New York Journal of Mathematics
New York J. Math. 21 (2015) 759–781.
On geometrical properties of
noncommutative modular function spaces
Ghadir Sadeghi and Reza Saadati
Abstract. We introduce and study the noncommutative modular function spaces of measurable operators affiliated with a semifinite von Neumann algebra and show that they are complete with respect to their
modular. We prove that these spaces satisfy the uniform Opial condition
with respect to ρe-a.e.-convergence for both the Luxemburg norm and the
Amemiya norm. Moreover, these spaces have the uniform Kadec–Klee
property with respect to ρe-a.e.-convergence when they are equipped with
the Luxemburg norm. The above geometric properties enable us to obtain some results in noncommutative Orlicz spaces.
Contents
1. Introduction
2. Preliminaries
3. Noncommutative modular function spaces
4. Uniform Opial and uniform Kadec–Klee properties
References
759
760
765
773
780
1. Introduction
The first attempts to generalize the classical function spaces of Lebesgue
type Lp were made in the early 1930’s by Orlicz and Birnbaum in connection
with orthogonal expansions. The possibility of introducing the structure of
a linear metric in Orlicz spaces Lϕ as well as the interesting properties of
these spaces and many applications to differential and integral equations
with kernels of nonpower type were among the reasons for the development
of the theory of Orlicz spaces.
We note two principal directions of further development. The first one
is a theory of Banach function spaces initiated in 1955 by Luxemburg [L55]
and then developed in a series of joint papers with Zaanen [LZ63]. The
main idea consists of considering the function space L of all real functions
Received May 7, 2014.
2010 Mathematics Subject Classification. Primary 46L52. Secondary 46B20, 46A80.
Key words and phrases. Measurable operator, von Neumann algebra, generalized singular value function, uniform Opial condition, Kadec–Klee property, modular function
space.
ISSN 1076-9803/2015
759
760
GHADIR SADEGHI AND REZA SAADATI
f ∈ L0 (X , Σ, ν) such that kf k < ∞, where (X , Σ, ν) is a measure space,
L0 (X , Σ, ν) denotes the space of all real measurable functions on X , and
k · k is a function norm which satisfies
kf k ≤ kgk
whenever
|f (x)| ≤ |g(x)| ν-a.e.
The noncommutative version of this approach, which is called the noncommutative symmetric space of τ -measurable operators affiliated with a von
Neumann algebra, was considered for the first time by Ovchinnikov [Ov71].
Some properties of these spaces were examined in [DDP89, DDS89, CDSF96,
P07].
The other direction, also inspired by the successful theory of Orlicz spaces,
is based on replacing the integral form of a nonlinear functional with an abstract given functional with some suitable properties. This idea was the basis
behind the theory of modular spaces initiated by Nakano [Na50] in connection with the theory of order spaces, which was redefined and generalized
by Musielak and Orlicz [MO59].
Presently the theory of modulars and modular spaces is applied extensively, in particular in the study of various Orlicz spaces [Or88] and interpolation theory [Kr82]. Recently, the author of this paper studied noncommutative Orlicz spaces from the point of view of modulars [Sa12]. The main
objective of the current paper is to investigate the theory of noncommutative
modular function spaces.
The organization of the paper is as follows. In the second section we
provide some necessary preliminaries related to the theory of τ –measurable
operators affiliated with a von Neumann algebra and the classical theory
of modular spaces. In Section 3, we introduce the definition of noncommutative modular function spaces associated to a modular on τ -measurable
operators and give some relations between the ρe-a.e.-convergence and the
convergence of the modular. In the last section, we prove that these spaces
satisfy the uniform Opial condition with respect to ρe-a.e.-convergence for
both the Luxemburg norm and the Amemiya norm. Moreover, these spaces
have the uniform Kadec–Klee property with respect to ρe-a.e.-convergence
when they are equipped with the Luxemburg norm.
2. Preliminaries
In this section, we collect some basic facts and give some notations related
to τ –measurable operators and modular function spaces. We denote by M a
semifinite von Neumann algebra on a Hilbert space H, with a fixed faithful
and normal semifinite trace τ . For standard facts concerning von Neumann
algebras we refer the reader to [Ta79]. The identity in M is denoted by 1
and we denote by P(M) the complete lattice of all self–adjoint projections
in M. A linear operator x : D(x) → H with domain D(x) ⊆ H is affiliated
0
with M if ux = xu for all unitaries u in the commutant M of M, and this
is denoted by xηM. Note that the equality ux = xu involves the equality
NONCOMMUTATIVE MODULAR FUNCTION SPACES
761
of the domains of the operators ux and xu, that is, D(x) = u−1 (D(x)). If x
is in the algebra B(H) of all bounded linear operators on the Hilbert space
H, then x is affiliated with M if and only if x ∈ M. If x is a self–adjoint
operator in H affiliated with M, then the spectral projection ex (B) is an
element of M for any Borel set B ⊆ R.
A closed and densely defined operator x affiliated with M is called τ measurable if there exists a number λ ≥ 0 such that
|x|
τ e (λ, ∞) < ∞.
f With the
The collection of all τ -measurable operators is denoted by M.
sum and product defined as the respective closure of the algebraic sum and
f is a ∗-algebra [Ne74]. Given positive real
product, it is well known that M
f for which there
numbers ε, δ, we define V(ε, δ) to be the set of all x ∈ M
exists p ∈ P(M) such that kxpkB(H) ≤ ε and τ (1 − p) ≤ δ. An alternative
description of this set is
n
o
f : τ e|x| (ε, ∞) < δ .
V(ε, δ) = x ∈ M
The collection {V(ε, δ)}ε,δ>0 is a neighborhood base at 0 for a vector space
f For x ∈ M,
f the generalized singular value function
topology τm on M.
µ(x) = µ(|x|) is defined by
n
o
µt (x) = inf λ ≥ 0 : τ e|x| (λ, ∞) ≤ t
(t ≥ 0).
It follows directly that the generalized singular value function µ(x) is a decreasing right-continuous function on the positive half-line [0, ∞). Moreover,
f
for all u, v ∈ M and x ∈ M,
µ(uxv) ≤ kukkvkµ(x)
and
µ(f (x)) = f (µ(x))
f and f is an increasing continuous function on [0, ∞)
whenever 0 ≤ x ∈ M
f
with f (0) = 0. M is a partially ordered vector space under the ordering
f then
x ≥ 0 defined by hxξ, ξi ≥ 0, ξ ∈ D(x). If 0 ≤ xα ↑ x holds in M,
sup µt (xα ) ↑α µt (x) for each t ≥ 0. The trace τ is extended to the positive
f as a nonnegative extended real–value functional which is positively
cone of M
homogeneous, additive, unitary invariant and normal. Furthermore,
τ (x∗ x) = τ (xx∗ )
f and
for all x ∈ M
Z
(2.1)
τ (f (x)) =
∞
f (µt (x))dt
0
762
GHADIR SADEGHI AND REZA SAADATI
f and f is a nonnegative Borel function which is bounded
whenever 0 ≤ x ∈ M
on a neighborhood of 0 and satisfies f (0) = 0. In the following proposition,
we list some properties of the rearrangement mapping µt (·).
