlicz spaces

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by  Lining Jiang

Closed subspaces and some basic topological properties
✩
of noncommutative Orlicz spaces
Lining Jianga, Zhenhua Maa,b,∗
6
1 aSchool of Mathematics and Statistics, BeijingInstitute of Technology, Beijing,100081, P.
0 R. China
2 bDepartment of Mathematics and Physics, Hebei Universityof Architecture, Zhangjiakou,
075024, P. R. China
n
a
J

6

] Abstract
A
O In this paper, we study the noncommutative Orlicz space L (M,τ), which
ϕ
h. generalizes the concept of noncommutative Lp space, where M isfa von Neu-
t
a mann algebra, and ϕ is an Orlicz function. As a modular space, the space
m
L (M,τ) possessesthe Fatouproperty,andconsequently,it is a Banachspace.
[ ϕ

1 In afddition, a new description of the subspace Eϕ(M,τ) = M Lϕ(M,τ)
v T
1 in Lϕ(M,τ), which is closed under the norm topolofgy and dense undefr the
4 measureftopology, is given. Moreover, if the Orlicz function ϕ satisfies the
9
2 ∆ -condition, then L (M,τ) is uniformly monotone, and the convergence in
2 ϕ
0
. the norm topology and mfeasure topology coincide on the unit sphere. Hence,
1
0 E (M,τ)=L (M,τ) if ϕ satisfies the ∆ -condition.
ϕ ϕ 2
6
1 Keywfords: Noncfommutative Orlicz spaces, τ-measurable operator, von
:
v
Neumann algebra, Orlicz function
i
X
2010 MSC: 46B20,46B25
r
a
✩ TheresearchissupportedbyNationalScienceFoundationofChina(GrantNo. 11371222)
andbythescientificresearchfoundationofEducationBureauofHebeiProvinceandbyGrad-
uatestudentinnovationprogramofBeijingInstituteofTechnology(GrantNo. 2015CX10037
).
∗Correspondingauthor
Email addresses: [email protected] (LiningJiang),[email protected]
(ZhenhuaMa)
Preprint submitted toElsevier January 13, 2016
1. Preliminaries
Noncommutative integration theory was first introduced by I.E. Segal [5]
and is a fundamental tool in many theories, such as operator theory and non-
commutative probability theory. Since the noncommutative space Lp(M,τ)
5 (1 ≤p ≤ ∞) , where τ is a faithful semifinite normal trace on a von Neumann
algebraM, has been defined [3, 5], many scholarshave conducted a systematic
study of these spaces, and obtained many interseting results [14, 15, 19]. As
a natural extension of the space, the theory of noncommutative Orlicz space
associatedtoatracewasintroducedandstudiedbymanymathematicians. For
10 details, one can see [8], [10], [16], [20], and so on. In this paper we will take
Sadeghi’sapproach[16]andstudythetopologicalpropertiesofnoncommutative
Orlicz spaces.
Tobeginwith,wecollectsomedefinitionsandfactsrelatedtovonNeumann
algebras. Suppose that M is a semi-finite von Neumann algebra acting on a
15 Hilbert space H with a normal semi-finite faithful trace τ. The identity in M
is denoted by 1 and the set of all self-adjoint projections on M is denoted by
P(M).
Definition 1.1. A densely-defined closed linear operator x : D(x) → H with
domain D(x) ⊆ H is called affiliated with M if and only if u∗xu = x for all
20 unitary operators u belonging to the commutant M′ of M.
Definition1.2. [1]SupposethatxaffiliatedwithM. Wecallxisτ-measurable,
if there exists a number λ≥0 such that
τ(e (|x|))<∞,
(λ,∞)
where e (|x|) is the spectral projection of |x| corresponding to the interval
(λ,∞)
(λ,∞). The collection of all τ-measurable operators is denoted by M.
f
Given 0<ε,δ ∈R, set
V(ε,δ)={x∈M: there exists e∈P(M) such that
e(H)∈fD(x),kxek ≤ε and τ(1−e)≤δ}.
B(H)
2
Here, ε,δ run over all strictly positive numbers [1]. An alternative description
of the set is given by
V(ε,δ)={x∈M:τ(e (|x|))<δ}.
