On the Generalized Weighted Lebesgue Spaces of Locally Compact Groups

Abstract

Let 𝐺 be a locally compact group with a fixed left Haar measure 𝜆 and Ω be a system of weights on 𝐺. In this paper, we deal with locally convex space 𝐿^𝑝(𝐺,Ω) equipped with the locally convex topology generated by the family of norms (‖.‖_𝑝,𝜔)_𝜔∈Ω. We study various algebraic and topological properties of the locally convex space 𝐿^𝑝(𝐺,Ω). In particular, we characterize its dual space and show that it is a semireflexive space. Finally, we give some conditions under which 𝐿^𝑝(𝐺,Ω) with the convolution multiplication is a topological algebra and then characterize its closed ideals and its spectrum.

1. Introduction

Throughout this paper, let 𝐺 denote a locally compact Hausdorff group with a fixed left Haar measure 𝜆. By a weight function on 𝐺, we mean an arbitrary strictly positive measurable function on 𝐺, and, by a system of weights on 𝐺, a set of weight functions Ω such that given 𝜔₁, 𝜔₂ in Ω and 𝑐>0, there is an 𝜈∈Ω such that 𝑐𝜔_𝑖(𝑥)≤𝜈(𝑥) (𝑖=1,2) for locally almost all 𝑥∈𝐺.

For a weight function 𝜔 and 1≤𝑝<∞, let 𝐿^𝑝(𝐺,𝜔) denote the space of all complex-valued measurable functions 𝑓 on 𝐺 such that 𝑓𝜔∈𝐿^𝑝(𝐺), the usual Lebesgue space on 𝐺 with respect to 𝜆; see [1] for more details. Then, 𝐿^𝑝(𝐺,𝜔) with the norm ‖⋅‖_𝑝,𝜔 defined by ‖𝑓‖_𝑝,𝜔∶=‖𝑓𝜔‖_𝑝 for all 𝑓∈𝐿^𝑝(𝐺,𝜔) is a Banach space. We also denote by 𝐿^∞(𝐺,1/𝜔) the space of all measurable complex-valued functions 𝑓 on 𝐺 such that 𝑓/𝜔∈𝐿^∞(𝐺), the space defined in [1]. Then, 𝐿^∞(𝐺,1/𝜔) with the norm ‖⋅‖_∞,𝜔 defined by ‖𝑓‖_∞,𝜔∶=‖𝑓/𝜔‖_∞ for all 𝑓∈𝐿^∞(𝐺,1/𝜔) is a Banach space. Furthermore, for 1≤𝑝<∞, the topological dual of 𝐿^𝑝(𝐺,𝜔) coincides with 𝐿^𝑞(𝐺,1/𝜔), where 𝑞 is the exponential conjugate to 𝑝 defined by 1/𝑝+1/𝑞=1. In fact, the mapping 𝑇 from 𝐿^𝑞(𝐺,1/𝜔) to 𝐿^𝑝(𝐺,𝜔) defined by 𝑇𝑓,𝑔=_𝐺𝑓(𝑥)𝑔(𝑥)𝑑𝜆(𝑥) (1.1) is an isometric isomorphism; see for example [2]. For measurable functions 𝑓 and 𝑔 on 𝐺, the convolution multiplication 𝑓∗𝑔(𝑥)=_𝐺𝑓𝑦𝑔𝑦⁻¹𝑥𝑑𝜆𝑦 (1.2) is defined at each point 𝑥∈𝐺 for which this makes sense. The algebraic and topological properties of weighted 𝐿^𝑝-spaces have been studied extensively; see for example [2–5].

Let 1≤𝑝<∞ and Ω be a system of weights on 𝐺, we set 𝐿^𝑝(𝐺,Ω)=_𝜔∈Ω𝐿^𝑝(𝐺,𝜔). (1.3)

In this paper, we equip the space 𝐿^𝑝(𝐺,Ω) with the natural locally convex topology generated by the family of norms ‖⋅‖_𝑝,𝜔, where 𝜔 runs through Ω. For a similar study in other contexts, see [6–8]. We investigate certain algebraic and topological properties of the locally convex space 𝐿^𝑝(𝐺,Ω). Our results generalize and improve some interesting results of [5] and partially answer a question raised in [3].

