# The Arens Algebras of Vector-Valued Functions

Authors: I. G. Ganiev; O. I. Egamberdiev
Journal: Journal of Function Spaces
 (2014)

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## Abstract

A class of Arens algebras of vector Banach algebra valued functions is considered. It is shown that Arens algebra of vector-valued functions is complete locally convex metrizable
algebra.


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## Body

## 1. Introduction

Let(Ω,Σ,μ) be a measure space with finite measure μ and let Lp=Lp(Ω,Σ,μ) be the Banach space of all measurable functions f on (Ω,Σ,μ) such that |f|p=(∫Ω|f|pdμ)1/p<∞, (1≤p<∞). In [1] the set
(1)Lω=⋂p≥1‍Lp
is introduced and it is shown that Lω is locally convex algebra in topology τ generated by the norms ∥·∥p. Such algebras are called Arens algebras. In [2] it is shown that (Lω,τ) is GB* algebra. Properties of algebra Lω are considered in [3, 4]. Moreover, it is shown that dual of Lω is ⋃1<q≤∞Lq and Lω is reflexive. In [5] a class of noncommutative analogue of Arens algebra Lω(M,φ) associated with semifinite von Neumann algebra and semifinite trace φ is introduced and in [6] it is shown that Lω(M,φ) is EW* algebra. In [6–8], properties of noncommutative Arens algebras are considered. Noncommutative Arens algebras Lω(M,Φ) associated with semifinite von Neumann algebra and center-valued trace Φ are introduced in [9] and in [10] they are generalized for the case when Φ is the Maharam trace.LetLp(Ω,X) be of Bochner maps from Ω into a unital Banach algebra (X,∥·∥X), integrable with degree p, 1≤p<∞, that is, Lp space of vector-valued functions, and
(2)Lω(Ω,X)=⋂p≥1Lp(Ω,X).
In the present paper we prove that Lω(Ω,X) is complete locally convex metrizable algebra, which we call Arens algebra of vector-valued functions. It is shown that if dual of X has Radon-Nikodym property, then dual space of Lω(Ω,X) is ⋃1<q≤∞Lq(Ω,X*); if X is reflexive, then Lω(Ω,X) also is reflexive.