(ε,∞)
f
Definition 1.3. ([1]) Suppose that x , x∈M. We say that x converges to x
n n
25 in measure (xn −τ−m→x for short ), if for all ε,fδ >0, there exists an n0 such that
x −x∈V(ε,δ), n≥n .
n 0
Remark 1. 1) Using the definition of V(ε,δ), one can get that x −τ−m→x if and
n
only if
lim τ(e (|x −x|))=0
(ε,∞) n
n→∞
for any ε>0.
2) It is known that the collection {V(ε,δ)} is a neighborhood base at
ε,δ>0
0 for a vector space topology τ on M and that M is a complete topological
m
30 ∗-algebra. f f
Inthesettingofτ-measurableoperators,thegeneralizedsingularvaluefunc-
tions are the analogue (and actually, generalization) of the decreasing rear-
rangements of functions in the classical settings, and is more importantly the
cornerstone for the theory of noncommutative rearrangementinvariant Banach
35 function spaces [12].
Definition 1.4. ([18]) For x∈M, the distribution function λ (x):[0,∞)→
(·)
[0,∞] is defined by f
λ (x)=τ(e (|x|)), s≥0.
s (s,∞)
Since the operator x is τ-measurable, λ (x) < ∞ for s large enough and
s
lim λ (x)=0 as noted before. Furthermore, the function λ (x) is decreas-
s→∞ s s
ingandright-continuoussinceτ isnormalande (|x|)↑e (|x|)strongly
(sn,∞) (s,∞)
as s ↓s.
n
Definition 1.5. ([18]) Let L0(X,Σ,m) be the space of measurable functions on
someσ-finitemeasurespace (X,Σ,m). Give an element x∈M, the generalized
f
3
singular value function µ (x):[0,∞]→[0,∞] is defined by
(·)
µ (x)=inf{s≥0:λ (x)≤t}, t>0,
t s
40 where λs(x) is the distribution function.
It is known that the infimum can be attained and that λ (x)≤t, t>0.
µt(x)
For details on the generalised singular value see [18]. We proceed to briefly
reviewtheconceptofaBanachfunctionspaceofmeasurablefunctionson(0,∞)
(see [13].) A function norm ρ on L0(0,∞) is defined to be a mapping ρ:L0 →
+
45 [0,∞] satisfying
1. ρ(f)=0 iff f =0 a.e.
2. ρ(λf)=λρ(f) for all f ∈L0,λ>0.
+
3. ρ(x+y)≤ρ(x)+ρ(y) for all.
4. f ≤g implies ρ(f)≤ρ(g) for all f,g ∈L0.
+
50 Such a ρ may be extended to all of L0 by setting ρ(f) = ρ(|f|), in which
case we may define Lρ(0,∞) = {f ∈ L0(0,∞) : ρ(f) < ∞}. It now Lρ(0,∞)
turns out to be a Banach space when equipped with the norm ρ(·), we refer to
it as a Banach function space.
Using the above context Dodds, Dodds and de Pagter [13] formally defined
the noncommutative space Lρ(M) to be
f
Lρ(M)={f ∈M :µ(f)∈Lρ(0,∞)}
f f
and showed that if ρ is lower semicontinuous and Lρ(0,∞) rearrangement-
55 invariant, Lρ(M) is a Banach space when equipped with the norm kfkρ =
ρ(µ(f)). f
Remark2. IfM is acommutativevon Neumannalgebra, then Mcan beiden-
tified with L∞(X,µ) and τ(f) = fdµ, where (X,µ) is a localizable measure
X
R
space, and where the distribution function and the generalized singular value
60 function defined above are exactly the usual distribution function and classical
rearrangement [2].
4
Next we recall the definition and some basic properties of noncommutative
Orlicz spaces.
Definition 1.6. ([17]) The function ϕ : [0,∞) → [0,∞] is called an Orlicz
function if
|u|
ϕ(u)= p(t)dt,
Z
0
where the real-valued function p defined on [0,∞) has the following properties:
65 (1) p(0)=0,p(t)>0 for t>0 and limt→∞p(t)=∞;
(2) p is right continuous;
(3) p is nondecreasing on (0,∞).