2. Preliminaries and Some Basic Results

Let 𝐺 be a locally compact Hausdorff group with a fixed left Haar measure 𝜆 and Ω be a system of weights on 𝐺. We equip 𝐿^𝑝(𝐺,Ω) with the locally convex topology generated by the family of norms (‖⋅‖_𝑝,𝜔)_𝜔∈Ω and denote this topology by 𝜏_Ω. So (𝐿^𝑝(𝐺,Ω),𝜏_Ω) has a basis of closed absolutely convex neighbourhoods at the origin of the form 𝑉_𝑝,𝜔=𝑓∈𝐿^𝑝(𝐺,Ω)∶𝑓_𝑝,𝜔≤1,(𝜔∈Ω). (2.1)

Note that the topology 𝜏_Ω is Hausdorff, because if 𝑓∈𝐿^𝑝(𝐺,Ω) and 𝑓≠0, we have 𝜆({𝑥∈𝐺∶𝑓(𝑥)≠0})>0. Put 𝐸={𝑥∈𝐺∶𝑓(𝑥)≠0} and fix 𝜔∈Ω. Then, 𝑓_𝑝,𝜔=_𝐺𝑓𝜔^𝑝𝑑𝜆^1/𝑝≥_𝐸𝑓𝜔^𝑝𝑑𝜆^1/𝑝>0, (2.2) and thus 𝜏_Ω is Hausdorff.

If Ω and Γ are two systems of weights on 𝐺 and for every 𝜔∈Ω, there is a 𝜈∈Γ such that 𝜔≤𝜈 (pointwise locally almost everywhere on 𝐺), then we write Ω≤Γ. In the case which Γ≤Ω and Ω≤Γ, we write Ω∼Γ.

Proposition 2.1. Let Ω and Γ be two systems of weights on 𝐺 and 𝑇∶𝐺→𝐺 be a measurable mapping such that Ω≤Γ∘𝑇∶={𝜈∘𝑇∶𝜈∈Γ}. If the Radon-Nikodym function ℎ=𝑑(𝜆∘𝑇⁻¹)/𝑑𝜆 belongs to 𝐿^∞(𝐺), then the mapping 𝑓↦𝑓∘𝑇 is a continuous linear map from (𝐿^𝑝(𝐺,Γ),𝜏_Γ) into (𝐿^𝑝(𝐺,Ω),𝜏_Ω).

Proof. Given 𝑓∈𝐿^𝑝(𝐺,Γ) and 𝜔∈Ω, choose 𝜈∈Γ such that 𝜔≤𝜈∘𝑇. Then we have 𝑓∘𝑇_𝑝,𝜔=_𝐺𝑓∘𝑇(𝑥)𝜔(𝑥)^𝑝𝑑𝜆(𝑥)^1/𝑝≤_𝐺𝑓∘𝑇(𝑥)(𝜈∘𝑇)(𝑥)^𝑝𝑑𝜆(𝑥)^1/𝑝=_𝐺𝑓(𝑥)𝜈(𝑥)^𝑝𝑑𝜆∘𝑇⁻¹(𝑥)^1/𝑝≤_𝐺𝑓(𝑥)𝜈(𝑥)^𝑝ℎ(𝑥)𝑑𝜆(𝑥)^1/𝑝≤‖ℎ‖_∞𝑓_𝑝,𝜈<∞. (2.3) Hence, 𝜔(𝑓∘𝑇)∈𝐿^𝑝(𝐺). Since 𝜔∈Ω was arbitrary, 𝑓∘𝑇∈𝐿^𝑝(𝐺,Ω). Continuity also follows from the above relations.

The space of all bounded Borel measurable functions on 𝐺 with compact support will be denoted by 𝐵_𝑐(𝐺). Let us remark that if 𝐵_𝑐(𝐺)⊆𝐿^𝑝(𝐺,Ω), then 𝐵_𝑐(𝐺) is norm dense in 𝐿^𝑝(𝐺,𝜔) for any weight 𝜔 on 𝐺; see for example [9].