Proposition 2.1. Let x, y and z be τ -measurable operators. Then the following hold true:
(i) The map t ∈ (0, ∞) 7→ µt (x) is nonincreasing and continuous from
the right. Moreover,
lim µt (x) = kxk ∈ [0, ∞].
t↓0
(ii)
(iii)
(iv)
(v)
(vi)
µt (x) = µt (|x|) = µt (x∗ ).
µt (x) ≤ µt (y), for t > 0, if 0 ≤ x ≤ y.
µt+s (x + y) ≤ µt (x) + µs (y), t, s > 0.
µt (zxy) ≤ kzkkykµt (x), t > 0.
µt+s (xy) ≤ µt (x)µs (y), t, s > 0.
For further details and proofs we refer the reader to [FK86, DDP89].
Lemma 2.2. [Ku90] Let x, y be τ -measurable operators. Then there exist
partial isometries u, v in M such that
|x + y| ≤ u|x|u∗ + v|y|v ∗ .
The proof for measurable operators is straightforward using the fact that
the square root function is operator monotone; see [AAP82].
In the sequel, we recall some basic concepts about modular function
spaces. Let Ω be a nonempty set and Σ be a nontrivial σ-algebra of subsets
of Ω. Let P be a δ-ring of subsets of Ω, such that E ∩ A ∈ P for any E ∈ P
and A ∈ Σ. Let us assume
S that there exists an increasing sequence of sets
Kn ∈ P such that Ω = Kn . In other words, the family P plays the role
of the δ-ring of subsets of finite measure. By E we denote the linear space
of all simple functions with supports from P. By M we denote the space of
all measurable functions, i.e., all functions f : Ω → R such that there exists
a sequence {sn } ⊂ E, |sn | ≤ |f | such that sn (ω) → f (ω) for all ω ∈ Ω. By
χA we denote the characteristic function of the set A.
Let us recall that a set function ν : Σ → [0, +∞] is called a σ-subadditive
measure if ν(∅) = 0, ν(A) ≤ ν(B) for any A ⊆ B and ν(∪n An ) ≤ Σn ν(An )
for any sequence of sets {An } ⊆ Σ.
Definition 2.3. A functional ρ : E ×Σ → [0, ∞] is called a function modular
if:
(i) ρ(0, A) = 0 for any A ∈ Σ.
(ii) ρ(f, A) ≤ ρ(g, A) whenever |f (ω) ≤ |g(ω)|, for ω ∈ Ω, f, g ∈ E,
A ∈ Σ.
(iii) ρ(f, ·) : Σ → [0, +∞] is a σ-subadditive measure for every f ∈ E.
(iv) ρ(α, A) → 0 as α decreases to 0 for every A ∈ P, where
ρ(α, A) = ρ(αχA , A).
NONCOMMUTATIVE MODULAR FUNCTION SPACES
763
(v) If there exists α > 0 such that ρ(α, A) = 0, then ρ(β, A) = 0 for
every β > 0.
(vi) For any α > 0, ρ(α, ·) is order continuous on P, i.e., ρ(α, An ) → 0
if {An } ⊆ P and decreases to ∅.
When ρ satisfies
(iii0 ) ρ(f, ·) : Σ → [0, +∞] is a σ-subadditive measure,
we say that ρ is additive if ρ(f, A∪B) = ρ(f, A)+ρ(f, B) whenever A, B ∈ Σ
such that A ∩ B = ∅ and f ∈ M.
The definition of ρ can be extended to all f ∈ M by
ρ(f, A) = sup{ρ(s, A); s ∈ E, |s(x)| ≤ |f (x)| for every ω ∈ Ω}.
Similarly as in the case of measure spaces, a set A ∈ Σ is called ρ-null
if ρ(α, A) = 0 for every α > 0. A property p(ω) is said to hold ρ-almost
everywhere (ρ-a.e.) if the set {ω ∈ Ω : p(ω) does not hold} is ρ-null. We say
that fn → f ρ-a.e. if {ω ∈ Ω : f (ω) 6= limn→∞ fn (ω)} is ρ-null. As usual,
we identify any pair of measurable sets whose symmetric difference is ρ-null,
as well as any pair of measurable functions differing only on a ρ-null set.
In the above conditions, we define the functional ρ : M → [0, +∞] by
ρ(f ) = ρ(f, Ω). Then it is easy to check that ρ is a modular, that is, ρ
satisfies the following properties:
(i) ρ(f ) = 0 if and only if f = 0.
(ii) ρ(αf ) = ρ(f ) for every scalar α with |α| = 1.
(iii) ρ(αf + βg) ≤ ρ(f ) + ρ(g) if α + β = 1 and α, β ≥ 0.
If (iii) is replaced by
(iii0 ) ρ(αf + βg) ≤ αρ(f ) + βρ(g) if α + β = 1 and α, β ≥ 0,
then we say that ρ is a convex modular.
A modular ρ defines a corresponding modular function space, i.e., the
vector space Lρ given by
Lρ = {x ∈ M :
ρ(λx) → 0
When ρ is convex, the formulas
kf kρ = inf λ > 0 :
and
ρ
as λ → 0} .
f
≤1
λ
1
= inf
(1 + ρ(λf )) : λ > 0
λ
define two complete norms on Lρ which are called the Luxemburg norm and
the Amemiya norm, respectively. The spaces (Lρ , k · kρ ) and (Lρ , k · kA
ρ)
are complete. Moreover, the Luxemburg norm and the Amemiya norm are
equivalent. Indeed,
kf kA
ρ
(2.2)
for every f ∈ Lρ .
kf kρ ≤ kf kA
ρ ≤ 2kf kρ
764
GHADIR SADEGHI AND REZA SAADATI
A function modular is said to satisfy the ∆2 -condition if
sup ρ(2fn , Dk ) → 0
n
as k → ∞, whenever {fn } ⊆ M, Dk ∈ M decreases to ∅ and
sup ρ(fn , Dk ) → 0.
n
Definition 2.4. A function modular is said to satisfy the ∆2 -type condition
if there exists k > 0 such that for any f ∈ Lρ it holds that ρ(2f ) ≤ kρ(f ).
It is clear that the ∆2 -type condition implies the ∆2 -condition, but in
general, they are not equivalent. Note that the ∆2 -type condition ensures
that 0 < ρ(λf ) < ∞ for every λ > 0 provided that 0 < ρ(f ) < ∞.
Let us recall an example of a modular function space.