For every Orlicz function ϕ, there is a complementary Orlicz function ψ :
[0,∞)→[0,∞] defined by
ψ(u)=sup{uv−ϕ(v):v ≥0}.
A pair of complementary Orlicz functions (ϕ,ψ) fulfils the following Young
Inequality:
uv ≤ϕ(u)+ψ(v), u,v∈[0,∞),
and equality holds if and only if u = ψ(v) or v = ϕ(u). For background on
Orlicz functions and Orlicz spaces one can see [9, 17].
Suppose that ϕ is an Orlicz function. For x∈M, set
f
ρ (x)=τ(ϕ(|x|)),
ϕ
e
70 then τ(ϕ(|x|)) is a convex modular on M [16].
f
Definition 1.7. Set
L (M,τ)= x∈M:τ(ϕ(λ|x|))<∞ for some λ>0 ,
ϕ
n o
f f
and equip the space with the Luxemburg norm
|x|
kxk=inf λ>0:τ ϕ ≤1 .
(cid:26) (cid:18) (cid:18) λ (cid:19)(cid:19) (cid:27)
Such a space is called a noncommutative Orlicz space.
5
Remark 3. Notice that if ϕ(x) = |x|p, 1 ≤ p < ∞ for any τ-measurable
operator x ∈ M, then L (M,τ) is nothing but the noncommutative Lp space
ϕ
Lp(M,τ) andfthe Luxembufrg norm generated by this function is expressed by
the fformula
1
kxkp =(τ(|x|p))p .
Similar to the commutative case, for x,y ∈ L (M,τ) one can define the
ϕ
following Orlicz norm: f
kxko =sup{τ(|xy|):τ(ψ(|y|))≤1},
where ψ is the complementary function of ϕ. Moreover, we have the following
relation between the two norms [16],
kxk≤kxko ≤2kxk.
We also can get the Young Inequality in noncommutative Orlicz spaces:
Lemma 1.1. ([20]) For a pair (ϕ,ψ) of complementary Orlicz functions we
have:
τ(|xy|)≤τ(ϕ(|x|))+τ(ψ(|y|)), for all x,y ∈M.
f
Moreover, if 0≤x∈M with τ(ϕ(x))<∞, then there is a 0≤y ∈M with
f f
τ(xy)=τ(ϕ(x))+τ(ψ(y)) and τ(ψ(y))≤1.
For further information on the theory of noncommutative Orlicz spaces we
refer the reader to [8, 10, 11, 16, 20].
75 2. Closed linear subspaces of Lϕ(M,τ)
f
In this section, we prove that the noncommutative Orlicz spaces L (M,τ)
ϕ
with the Luxemburg norm have the Fatou property. Consequently, the fspace
is complete. In addition, we give a new description of the subspace E (M,τ)
ϕ
given in [16], and prove that this is a closed linear subspace in norm topfology
80 and a dense subspace in measure topology of the Lϕ(M,τ).
Firstly we give the definition of rearrangementinvafriant as follows.
6
Definition 2.1. A linear subspace E of M is called rearrangement invariant
if and only if x ∈ E, y ∈ M and for all t >f0, µt(y) ≤ µt(x) imply that y ∈ E
and kykE ≤kxkE. f
85 ItiswellknownthatLϕ(M,τ)arenormedrearrangementinvariantoperator
spaces [4]. From Corollary 2f.4 in [13] we know that a normed rearrangement
invariant operator space with the Fatou property is a Banach space. Hence,
in order to prove that L (M,τ) are Banach spaces, it suffices to show that
ϕ
Lϕ(M,τ) satisfy the Fatou pfroperty.
f
90 Theorem 2.1. (Fatou property) Suppose that x ∈ M,xn ∈ Lϕ(M,τ). If
supnkxnk<∞ and 0≤xn ↑n x, then x∈Lϕ(M,τ) anfd kxk=supnfkxnk.
f
Proof. Sincex ∈L (M,τ),onehasthatµ(x )∈L (0,∞). FromProposition
n ϕ n ϕ
1.7 in [13], if xn, x ∈ Mf and 0 ≤ xn ↑n x then µt(xn) ↑n µt(x) holds for all
t≥0. Since supnkxnkf<∞=supnkµ(xn)k<∞.