Corollary 2.2. Let Ω and Γ be two systems of weights on 𝐺. Then,(i)If Ω≤Γ, then the induced topology 𝜏_Ω on 𝐿^𝑝(𝐺,Γ) is weaker than 𝜏_Γ.(ii)If 𝐵_𝑐(𝐺)⊆𝐿^𝑝(𝐺,Γ)⊆𝐿^𝑝(𝐺,Ω) and 𝜏_Ω⊆𝜏_Γ, then Ω≤Γ. In particular, Ω∼Γ if and only if 𝐿^𝑝(𝐺,Γ)=𝐿^𝑝(𝐺,Ω).

Proof. (i) is trivial. For (ii), we observe that for any 𝜔∈Ω, there is a 𝜈∈Γ such that 𝑉_𝑝,𝜈⊆𝑉_𝑝,𝜔∩𝐿^𝑝(𝐺,Γ). So the identity map 𝐼 from (𝐿^𝑝(𝐺,Ω),‖⋅‖_𝑝,𝜈) into (𝐿^𝑝(𝐺,𝜔),‖⋅‖_𝑝,𝜔) is continuous. Since 𝐿^𝑝(𝐺,Γ) is dense in (𝐿^𝑝(𝐺,𝜈),‖⋅‖_𝑝,𝜈), 𝐼 can be extended continuously to a continuous linear mapping on 𝐿^𝑝(𝐺,𝜈). The extension map is again the identity map. So 𝐿^𝑝(𝐺,𝜈)⊆𝐿^𝑝(𝐺,𝜔). Hence, there exists a constant 𝑐>0 such that 𝜔(𝑥)≤𝑐𝜈(𝑥) locally almost everywhere; see Lemma 2.1 in [10]. This proves that Ω≤Γ.

Let us recall the definition of the projective limit of a family of locally convex spaces. Let (Λ,≤) be a partially ordered set and {𝑋_𝛼∶𝛼∈Λ} be a family of locally convex spaces, and for 𝛼≤𝛽, 𝑓_𝛼,𝛽 be a linear map from 𝑋_𝛽 into 𝑋_𝛼. Suppose that 𝑓_𝛼𝛾=𝑓_𝛼𝛽∘𝑓_𝛽𝛾 for all 𝛼≤𝛽≤𝛾 and 𝑓_𝛼𝛼 be the identity map on 𝑋_𝛼 for all 𝛼∈Λ. Then, the projective limit of the family (𝑋_𝛼,𝑓_𝛼,𝛽) is defined as lim_𝛼𝑋_𝛼,𝑓_𝛼,𝛽=𝑥_𝛼∈_𝛼∈Λ𝑋_𝛼∶𝑥_𝛼=𝑓_𝛼,𝛽𝑥_𝛽,whenever𝛼≤𝛽; (2.4) for more details see for example [11].

Proposition 2.3. Let Ω be a system of weights on 𝐺. Then (𝐿^𝑝(𝐺,Ω),𝜏_Ω) is a complete space.

Proof. We note that for any two weights 𝜔,𝜈∈Ω with 𝜔≤𝜈, 𝐿^𝑝(𝐺,𝜈)⊆𝐿^𝑝(𝐺,𝜔). Let the mapping 𝐼_𝜔,𝜈∶𝐿^𝑝(𝐺,𝜈)→𝐿^𝑝(𝐺,𝜔) be the canonical injection. Then, it is clear that (𝐿^𝑝(𝐺,Ω),𝜏_Ω) is isomorphic to the projective limit system lim_𝜔(𝐿^𝑝(𝐺,𝜔),𝐼_𝜔,𝜈) of the Banach spaces (𝐿^𝑝(𝐺,𝜔),‖⋅‖_𝑝,𝜔), 𝜔∈Ω, and, hence, is complete; see Lemma 3.2.1 in [12].

Proposition 2.4. The locally convex space (𝐿^𝑝(𝐺,Ω),𝜏_Ω) is normable if and only if the topology 𝜏_Ω is generated by ‖⋅‖_𝑝,𝜔 for some 𝜔∈Ω.

Proof. If 𝐿^𝑝(𝐺,Ω) is normable, then it has a neighbourhood 𝑉 of zero that is norm bounded with respect to ‖⋅‖_𝑝,𝜔 for every 𝜔∈Ω. Hence, there is 𝜔^′∈Ω so that 𝑉_𝑝,𝜔′={𝑓∈𝐿^𝑝(𝐺,Ω)∶‖𝑓‖_𝑝,𝜔′≤1} is norm bounded in the space (𝐿^𝑝(𝐺,𝜈),‖⋅‖_𝑝,𝜈) for every 𝜈∈Ω. This implies that there is a positive constant 𝑐_𝜈 so that 𝑉_𝑝,𝜔′⊆𝑐_𝜈𝑉_𝑝,𝜈, and our claim is proved. The converse is clear.