Example 2.5 (Orlicz–Musielak spaces). Let (Ω, Σ, ν) be a measure space,
where ν is a positive σ-finite measure. Let us denote by P the δ-ring of all
sets of finite measure. Define the modular ρ by the formula
Z
ϕ(t, |f (t)|)dν(t)
ρϕ (f, E) =
E
provided ϕ belongs to the class Φ. For the precise definitions of the class
Φ and properties of Orlicz–Musielak spaces see e.g., [Mu83]. For an Orlicz–
Musielak space, the ρ-null sets coincide with the sets of measure zero in
the sense of ν [Ko88, Proposition 4.1.9]. Thus the ρ-a.e.-convergence is
equivalent to the convergence almost everywhere. If the modular is given by
Z
ρϕ (f, E) =
ϕ(|f (t)|)dν(t),
E
where ϕ is an Orlicz function, we obtain the notion of Orlicz space. In
particular, if ϕ(t) = tp for 1 ≤ p < +∞, we obtain the Lebesgue space
Lp (ν), where the Luxemburg norm is the classic norm k · kp .
An Orlicz function ϕ is said to satisfy the ∆2 -condition if there exists
k > 0 such that ϕ(2α) ≤ kϕ(α) for every α ≥ 0. If ϕ is an Orlicz function
satisfying the ∆2 -condition it is clear that the modular ρϕ satisfies the ∆2 type condition.
In the following theorem we recall some of the properties of modular
function spaces that will be used later on in this paper. For proofs and
details the reader is referred to [Mu83, Ko88].
Theorem 2.6.
(i) (Lρ , k · kρ ) is complete and the norm k · kρ is monotone with the
natural order in M.
(ii) kf kρ → 0 if and only if ρ(αf ) → 0 for every α > 0.
(iii) If ρ(αf ) → 0 for an α > 0 then there exists a subsequence {gn } of
{fn } such that gn → 0 ρ-a.e.
NONCOMMUTATIVE MODULAR FUNCTION SPACES
765
(iv) If fn → f ρ-a.e., there exists a nondecreasing sequence of sets En ∈
P such that En ↑ Ω and {fn } converges uniformly to f on every En
(Egoroff theorem).
(v) ρ(f ) ≤ lim inf n→∞ ρ(fn ) whenever fn → f ρ-a.e. (This property is
equivalent to the Fatou property).
Let X be a Banach space and τ be a topology on X. We say that X
satisfies the uniform Opial condition with respect to τ if oτ (α) > 0 for every
α > 0, where oτ (·) is the Opial modulus defined as
n
o
oτ (α) = inf lim inf kxn + xk − 1 ,
n→∞
and the infimum is taken over all x ∈ X with kxk ≥ α and all sequences
{xn } such that τ -limn→∞ xn = 0 and lim inf n→∞ kxn k ≥ 1 (see [APP13]).
A space X is said to have the uniform Kadec–Klee property with respect
to τ (U KK(τ )) if for every 0 < ε ≤ 2 there exists δ > 0 such that if {xn } is a
sequence in the unit ball of X with sep{xn } = inf {kxn − xm k : n 6= m} > ε
and {xn } is τ -convergent to x ∈ X, then kxk < 1 − δ.
In connection to the uniform Kadec–Klee property, we can define the
following modules:
n
o
kτ (ε) = inf 1 − kxk : {xn } ∈ BX , τ − lim xn = x and sep{xn } > ε .
n→∞
It is clear that X has the uniform Kadec–Klee property with respect to τ
iff kτ (ε) > 0 for every ε > 0.
In the particular case that X is a modular function space, we can replace
the convergence with respect to a topology τ by the ρ-a.e.-convergence. The
uniform Opial condition and the uniform Kadec–Klee property are geometric
properties, which are connected to the existence of fixed points for some
kinds of mappings.
3. Noncommutative modular function spaces
In this section, we assume that M is a semifinite von Neumann algebra on
a Hilbert space H with a normal faithful trace τ . We define noncommutative
modular function spaces of τ -measurable operators affiliated to M. To this
end, we assume Ω = [0, ∞) and ν is a Lebesgue measure on Ω. By L0 (ν) we
denote the linear space of all (equivalence classes of) real valued Lebesgue
measurable functions on Ω, Σ is the σ-algebra of Lebesgue measurable sets
and P is the family of such sets of finite measure. Let ρ(f ) := ρ(f, Ω) be a
convex modular on L0 (ν) such that for every f, g ∈ L0 (ν), f ≺≺ g (i.e., f
is submajorized by g in the sense of Hardy, Littlewood and Polya) implies
that ρ(f ) ≤ ρ(g), and let Lρ denote the modular function space associated
to ρ on L0 (ν).
f as follows:
We now define the functional ρe on M
f → [0, ∞],
ρe : M
ρe(x) = ρ(µ(|x|)).
766
GHADIR SADEGHI AND REZA SAADATI
f
Proposition 3.1. ρe is a convex modular functional on M.
Proof. It is sufficient to show that ρe satisfies
ρe(αx + βy) ≤ αe
ρ(x) + β ρe(y)
(3.1)
f It is known
for α + β = 1 with α, β ≥ 0. Let x, y be two operators in M.
that µ(x + y) ≺≺ µ(x) + µ(y) [FK86, Theorem 4.4]. By Lemma 2.2 there
f such that
exist partial isometries u, v in M
|x + y| ≤ u|x|u∗ + v|y|v ∗ .
Hence
ρe(αx + βy) = ρ(µ(|αx + βy|)) ≤ ρ(µ(αu|x|u∗ + βv|y|v ∗ ))
≤ αρ(µ(|x|)) + βρ(µ(|y|))
= αe
ρ(x) + β ρe(y).
f
Thus, ρe is a convex modular on M.
f τ ) is defined by
The noncommutative modular function space Lρe(M,
n
o
f τ) = x ∈ M
f : ρe(λx) → 0 as λ → 0 ,
Lρe(M,
or equivalently,
n
f τ) = x ∈ M
f:
Lρe(M,
n
f:
= x∈M
ρ(λµ(x)) → 0
o
µ(x) ∈ Lρ .
as
λ→0
o
f τ ) can be equipped with the Luxemburg norm deThe vector space Lρe(M,
fined by
o
n
x
kxkρe = inf λ > 0 : ρe
≤1
λ
µ(|x|)
= inf λ > 0 : ρ
≤1
λ
= kµ(|x|)kρ .
Similar to the commutative case, one can define the Amemiya norm as
follows:
1
A
kxkρe = inf
(1 + ρe(λx)) : λ > 0
λ
1
= inf
(1 + ρ (λµ(|x|))) : λ > 0
λ
= kµ(|x|)kA
ρ.
Moreover, by (2.2), these norms satisfy
kxkρe ≤ kxkA
ρe ≤ 2kxkρe.