## 2. Preliminaries

Let(Ω,Σ,μ) be a measure space with finite measure; let L0(Ω) be the classes of all measurable functions on Ω.Throughout this paper,X stands for a fixed initial Banach algebra with the norm ∥·∥X. By L0(Ω,X) we denote the classes of all measurable by Bochner mappings from Ω to X; that is,
(3)L0(Ω,X)={f:Ω⟶X:∥f(ω)∥X∈L0(Ω)}.
It is known [11] that L0(Ω,X) is algebra with respect to multiplication operation (fg)(ω)=f(ω)g(ω) for any f,g∈L0(Ω,X). By Lp(Ω,X), 1≤p<∞, we denote the Banach space of X valued measurable functions f on (Ω,Σ,μ) with the norm defined as
(4)∥f∥p=(∫Ω∥f(ω)∥Xpdμ)1/p=|∥f(ω)∥X|p;
that is, Lp(Ω,X)={f∈L0(Ω,X):∥f(ω)∥X∈Lp} and L∞(Ω,X)={f∈L0(Ω,X):∃λ>0,∥f(ω)∥X≤λa.e.}={f∈L0(Ω,X):∥f(ω)∥X∈L∞} with the norm defined as
(5)∥f∥∞=inf⁡{λ>0:∥f(ω)∥X≤λa.e.}.
It is known [11] that the pair (L∞(Ω,X),∥·∥∞) is a Banach algebra.The following well known properties hold true.(a)
Forf,g∈Lp(Ω,X) the Minkowski inequality
(6)∥f+g∥p≤∥f∥p+∥g∥pis valid.(b)
Forh∈Lp and f∈Lq(Ω,X), where q:=p/(p-1) for p≠1 and q:=∞ for p=1, the Hölder inequality
(7)∥hf∥1≤|h|p∥f∥qis valid.Proposition 1.
(i) Iff∈L∞(Ω,X) and g∈Lp(Ω,X) then fg∈Lp(Ω,X) and
(8)∥fg∥p≤∥f∥∞∥g∥p.(ii) If p, q, r≥1, 1/r=(1/p)+(1/q), f∈Lp(Ω,X), and g∈Lq(Ω,X), then fg∈Lr(Ω,X) and ∥fg∥r≤∥f∥p∥g∥q.
(iii) If∥f(ω)∥X≤∥g(ω)∥X a.e., f∈L0(Ω,X), and g∈Lp(Ω,X), then f∈Lp(Ω,X) and ∥f∥p≤∥g∥p.Proof.
(i) Letf∈L∞(Ω,X) and g∈Lp(Ω,X). Since ∥f(ω)g(ω)∥X≤∥f(ω)∥X∥g(ω)∥X≤∥f∥∞∥g(ω)∥X and ∥g(ω)∥X∈Lp we have that fg∈Lp(Ω,X) and
(9)∥fg∥p=(∫Ω∥f(ω)g(ω)∥Xpdμ)1/p≤∥f∥∞(∫Ω∥g(ω)∥Xpdμ)1/p=∥f∥∞∥g∥p.
(ii) Letp, q, r≥1, 1/r=(1/p)+(1/q), f∈Lp(Ω,X), and g∈Lq(Ω,X). As ∥f(ω)∥X∈Lp and ∥g(ω)∥X∈Lq from inequality ∥f(ω)g(ω)∥X≤∥f(ω)∥X∥g(ω)∥X we get that fg∈Lr(Ω,X) and ∥fg∥r=(∫Ω∥f(ω)g(ω)∥Xrdμ)1/r≤(∫Ω(∥f(ω)∥X∥g(ω)∥X)rdμ)1/r≤(∫Ω∥f(ω)∥Xpdμ)1/p(∫Ω∥g(ω)∥Xqdμ)1/q= ∥f∥p∥g∥q.
(iii) Let∥f(ω)∥X≤∥g(ω)∥X a.e., f∈L0(Ω,X), and g∈Lp(Ω,X). Then ∫Ω∥f(ω)∥Xpdμ≤∫Ω∥g(ω)∥Xpdμ<∞. This means that f∈Lp(Ω,X) and ∥f∥p≤∥g∥p.