95 Then µ(x)∈Lϕ(0,∞) and kµ(x)k=supnkµ(xn)k by the classical counter-
part of Theorem 2.1.
Hence, x∈L (M,τ) and kxk=sup kx k.
ϕ n n
f
In [20], Kunze considers the properties of the space
k·k
E (M,τ)=M L (M,τ) .
ϕ ϕ
\
f f
Now we give another characterizationof this space. Indeed, set
A (M,τ)= x∈M:τ(ϕ(λ|x|))<∞ for all λ>0 .
ϕ
n o
f f
ItiseasytoverifythatA (M,τ)isalinearsubspaceofL (M,τ). Thefollowing
ϕ ϕ
theoremshowsthat Aϕ(M,fτ) is a closedlinear subspace infnormtopologyand
100 a dense subspace in measfure topology of Lϕ(M,τ).
f
Theorem 2.2. The following statements are true:
1. A (M,τ)isaclosedlinearsubspaceofL (M,τ)underthenormtopology;
ϕ ϕ
2. A (Mf,τ) is a dense subspace of L (M,τ)funder the measure topology.
ϕ ϕ
f f
7
Proof. (1) Let x ∈ A (M,τ) and x ∈ L (M,τ) be given with x → x in
n ϕ ϕ n
105 follows from Lemma of [6f] that any z ∈ M bfelongs to Aϕ(M) if and only if
µ(z)∈Aϕ(0,∞). f
By corollary 4.3 of [12] we now have that
kµ(x )−µ(x)k≤kµ(x −x)k=kx −xk→0.
n n n
BytheclassicalcounterpartofTheorem2.1wenowhavethatµ(x)∈A (0,∞).
ϕ
Hence x∈A (M,τ) as required.
ϕ
(2) Now for anyfx∈Lϕ(M,τ), set
f ∞
x=u|x|=u λde (|x|)
λ
Z
0
n
be the polar decomposition of x. For each n∈N, set xn =u 0 λdeλ(|x|), then
R
it is obvious that x ∈A (M,τ) and
n ϕ
f ∞
x−x =u|x|e (|x|)=u λde (|x|).
n (n,∞) λ
Z
n
For any ε>0,
e (|x|), ε<n,
e (|x−x |)= (n,∞)
(ε,∞) n e (|x|), ε≥n.
(ε,∞)

Since x is a τ-measurable operator, lim τ(e (|x|)) = 0, which means that
n (n,∞)
110 limnτ(e(ε,∞)(|x−xn|))=0foranyε>0. Hence,Aϕ(M,τ)isadensesubspace
of Lϕ(M,τ) under the measure topology. f
f
In order to study the further properties of A (M,τ), we need the following
ϕ
lemma. f
k·k
Lemma 2.1. By E (M,τ) = E we denote the set M L (M,τ) . If
ϕ ϕ ϕ
T
115 x ∈ Lϕ(M,τ) and τ(ϕf(|x|)) < ∞, then the distance d(x,Eϕ) frofm x to Eϕ
is no morfe than 1, where d(x,Eϕ)=inf{kx−yk: y ∈Eϕ}.
∞
Proof. Letx=u|x| be the polardecompositionofx, where|x|= λde (|x|).
0 λ
n R
For each n ∈ N, set xn = u 0 λdeλ(|x|). Since τ(ϕ(|x|)) < ∞, for any ε > 0
R
8
one can choose an n0 ∈N such that
∞
τ(ϕ(|x−x |)= ϕ(λ)dτ(e )<ε.
n0 Z λ
n0
Since x ∈E , the Young Inequality implies
n ϕ
d(x,E )≤kx−x k≤kx−x ko ≤1+τ(ϕ(|x−x |)<1+ε.
ϕ n0 n0 n0
Therefore, d(x,E )≤1 since ε is arbitrary.