NONCOMMUTATIVE MODULAR FUNCTION SPACES
767
f τ ) is a Banach space since
It follows from [DDP89, Theorem 4.5] that Lρe(M,
the norm k · kρe has the Fatou property.
Theorem 3.2. Let ρ be a convex function modular with the ∆2 -type conf τ ) is ρedition. Then the noncommutative modular function space Lρe(M,
complete.
f τ ). We first
Proof. Suppose that {xn } is a ρe–Cauchy sequence in Lρe(M,
show that {xn } is a Cauchy sequence in the measure topology τm . If ε, δ > 0
are given, then there exist η > 0, n0 ∈ N such that kf kρ < ε if ρ(f ) < η for
any f ∈ L0 (ν) and
ρe(xn − xm ) < η
for any n, m ≥ n0 . It is known that µ(x) is a nonincreasing function, so we
have
µ(xn − xm ) ≥ µδ (xn − xm )χ[0,δ) + µ(xn − xm )χ[δ,∞) .
It follows that
ρ(µδ (xn − xm )χ[0,δ) ) ≤ ρe(xn − xm ) < η
for all n, m ≥ n0 . This yields kµδ (xn − xm )χ[0,δ) kρ < ε. Hence
µδ (xn − xm ) < ε/kχ[0,δ) kρ
τm
f such that xn −→
for any n, m ≥ n0 . Consequently, there exists x ∈ M
x.
Moreover, by [DDP89, Theorem 3.4],
|µ(xn ) − µ(xm )| ≺≺ µ(xn − xm ),
whence
ρ (µ(xn ) − µ(xm ))) ≤ ρ (µ(xn − xm ))
for all n, m ∈ N. Therefore, {µ(xn )} is Cauchy sequence in Lρ . It follows
that there exists f ∈ Lρ such that kµ(xn ) − f kρ → 0 as n → ∞. By
Theorem 2.6(ii), we obtain
(3.2)
ρ (µ(xn ) − f )) → 0
as
n → ∞.
Using Theorem 2.6(iii), we may suppose (by passing to a subsequence if
necessary) that µ(xn ) → f ρ-a.e. We use a known version of the Egoroff
theorem for modular function spaces. There exists a nondecreasing sequence
of sets Ek with finite measure such that Ek ↑ Ω and {µ(xn )} converges
uniformly to f on every Ek . On the other hand, it follows from [FK86,
Lemma 3.4] that µ(xn ) → µ(x) almost everywhere on [0, ∞) and so
µ(x) = f ∈ Lρ ,
f τ ). Moreover, it follows from (3.2) that
that is x ∈ Lρe(M,
ρ(µ(xn ) − µ(x))) → 0
ρ
as n → 0, i.e., µ(xn ) −→ µ(x). Observe that
(xn − xm ) → (xn − x)
768
GHADIR SADEGHI AND REZA SAADATI
f Similar to the above argument one can
for the measure topology τm in M.
ρ
show that µ(xn − xm ) −→ µ(xn − x). Since ρ has the Fatou property, it
follows that
ρ (µ(xn − x)) ≤ lim inf ρ (µ(xn − xm )) .
m→∞
Hence limn→∞ ρe(xn − x) = 0.
Let us give an example of a noncommutative modular function space.
Example 3.3. Let Lϕ (Ω, ν) be an Orlicz space on Ω = [0, ∞) with respect
to the modular functional
Z ∞
ρϕ (f ) =
ϕ(|f (t)|)dν(t)
0
for every f ∈ Lϕ , where ν is the Lebesgue measure on Ω. We can consider
the noncommutative Orlicz spaces from the point of view of modulars as
follows:
n
o
f τ) = x ∈ M
f : ρeϕ (λx) → 0
Lϕ (M,
as λ → 0 ,
f → [0, ∞] is defined by
where ρeϕ : M
Z
ρeϕ (x) = τ (ϕ(|x|)) =
∞
ϕ(µt (|x|))dt.
0
For more details, we refer the reader to [Sa12, AB14].
f τ ).
Proposition 3.4. Let x, y ∈ Lρe(M,
(i) If ρe(λx) ≤ ρe(λy) for every λ > 0, then kxkρe ≤ kykρe.
(ii) The function α 7→ kαxkρe is nondecreasing for α ≥ 0.
(iii) If kxkρe < 1, then ρe(x) ≤ kxkρe.
(iv) If ρ satisfies the ∆2 – type condition, then
ρe(x + y) ≤
k
(e
ρ(x) + ρe(y))
2
f with k > 0 satisfying ρ(2f ) ≤ kρ(f ) for any
for all x and y in M,
f ∈ Lρ .
Proof. The properties (i) and (ii) follow immediately from the definition of
k · kρe.
≤ 1 and
We prove now that (iii) holds. Let kxkρe < α < 1. Then ρ µ(x)
α
so
µ(|x|)
µ(|x|)
ρe(x) = ρ α
≤ αρ
≤ α.
α
α
Since α is arbitrary, we obtain that ρe(x) ≤ kxkρe.
For (iv), it follows from Lemma 2.2 that there exist partial isometries u, v
f such that
in M
|x + y| ≤ u|x|u∗ + v|y|v ∗ .
NONCOMMUTATIVE MODULAR FUNCTION SPACES
769
By Proposition 2.1(iii),(v) and the submajorization inequality
µ(x + y) ≺≺ µ(x) + µ(y)
f we have
for x, y ∈ M,
µ(|x + y|) ≤ µ(u|x|u∗ + v|y|v ∗ ) ≺≺ µ(u|x|u∗ ) + µ(v|y|v ∗ )
≤ µ(|x|) + µ(|y|).
Therefore
ρe(x + y) = ρ(µ(|x + y|)) ≤ ρ(µ(u|x|u∗ + v|y|v ∗ ))
≤ ρ(µ(|x|) + µ(|y|))
µ(|x|) + µ(|y|)
≤ kρ
2
k
≤ (ρ(µ(|x|) + ρ(µ(|y|)))
2
k
= (e
ρ(x) + ρe(y)).
2
Definition 3.5. The growth function ωρe of the modular ρe is defined as
follows:
ρe(αx)
: 0 < ρe(x) < +∞
ωρe(α) := sup
ρe(x)
ρ(αµ(|x|))
= sup
: 0 < ρ(µ(|x|)) < +∞
for all α ≥ 0.
ρ(µ(|x|))
The next lemma can be easily proved.
Lemma 3.6. Let ρ be a convex function modular with the ∆2 -type condition.
Then the growth function ωρe has the following properties:
(i) ωρe(α) < +∞ for every 0 ≤ α < +∞.
(ii) ωρe : [0, +∞) → [0, +∞) is a convex, strictly increasing function, so
it is continuous.
(iii) ωρe(αβ) ≤ ωρe(α)ωρe(β) for all α, β ≥ 0.