## 3. The Arens AlgebrasLω(Ω,X) of Vector-Valued Functions

In this section we introduce notion of the Arens algebras of vector-valued functions. Put(10)Lω(Ω,X)=⋂p≥1Lp(Ω,X);
that is, Lω(Ω,X)={f∈L0(Ω,X):∥f∥1<∞,∥f∥2<∞,…,∥f∥p<∞,…}, and will consider in Lω(Ω,X) local convex topology τX is generated by system of norms {∥·∥p}p≥1. From (ii) Proposition 1 we get that ∥f∥1≤∥f∥2≤⋯≤∥f∥n≤⋯, that is, the topology τX generated by countable system of norms {∥·∥n}n=1∞. By Theorem III.2.2 [12] it means that topological vector space (Lω(Ω,X),τX) is metrizable space with respect to the metric
(11)ρ(f,g)=∑k=1∞12k∥f-g∥k1+∥f-g∥k.Theorem 2.
(
L
ω
(
Ω
,
X
)
,
τ
X
) is a complete topological algebra.Proof.
Let{fn} be a fundamental sequence in (Lω(Ω,X),τX). Then {fn} is fundamental in Lp(Ω,X) for any 1≤p<∞. Since Lp(Ω,X) is complete there exist fp∈Lp(Ω,X) such that ∥fn-fp∥p→0 as n→∞, as
(12)∥fn-fp∥q≤μ(Ω)pq/p+q∥fn-fp∥p,
for all 1≤q<p<∞, implies that ∥fn-fp∥q→0. This means that fp=fq if 1≤q<p<∞. Therefore f=fp is limit of sequence {fn} and f is an element of space Lω(Ω,X). Since the inequality
(13)∥fg∥p≤∥f∥2p∥g∥2p
is valid the multiplication is continuous on Lω(Ω,X). Hence (Lω(Ω,X),τX) is a topological algebra.Theorem 3.
(i) Iff∈L0(Ω,X), g∈Lω(Ω,X), and ∥f(ω)∥X≤∥g(ω)∥X a.e. then f∈Lω(Ω,X) and ρ(f,0)≤ρ(g,0).
(ii) Ifg∈L∞(Ω,X) and f∈Lω(Ω,X) then gf∈Lω(Ω,X) and there exist m∈ℕ(14)ρ(gf,0)≤2mρ(f,0).Proof.
(i) Asg∈Lω(Ω,X) we have that g∈Lp(Ω,X) for any p≥1. Then from ∥f(ω)∥X≤∥g(ω)∥X a.e. by Proposition 1(iii) we get that f∈Lp(Ω,X) for any p≥1. This means that f∈Lω(Ω,X). Since ∥f∥p≤∥g∥p for any p≥1 we have
(15)ρ(f,0)=∑k=1∞12k∥f∥k1+∥f∥k≤∑k=1∞12k∥g∥k1+∥g∥k=ρ(g,0).
(ii) Letg∈L∞(Ω,X) and f∈Lω(Ω,X); then f∈Lp(Ω,X) for all p≥1. By Proposition 1(i) gf∈Lp(Ω,X) for all p≥1. Therefore gf∈Lω(Ω,X). We chose m∈ℕ that ∥g∥∞≤2m. Then using Proposition 1(i) we get
(16)ρ(gf,0)=∑k=1∞12k∥gf∥k1+∥gf∥k≤∑k=1∞12k∥g∥∞∥f∥k1+∥g∥∞∥f∥k≤∑k=1∞12k2m∥f∥k1+2m∥f∥k≤2m∑k=1∞12k∥f∥k1+∥f∥k=2mρ(f,0).Theorem 4.
L
∞
(
Ω
,
X
) is dense in Lω(Ω,X) and Lω(Ω,X) is dense in Lp(Ω,X).Proof.
Letf∈Lω(Ω,X). There exists sequence of idempotents en from L0(Ω) such that en↑1, where 1 is unit element on L0(Ω). Then ∥f(ω)en(ω)∥X=∥f(ω)∥Xen(ω)∈L∞(Ω); that is, f(ω)en(ω)∈L∞(Ω,X). Let fn=fen. Then fn∈L∞(Ω,X) and ∥f(ω)-fn(ω)∥X=∥f(ω)-f(ω)en(ω)∥X=∥f(ω)(1(ω)-en(ω))∥X=∥f(ω)∥X(1(ω)-en(ω))↓0. As norm |·|p is order continuous in Lp space for all p≥1 we get that ∥f-fn∥p=|∥f(ω)∥X(1(ω)-en(ω))|p→0 for all p≥1 as n→∞. Therefore
(17)ρ(fn,f)=∑k=1∞12k∥fn-f∥k1+∥fn-f∥k⟶0,
as n→∞. The density Lω(Ω,X) in Lp(Ω,X) follows from density L∞(Ω,X) in Lω(Ω,X).
Letφ1,φ2,…,φn∈Lω and x1,x2,…,xn∈X. Then we can define function f:Ω→X by setting
(18)f(ω)=∑i=1nφi(ω)xi,
for almost all ω∈Ω.
We will denoteLω⊗X={f∈L0(Ω,X):f(ω)=∑i=1nφi(ω)xi,φi∈Lω,xi∈X,i=1,n¯,n∈ℕ}. Lω⊗X is subspace of Lω(Ω,X), because ∫Ω∥f(ω)∥Xpdμ≤∑i=1n∫Ω|φi(ω)|p∥xi∥Xpdμ=∑i=1n∥xi∥Xp∫Ω|φi(ω)|pdμ<∞ for all p≥1.Theorem 5.
L
ω
⊗
X is dense in Lω(Ω,X).Proof.
Letf∈Lω(Ω,X). Then f∈Lp(Ω,X) for all p≥1.
Letsn(ω) be the sequence simple X valued functions; that is, every sn(ω) has form {∑i=1mχBi(ω)xi:Bi∈Σ,Bi∩Bj=Ø,xi∈X}, such that sn(ω)→f(ω) a.e. on Ω. Then ∥sn(ω)∥X→∥f(ω)∥X almost everywhere on Ω.
LetAn={ω:∥sn(ω)∥X≤2∥f(ω)∥X}. If we set gn(ω)=sn(ω)χAn(ω) we have that gn(ω)→f(ω) a.e. on Ω and
(19)sup⁡n∥gn(ω)-f(ω)∥X≤sup⁡n∥gn(ω)∥X+∥f(ω)∥X≤3∥f(ω)∥X.
By dominated convergence we have that
(20)∫Ω∥gn(ω)-f(ω)∥Xpdμ⟶0,
for all p≥1 and of course gn∈Lω⨂X.