ϕ
The following theorem shows that A (M,τ) is the closure (in the norm
ϕ
topology) of the set of all bounded τ-measurfable operators.
k·k
120 Theorem 2.3. Aϕ(M,τ)=Eϕ(M,τ)=M Lϕ(M,τ) .
T
f f f
Proof. For any x ∈ A (M,τ) and k ≥ 1, we have kx ∈ A (M,τ). Therefore
ϕ ϕ
d(kx,Eϕ)≤1 or d(x,Eϕ)f≤ k1. Since k is arbitrary, then we hfave x∈Eϕ, i.e.,
A (M,τ)⊆E .
ϕ ϕ
Ofn the other hand, observing that M is contained in Aϕ(M,τ) and that
125 Aϕ(M,τ) is a closed subspace of Lϕ(M,τ) by (1) of Theoremf2.2, then Eϕ is
contafined in Aϕ(M,τ), which impliesfthat Aϕ(M,τ)=Eϕ.
Moreover,by tfhe definition of Aϕ(M,τ), wefget
f k·k
A (M,τ)=M L (M,τ) =E (M,τ).
ϕ ϕ ϕ
\
f f f
Inthefollowing,similartotheclassicalcase,westilluseE (M,τ)todenote
ϕ
the set x∈M:τ(ϕ(λ|x|))<∞ for all λ>0 . f
n o
f
130 Theorem 2.4. Let x∈M be given, the following statements are equivalent:
f
1. x∈E (M,τ).
ϕ
n
2. lim kfx−x k=0, where x =u λde (|x|).
n→∞ n n 0 λ
R
Proof. (1)⇒(2). Given ε>0, we have
|x| ∞ λ
τ ϕ = ϕ dτ(e (|x|))<∞,
λ
(cid:18) (cid:18) ε (cid:19)(cid:19) Z (cid:18)ε(cid:19)
0
9
this is because x∈E (M,τ).
ϕ
n
Since xn =u 0 λdeλf(|x|), when n is large enough it follows that
R
|x−x | ∞ λ
n
τ ϕ = ϕ dτ(e (|x|))≤1.
λ
(cid:18) (cid:18) ε (cid:19)(cid:19) Z (cid:18)ε(cid:19)
n
Hence, by the Young Inequality,
o
x−x x−x |x−x |
n n n
≤ ≤1+τ ϕ ≤2
(cid:13) ε (cid:13) (cid:13) ε (cid:13) (cid:18) (cid:18) ε (cid:19)(cid:19)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
for such n∈N. This yields kx−xnk→0 as n→∞, since ε is arbitrary.
n
135 (2)⇒(1). Since xn =u 0 λdeλ(|x|) , then xn ∈M. If limn→∞kx−xnk=
R
0, then x∈E =E (M,τ) by definition.
ϕ ϕ
f
3. The properties of Lϕ(M,τ) for ϕ ∈ ∆2
f
In this section, we will prove that if the Orlicz function ϕ satisfies the ∆
2
condition(forshort,denoteitbyϕ∈∆ ),namely,thereexistsaconstantk >0
2
such that for all u>0,
ϕ(2u)≤kϕ(u),
then E (M,τ) is uniformly monotone, and
ϕ
f
E (M,τ)=L (M,τ).
ϕ ϕ
f f
Using Lemma 2.1 of [6], and the fact that kµ(x)k = kxk we can get following
Lemmas from the classical counterparts of these Lemmas applied to µ(x).
140 Lemma 3.1. Suppose ϕ∈∆2 and x∈Lϕ(M,τ). For any ε>0, there exists a
δ(ε)>0 such that τ(ϕ(|x|))≥δ whenever kxfk≥ε.
Lemma 3.2. Suppose ϕ ∈ ∆ and x ∈ L (M,τ). For any ε ∈ (0,1), there
2 ϕ
exists a δ(ε)∈(0,1) such that kxk≤1−δ whefnever τ(ϕ(|x|)) ≤1−ε.
Lemma 3.3. Suppose ϕ ∈ ∆ and x ∈ L (M,τ). For any ε ∈ (0,1), there
2 ϕ
145 exists a δ(ε)∈(0,1) such that kxk≥1+δ whefnever τ(ϕ(|x|)) ≥1+ε.
10