−1
−1
−1
(iv) ωρ−1
e (α)ωρe (β) ≤ ωρe (αβ) for all α, β ≥ 0, where ωρe is the inverse
function of ωρe.
The following lemma shows that the growth function can be used to give
an upper bound for the norm of an operator.
Lemma 3.7. Let ρ be a convex function modular with the ∆2 -type condition.
Then
1
kxkρe ≤
−1
1
ωρe ρe(x)
f τ ).
for every x ∈ Lρe(M,
770
GHADIR SADEGHI AND REZA SAADATI
Proof. Assume that α < kxkρe = kµ(x)kρ . We have ρe
1
implies ρe(x)
< ωρe α1 . Hence
1
1
−1
< .
ωρe
ρe(x)
α
Letting α → kxkρe, we obtain kxkρe ≤
ωρ−1
e
1
1
ρ
e(x)
x
α
> 1, which
.
Proposition 3.8. Let ρ be a convex function modular with the ∆2 -type
condition. Then for every ε > 0, there exists δ > 0 such that
e(x) < δ.
kxkρe < ε (kxkA
ρe < ε) if ρ
Proof. For ε > 0, we choose δ =
1
1 .
ωρ−1
e (ε)
For the Amemiya norm we use
the fact that it is equivalent to the Luxemburg norm.
Theorem 3.9. Let ρ be a convex function modular with the ∆2 -type condition. Let {xn } and {yn } be two sequences in the noncommutative modular
f τ ) such that ρe(yn ) → 0. Then
function space Lρe(M,
lim sup ρe(xn + yn ) = lim sup ρe(xn ).
n→∞
n→∞
Proof. It is known that µ(xn + yn ) ≺≺ µ(xn ) + µ(yn ) [FK86, Theorem 4.4].
For every ε ∈ (0, 1) we have
ρe(xn + yn ) = ρ(µ(xn + yn )) ≤ ρ(µ(xn )) + µ(yn ))
µ(xn )
µ(yn )
≤ρ
+ρ
1−ε
ε
1
1
ρe(xn ) + ωρe
ρe(yn ),
≤ ωρe
1−ε
ε
and so
lim sup ρe(xn + yn ) ≤ ωρe
n→∞
1
1−ε
lim sup ρe(xn ).
n→∞
Since ε > 0 is arbitrary and
1
ωρe
→ 1 as ε → 0+
1−ε
we get
lim sup ρe(xn + yn ) ≤ lim sup ρe(xn ).
n→∞
n→∞
Moreover, the same argument shows that
lim sup ρe(xn ) = lim sup ρe(xn + yn − yn ) ≤ lim sup ρe(xn + yn ).
n→∞
n→∞
n→∞
NONCOMMUTATIVE MODULAR FUNCTION SPACES
771
Corollary 3.10. Let ϕ satisfy the ∆2 -condition. Let {xn } and {yn } be
two sequences of τ -measurable operators in the noncommutative Orlicz space
f τ ) such that τ (ϕ(yn )) → 0. Then
Lϕ (M,
lim sup τ (ϕ(|xn + yn |)) = lim sup τ (ϕ(|xn |)).
n→∞
n→∞
Before we give the main theorem of this section, we need the following
lemma.
Lemma 3.11. Let ε > 0 and λ > 1 be such
that λε < 1. Then for every
1
f τ ) such that ρe(λx) < ∞ and ρe
x, y ∈ Lρe(M,
ε(λ−1) y < ∞, we have
|e
ρ(x + y) − ρe(x)| ≤ ε [e
ρ(λx) − λe
ρ(x)] + ρe(cε y),
where cε =
1
ε(λ−1) .
Proof. The proof of this lemma is essentially the same as that of [BL83,
f τ ) and j(x) = ρe(x).
Lemma 3] if C is replaced by Lρe(M,
Corollary 3.12. Let ε > 0 and λ > 1 be such that λε < 1. Then
for every
|y|
ϕ
f
x, y ∈ L (M, τ ) such that τ (ϕ(λ|x|)) < ∞ and τ ϕ ε(λ−1)
< ∞, we
have
|τ (ϕ(|x + y|)) − τ (ϕ(|x|))| ≤ ε [τ (ϕ(λ|x|)) − λτ (ϕ(|x|))] + τ (ϕ(cε y)),
where cε =
1
ε(λ−1) .
f Then the sequence {xn } is said
Definition 3.13. Let {xn } and x be in M.
to be ρe-a.e. convergent to x if µ(xn − x) → 0 ρ-a.e.
Theorem 3.14. Let ρ be an additive convex modular on L0 (ν) and {xn } be
f τ ) which is ρe-a.e.-convergent to 0. Assume that there
a sequence in Lρe(M,
exists λ > 1 such that supn ρe(λxn ) < ∞. Then
lim (e
ρ(xn + y) − ρe(xn )) = ρe(y)
n→∞
f τ ).
for all y ∈ Lρe(M,
Proof. By the definition of the ρe-a.e.-convergence we have that {µ(xn )} → 0
ρ-a.e. It follows from Egoroff’s theorem
S that there exists an increasing sequence of sets Ek ∈ P such that Ω = k Ek and {µ(xn )} converges uniformly
to 0 on every Ek . On the other hand we have
|e
ρ(xn + y) − ρe(xn ) − ρe(y)|
≤ |ρ(µ(xn + y), Em ) − ρ(µ(xn ), Em ) − ρ(µ(y), Em )|
c
c
c
)|
+ |ρ(µ(xn + y), Em
) − ρ(µ(xn ), Em
) − ρ(µ(y), Em
772
GHADIR SADEGHI AND REZA SAADATI
c denotes the complement of the subset E . Using Lemma 3.11 we
where Em
m
get
(3.3) |ρ(µ(xn + y), Em ) − ρ(µ(y), Em )| ≤ ε|ρ(λµ(y), Em ) − λρ(µ(y), Em )|
+ ρ(cε µ(xn ), Em ),
for every ε > 0 such that λε < 1. Since {µ(xn )} converges uniformly to 0
on every Em , we have
lim sup |ρ(µ(xn + y), Em ) − ρ(µ(xn ), Em ) − ρ(µ(y), Em )| ≤ ερ(λµ(y)).
n→∞
Using the same strategy we get
c
c
c
lim sup |ρ(µ(xn + y), Em
) − ρ(µ(xn ), Em
) − ρ(µ(y), Em
)|
n→∞
c
c
c
≤ ε lim sup ρ(λµ(xn ) + ρ(cε µ(y), Em
) + ρ(cε µ(y), Em
) + ρ(µ(y), Em
).
n→∞
Hence
lim sup |ρ(µ(xn + y) − ρ(µ(xn ) − ρ(µ(y)
n→∞
c
c
≤ ερ(λµ(y)) + ε sup ρ(λµ(xn )) + ρ(cε µ(y), Em
) + ρ(µ(y), Em
).
n
f τ ) to get
Let m tend to ∞ and use the fact that y ∈ Lρe(M,
lim sup |ρ(µ(xn + y)) − ρ(µ(xn ) − ρ(µ(y)|
n→∞
≤ ερ(λµ(y)) + ε sup ρ(λµ(xn )).
n
Finally, we let ε approach 0 to get
lim sup |e
ρ(xn + y) − ρe(xn ) − ρe(y)| ≤ 0,
n→∞
which completes the proof.