## 4. Dual of Arens AlgebrasLω(Ω,X)

In this section we study dual of Arens algebras of vector-valued functions. It is shown that if dual ofX has Radon-Nikodym property, then dual space of Lω(Ω,X) is ⋃1<q≤∞Lq(Ω,X*); if X is reflexive, then Lω(Ω,X) also is reflexive.Definition 6 (see [13]).
Recall that a Banach spaceY is said to possess the Radon-Nikodym property, with respect to μ, if for any μ-continuous vector measure m:Σ→Y of bounded total variation, there exists a Bochner integrable function f:Ω→Y such that
(21)m(A)=∫Afdμ,
for all A∈Σ. It is known [13] that dual of Banach algebra c0, c0*=l1 has the Radon-Nikodym property.
A class of Banach algebrasX whose duals X* have the Radon-Nikodym property is studied in [14].Theorem 7.
If the dual spaceX* of Banach algebra X has Radon-Nikodym property with respect to μ, then the dual of Lω(Ω,X) is the space
(22)E=⋃1<q≤∞Lq(Ω,X*),
and for any ℓ∈(Lω(Ω,X))* there exists g∈E such that
(23)ℓ(f)=∫Ωgfdμ(f∈Lω(Ω,X)).Proof.
SinceLω(Ω,X) is countably normed space by [15] dual of Lω(Ω,X) is the space ⋃1<q≤∞(Lp(Ω,X))*, where (1/p)+(1/q)=1. As X* has the Radon-Nikodym property with respect to μ by Theorem IV.1.1 [13] (Lp(Ω,X))*=Lq(Ω,X*), where (1/p)+(1/q)=1. Hence, the dual of Lω(Ω,X) is the space
(24)E=⋃1<q≤∞Lq(Ω,X*),
where (1/p)+(1/q)=1.
Letℓ be a continuous linear functional on Lω(Ω,X). Then by [15] there exists natural number p  (1≤p<∞) and C>0 such that
(25)|ℓ(f)|≤C∥f∥p.
It means ℓ∈(Lp(Ω,X))*. Using the general form of continuous linear functionals in Lp(Ω,X) (see [13], IV.1) we get that there exists g∈Lq(Ω,X*)⊂E such that
(26)ℓ(f)=∫Ωgfdμ.Theorem 8.
IfX is a reflexive Banach algebra then Lω(Ω,X) is reflexive.Proof.
LetX be a reflexive Banach algebra. Since every reflexive space has Radon-Nikodym property with respect to μ, by Theorem 7,
(27)Lω(Ω,X)**=(⋃1<q≤∞Lq(Ω,X*))*=⋂p≥1Lp(Ω,X**)=⋂p≥1Lp(Ω,X)=Lω(Ω,X).
This means that Lω(Ω,X) is semireflexive space. By Theorem 2  Lω(Ω,X) is (F)-space. Since by Corollary II.7.1 [16] every (F)-space is barreled we have that Lω(Ω,X) is barreled. Hence by IV.5. [16] Lω(Ω,X) is reflexive.



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