Remark 3.15. It is known that, for an Orlicz space, the ρ-null sets coincide
with the sets of measure zero. Thus the ρ-a.e.-convergence is equivalent to
the convergence almost everywhere. It follows from [FK86, Lemma 3.1] that
the ρe-a.e.-convergence in noncommutative Orlicz spaces as well as noncommutative Lp –spaces is equivalent to the convergence in the measure topology
τm .
Corollary 3.16. Let {xn } be a sequence in the noncommutative Orlicz space
f τ ) such that {xn } converges to x in the measure topology. Assume
Lϕ (M,
that there exists λ > 1 such that supn τ (ϕ(λ|xn |)) < ∞. Then
lim inf τ (ϕ(|xn |)) = lim inf τ (ϕ(|xn − x|)) + τ (ϕ(|x|)).
n→∞
n→∞
NONCOMMUTATIVE MODULAR FUNCTION SPACES
773
Brezis and Lieb [BL83] proved that if {fn } is a sequence of LP –uniformly
bounded functions on a measure space and if fn → f almost everywhere
then
lim inf kfn kp = lim inf kfn − f kp + kf kp ,
n→∞
n→∞
for all p ∈ (0, ∞). Now we establish an extension of this equality to the
noncommutative setting.
f τ ).
Corollary 3.17. Let p ≥ 1 and {xn } be a bounded sequence of Lp (M,
Assume that {xn } converges to x in the measure topology. Then
lim inf kxn kp = lim inf kxn − xkp + kxkp .
n→∞
n→∞
We will say that a Banach space X with the τ –topology on X satisfies the
Opial condition if for every bounded sequence {xn } in X which τ –converges
to x ∈ X we have
lim inf kxn − xk < lim inf kxn − yk
n→∞
n→∞
for every y 6= x.
f τ ) be ρe-a.e. convergent to 0.
Theorem 3.18. Let ε > 0 and {xn } ⊆ Lρe(M,
Assume there exists λ > 1 such that
sup ρe(λxn ) < ∞.
n
f τ ) such that ρe(x) > ε. Then
Let x ∈ Lρe(M,
lim inf ρe(x) + ε ≤ lim inf ρe(x + xn ).
n→∞
n→∞
Proof. The proof is obvious using the conclusion of Theorem 3.14. This is
a kind of Opial property.
Corollary 3.19. Let ε > 0 and {xn } be a sequence in the noncommutative
f τ ) such that {xn } converges to 0 in the measure topology.
Orlicz space Lϕ (M,
Assume that there exists λ > 1 such that sup τ (ϕ(λ|xn |)) < ∞. Let x ∈
f τ ) such that τ (ϕ(|x|)) > ε. Then
Lϕ (M,
lim inf τ (ϕ(|xn |)) + ε ≤ lim inf τ (ϕ(|xn + x|)).
n→∞
n→∞
4. Uniform Opial and uniform Kadec–Klee properties
In this section, we assume that ρ is a convex additive modular with the
∆2 -type condition on L0 (ν) such that for every f, g ∈ L0 (ν), f ≺≺ g implies
that ρ(f ) ≤ ρ(g). Note that the additive condition may seem strong, but
many interesting examples lead to additive modulars (e.g., any modular generated by a functional measure). In the paper [J04], the author showed that
modular function spaces Lρ satisfy the uniform Opial condition with respect
to the ρ-a.e.-convergence for both the Luxemburg norm and Amemiya norm,
and also showed that modular function spaces Lρ have the uniform Kadec–
Klee property with respect to the ρ-a.e.-convergence when equipped with
774
GHADIR SADEGHI AND REZA SAADATI
the Luxemburg norm. We prove these results for noncommutative modular
function spaces.
Before we give the main results of this section we need the following
lemma. It is a generalization of [J04, Lemma 2.5] which was given for
classical modular function spaces.
Lemma 4.1. Let {xn } be a sequence in the noncommutative modular funcf τ ) such that xn → 0 ρe-a.e. Let x be a given operator in
tion space Lρe(M,
f τ ). Then for every ε > 0 there exist a subsequence {xn }, a sequence
Lρe(M,
k
f
f
{yk } in Lρe(M, τ ) and an operator y ∈ Lρe(M, τ ) such that:
(i) limk→∞ kxnk − yk kρe = 0.
(ii) supp(µ(yk )) ∩ supp(µ(y)) = ∅ for every k ∈ N.
(iii) kx − ykρe < ε.
Proof. Since the given sequence {xn } converges to 0 ρe-a.e., µ(xn ) → 0
ρ-a.e. By using the version of the Egoroff theorem for modular function
spaces, we can assume that there exists a sequence of sets Ωk ∈ P such that
Ω = ∪k Ωk , Ωi ∩Ωj = ∅ and {µ(xn )} converges uniformly to the null function
on every Ωk . Moreover, the ∆2 –type condition of the modular implies that
f τ ). Also,
ρe(x) = ρ(µt (x)) < +∞ for every measurable operator x ∈ Lρe(M,
ρe(x) = ρ(µt (x)) = Σk ρ(µt (x), Ωk ) since ρ is additive. Let ε > 0 be given.
We will consider the corresponding δ > 0 given in Proposition 3.8. Then
there exists a positive integer k0 such that
X
ρ(µt (x), Ωk ) < δ.
k>k0
Let {εn } be a sequence in (0, 1) with limn→∞ εn = 0 and for every n ∈ N,
take the corresponding δn given by Proposition 3.8. By Definition 2.3(iv),
0
ρ(α, ∪kk=1
Ωk ) → 0 as α decreases to 0. Thus there exists a number α0 > 0
such that
!
k0
[
δ1
ρ α0 ,
Ωk < .
2
k=1
0
Due to the fact that {µt (xn )} converges uniformly to 0 in ∪kk=1
Ωk , there
exists n1 ∈ N such that for every n ≥ n1 , we have |µt (xn )| ≤ α0 for every t ∈
0
f τ ) and a positive
∪kk=1
Ωk . Consider the measurable operator xn1 ∈ Lρe(M,
integer k1 > k0 such that
X
δ1
ρ(µt (xn1 ), Ωk ) < .
2
k>k1
1
Define the operator y1 = xn1 ek1 where ek1 = e|x| (∪kk>k
Ω ). Since
0 k
y1 = xn1 ek1 = xn1 − xn1 e⊥
k1 ,
NONCOMMUTATIVE MODULAR FUNCTION SPACES
775
we have
⊥
µ(y1 ) = µ(xn1 − xn1 e⊥
k1 ) ≺≺ µ(xn1 ) + µ(xn1 ek1 ) ≤ 2µ(xn1 ).
The convex additive modular ρ satisfies the ∆2 -type condition. Hence
ρ(µ(y1 )) ≤ ρ(2µ(xn1 )) ≤ kρ(µ(xn1 ))
f τ ). So
f τ ) since xn ∈ Lρe(M,
for some k > 0. This ensures that y1 ∈ Lρe(M,
1
ρe(xn1 − y1 ) = ρ(µt (xn1 − y1 )) = ρ(µt (xn1 e⊥
k1 ))


k0
k1
∞
[
[
[
Ωk ∪
Ωk 
= ρ µt (xn1 e⊥
Ωk ∪
k1 ),
k=1

k0
[
= ρ µt (xn1 e⊥
k1 ),
k>k0
≤ ρ µt (xn1 ),
k0
[
≤ ρ α0 ,
k=1
∞
[
Ωk ∪

Ωk 
k>k1
k=1
k0
[
Ωk 
k>k1
k=1

∞
[
Ωk ∪
k>k1

!
Ωk
+
X
ρ (µt (xn1 ), Ωk )
k>k1
< δ1 .
Suppose that, by induction, we have two finite sequences of positive integers
k0 < k1 < · · · < kl−1 and n1 < n2 < · · · < nl−1 with ρe(xni − yi ) < δi for
i
Ω ).
i = 1, 2, . . . , l−1, where the operator yi is defined by yi = xni e|x| (∪kk>k
i−1 k
k
l−1
By Definition 2.3(iv), we can find αl such that ρ(α1 , ∪k=1
Ωk ) < δ2l . Since
kl−1
{µt (xn )} converges uniformly to the null function on ∪k=1 Ωk , we can find
kl−1
nl > nl−1 such that |µt (xnl )| ≤ αl for every t ∈ ∪k=1
Ωk . Moreover, we can
find kl > kl−1 with
X
δl
ρ (µt (xnl ), Ωk ) < .
2
k>kl
f τ ),
Using the above argument, we define the operator yl = xnl ekl ∈ Lρe(M,
kl
|x|
where ekl = e (∪k>kl−1 Ωk ), and obtain ρe(xnl − yl ) < δl . Therefore
kxnl − yl k < εl .
Consequently, limk→∞ kxnk − yk k = 0.
0
Now, define the operator y = xe0 where e0 = e|x| (∪kk=1
Ωk ). It is clear
that supp(µ(yk )) ∩ supp(µ(y)) = ∅ for every k ∈ N. Moreover
),
∪
Ω
ρe(x − y) = ρ(µ(x − xe0 )) = ρ µ(xe⊥
k>k0 k
0
776
GHADIR SADEGHI AND REZA SAADATI
≤ ρ (µ(x), ∪k>k0 Ωk ) =
X
ρ(µ(x), Ωk ) < δ,
k>k0
whence kx − ykρe < ε.
Corollary 4.2. Let ϕ satisfy the ∆2 -condition. Let {xn } be a sequence in
f τ ) converging to 0 in the measure
the noncommutative Orlicz space Lϕ (M,
f τ ). Then for every ε > 0 there
topology. Let x be a given operator in Lϕ (M,
f τ ) and an operator
exist a subsequence {xnk }, a sequence {yn } in Lϕ (M,
f τ ) such that:
y ∈ Lϕ (M,
(i) limk→∞ kxnk − yk kρeϕ = 0.
(ii) supp(µ(yk )) ∩ supp(µ(y)) = ∅ for every k ∈ N.
(iii) kx − ykρeϕ < ε.
f τ)
Theorem 4.3. The noncommutative modular function space Lρe(M,
equipped with the Luxemburg norm k · kρe satisfies the uniform Opial condition with respect to the ρe-a.e.-convergence. In particular for every c > 0,
1 + oρe(c) ≥ ωρ−1
e (1 + α),
where α =
1
.
ωρe( 1c )
f τ ) such that xn →
Proof. Let c > 0 and {xn } be a sequence in Lρe(M,
f τ)
0 ρe-a.e. and limn→∞ kxn kρe ≥ 1. Consider an operator x ∈ Lρe(M,
with kxkρe ≥ c. We apply Lemma 4.1 to the sequence {xn } and some
0 < ε < c to obtain a subsequence {xnk } of {xn }, a sequence {yk } ∈
f τ ) and an operator y ∈ Lρe(M,
f τ ) such that limk→∞ kxn − yk kρe = 0,
Lρe(M,
k
supp(µ(yk )) ∩ supp(µ(y)) = ∅ for k ∈ N and kx − ykρe < ε. We can also
assume that lim inf n→∞ kxn + xkρe = limn→∞ kxnk + xkρe. Consequently,
lim inf k→∞ kyk kρe ≥ 1 and kykρe ≥ c − ε.
Fix 0 < t < 1 and define the functions z = yt and zk = ytk for k ∈ N. We
know that kzkρe > c − ε and kzk kρe > 1 for k large enough. Set
1
1
γε := ωρ−1
1
+
where
ζ
=
ω
.
ρe
e
ζ
c−ε
Using the properties of the modular ρe, the definition of the growth function ωρe(·) and Lemma 3.6(iii), we have
zk + z
µ(zk + z)
µ(zk ) − µ(z)
ρe
=ρ
≥ρ
γε
γε
γε
µ(zk ) − µ(z)
=ρ
, supp(µ(zk )) ∪ supp(µ(z))
γε
µ(zk )
µ(z)
=ρ
, supp(µ(zk )) + ρ
, supp(µ(z))
γε
γε
µ(zk )
µ(z)
=ρ
+ρ
γε
γε
NONCOMMUTATIVE MODULAR FUNCTION SPACES
= ρe
zk
γε
1
+ ρe
z
γε
777
1
z
≥
ρe(zk ) +
ρe
ωρe(γε )
ωρe( γε )
c−ε
c−ε
1
1
>
+
= 1.
ωρe(γε ) ωρe(γε ) ωρe( 1 )
c−ε
Thus lim inf k→∞ kyk + ykρe ≥ tγε . Letting t tend to 1, we deduce that
lim inf k→∞ kyk + ykρe ≥ γε . On the other hand,
lim inf kxk + xkρe = lim kxnk + xkρe
k→∞
k→∞
≥ lim inf kyk + ykρe − lim kxnk − yk kρe − kx − ykρe
k→∞
k→∞
≥ γε − ε.
Since ε is arbitrary and the functions ωρe(·) and ωρ−1
e (·) are continuous, we
infer that
!
1
−1
1+
lim inf kxk + xkρe ≥ ωρe
,
k→∞
ωρe( 1c )
which yields the desired lower bound for 1 + oρe(c). This implies that
f τ ), k · kρe) has the uniform Opial condition with respect to the ρe(Lρe(M,
a.e.-convergence
f τ)
Theorem 4.4. The noncommutative modular function space Lρe(M,
equipped with the Amemiya norm k · kA
ρe satisfies the uniform Opial condition
with respect to the ρe-a.e.-convergence. In particular for every c > 0,
1 + oρe(c) ≥ ωρ−1
e (1 + α),
where α =
1
.
ωρe( 1c )
f τ ) such that xn → 0 ρe-a.e. and
Proof. Let {xn } be a sequence in Lρe(M,
A
f
lim inf kxn kA
ρe ≥ 1. Consider an operator x ∈ Lρe(M, τ ) with kxkρe ≥ c. By
Lemma 4.1 and using the same argument as in the proof Theorem 4.3, we
can assume that supp(µ(xn )) ∩ supp(µ(x)) = ∅ and kxkA
ρe ≥ 1 for every
n ∈ N. Recall that
1 + ρe(λx)
A
kxkρe = inf
:λ>0
λ
f τ ). Then if x is an operator in Lρe(M,
f τ ) with kxkA ≥ 1,
for every x ∈ Lρe(M,
ρe
we have that ρe(λx) ≥ λ − 1 for every λ > 0.
778
GHADIR SADEGHI AND REZA SAADATI
Set γ := ωρ−1
1+
e
1
ωρe( 1c )
and fix λ > 0. Using the properties of the
modular ρe and the growth function ωρe(·), we get
1
λ(xn + x)
1
λxn
λx
1 + ρe
=
1 + ρe
+ ρe
λ
γ
λ
γ
γ
#
"
)
1
ρe(λxn ) ρe( λx
c
≥
1+
+
λ
ωρe(γ)
ωρe( γc )
1
λ−1
λ−1
≥
1+
+
λ
ωρe(γ) ωρe( γc )
"
!#
1
1
1
1 + (λ − 1)
+
≥
λ
ωρe(γ) ωρe(γ)ωρe( 1c )
= 1.
Taking infimum over all λ > 0, we infer that lim inf n→∞ kxn + xkA
ρe ≥ γ,
which completes the proof.
Corollary 4.5. Let ϕ satisfy the ∆2 –condition. Then the noncommutative
f τ ) equipped with the Luxemburg and Amemiya norms
Orlicz space Lϕ (M,
satisfies the uniform Opial condition with respect to the convergence in measure.
f τ)
Theorem 4.6. The noncommutative modular function space Lρe(M,
equipped with the Luxemburg norm k · kρe has the uniform Kadec–Klee property with respect to the ρe-a.e.-convergence. Moreover, for every ε > 0,
kρe(ε) ≥ 1 −
where ζ =
1
1
ωρe( 1−ζ
)
,
1
.
2ωρe( 1ε )
f τ ) with
Proof. Let ε > 0 and {xn } be a sequence in the unit ball of Lρe(M,
f τ ) ρe-a.e.
sep(xn ) > ε. Assume that {xn } is convergent to some x ∈ Lρe(M,
xn
x
Consider any real number t > 0 and define yn := 1+t and y := 1+t
. In this
ε
case kyn kρe < 1 and ρe(yn ) ≤ kyn k < 1 for every n ∈ N. Put ξ = 1+t
. Since
yn −ym
kyn − ym k ≥ ξ for every n 6= m, we see that ρe
> 1 for every n 6= m.
ξ
This implies that
( x
)
yn −ym
ρ
e
ρ
e
(
)
ξ
1
1
ξ
ωρe
: 0 < ρe(x) < ∞ ≥
>
,
= sup
ξ
ρe(x)
ρe (yn − ym )
ρe (yn − ym )
NONCOMMUTATIVE MODULAR FUNCTION SPACES
779
and consequently ρe (yn − ym ) > ωρe 1ξ for every n 6= m. If we consider
m ∈ N and use Theorem 3.14, we obtain
1 1
≤ lim inf ρe (yn − ym ) = lim inf ρe ((yn − y) − (ym − y))
n→∞
n→∞
ωρe ξ
= lim inf ρe (yn − y) + ρe(ym − y).
n→∞
Letting m tend to infinity, we get
lim inf ρe (yn − y) ≥
n→∞
1
.
2ωρe( 1ξ )
On the other hand, using Theorem 3.14, we have
lim inf ρe(yn ) = lim inf ρe(yn − y + y) = lim inf ρe(yn − y) + ρe(y).
n→∞
n→∞
n→∞
Thus
0 ≤ ρe(y) = lim inf ρe(yn ) − lim inf ρe(yn − y) ≤ 1 −
n→∞
n→∞
1
.
2ωρe( 1ξ )
Employing Lemma 3.7 and the fact that the function ωρ−1
e is increasing we
get
1
,
kxkρe ≤ (1 + t)
1
)
ωρe( 1−ζ
where ζ =
1
.
2ωρe( 1ξ )
Since t > 0 is arbitrary and the functions ωρe and ωρ−1
e are
continuous, it follows that
kxkρe ≤
where ζ =
1
.
2ωρe( 1ξ )
1
1
ωρe( 1−ζ
)
:= 1 − δ(ε),
In order to finish the proof we have to check that 1−δ(ε) <
1. Taking into account that ωρe is an increasing function and that ωρe(1) = 1,
the above inequality holds if and only if ωρe( 1ε ) > 0. This condition is satisfied
due to the ∆2 –type condition. Indeed, since we are assuming that there
f τ ), we
exists some k > 0 such that ρe(2x) ≤ ke
ρ(x) for every x ∈ Lρe(M,
obtain
x
ρe( 2 )
1
1
ωρe
= sup
: 0 < ρe(x) < ∞ ≥ > 0.
2
ρe(x)
k
It follows from 0 < ε ≤ 2 that ωρe( 1ε ) ≥ ωρe( 12 ) > 0 as required.
Corollary 4.7. Let ϕ satisfy the ∆2 -condition. Then the noncommutative
f τ ) equipped with the Luxemburg norm has the uniform
Orlicz space Lϕ (M,
Kadec–Klee property with respect to the convergence in measure.
Acknowledgement. The authors are grateful to the reviewer and the area
editor for their valuable comments and suggestions.
780
GHADIR SADEGHI AND REZA SAADATI
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(Ghadir Sadeghi) Department of Mathematics and Computer Sciences, Hakim
Sabzevari University, P.O. Box 397, Sabzevar, IRAN
ghadir54@gmail.com
g.sadeghi@hsu.ac.ir
(Reza Saadati) Department of Mathematics, Iran University of Science and
Technology, Tehran, Iran
rsaadati@eml.cc
This paper is available via http://nyjm.albany.edu/j/2015/21-33.html.