3. The Dual and Bidual of 𝐿^𝑃(𝐺,Ω),1≤𝑝<∞

In this section we deal with the dual space of (𝐿^𝑝(𝐺,Ω),𝜏_Ω) and, among other things, characterize its equicontinuous subsets.

Theorem 3.1. If 1≤𝑝<∞ and 𝐵_𝑐(𝐺)⊆𝐿^𝑝(𝐺,Ω), then the dual space of (𝐿^𝑝(𝐺,Ω),𝜏_Ω) is Ω⋅𝐿^𝑞(𝐺)∶={𝜔𝑓∶𝜔∈Ω,𝑓∈𝐿^𝑞(𝐺)} with 1/𝑝+1/𝑞=1.

Proof. Let ℎ∈𝐿^𝑞(𝐺,1/𝜔). We define the linear functional 𝐹∶𝐿^𝑝(𝐺,Ω)→ℂ by 𝐹(𝑓)=∫_𝐺𝑓ℎ𝑑𝜆, then 𝐹∈(𝐿^𝑝(𝐺,Ω),𝜏_Ω)^∗.
Conversely, let 𝐹∈(𝐿^𝑝(𝐺,Ω),𝜏_Ω)^∗. First, we know that 𝐵_𝑐(𝐺)⊆𝐿^𝑝(𝐺,Ω)⊆𝐿^𝑝(𝐺,𝜔) for every 𝜔∈Ω. So there is a 𝜈∈Ω such that |𝐹(𝑓)|≤1 whenever 𝑓∈{ℎ∈𝐿^𝑝(𝐺,Ω)∶‖ℎ‖_𝑝,𝜈≤1}. As 𝐹 is bounded in the intersection of the unit ball of (𝐿^𝑝(𝐺,𝜈),‖⋅‖_𝑝,𝜈) with (𝐿^𝑝(𝐺,Ω),‖⋅‖_𝑝,𝜈), 𝐹 is continuous on 𝐿^𝑝(𝐺,Ω) with the topology induced by the norm ‖⋅‖_𝑝,𝜈. Since 𝐿^𝑝(𝐺,Ω) is dense in (𝐿^𝑝(𝐺,𝜈),‖⋅‖_𝑝,𝜈), 𝐹 can be extended continuously to a continuous linear form on 𝐿^𝑝(𝐺,𝜈) which we denote by 𝐹. Then, we have 𝐹∈(𝐿^𝑝(𝐺,𝜈),‖⋅‖_𝑝,𝜈)^∗, and hence there is a unique ℎ∈𝐿^𝑞(𝐺,1/𝜈) so that 𝐹𝑓=_𝐺𝑓ℎ𝑑𝜆𝑓∈𝐿^𝑝(𝐺,𝜈); (3.1) therefore, we obtain the following isomorphism: Φ∶_𝜔∈Ω𝐿^𝑞1𝐺,𝜔⟶𝐿^𝑝(𝐺,Ω),𝜏_Ω^∗, (3.2) defined by Φ(ℎ)=𝐹_ℎ, where 𝐹_ℎ(𝑓)=∫_𝐺𝑓ℎ𝑑𝜆 for all 𝑓∈𝐿^𝑝(𝐺,Ω).

Lemma 3.2. Let Ω be a system of weights on 𝐺. For every 𝜔∈Ω, define the mapping 𝑇_𝜔∶𝐿^𝑝(G,Ω)→𝐿^𝑝(𝐺) by 𝑇_𝜔(𝑓)=𝑓𝜔. Then, 𝑉^∘_𝑝,𝜔=𝑇^∗_𝜔(𝐵^∘) for 𝜔∈Ω, where 𝐵 is the closed unit ball of 𝐿^𝑝(𝐺) and 𝐵^∘ denotes its polar.

Proof. It is clear that 𝑇_𝜔 is a well-defined continuous linear map. Also, 𝑇_𝜔(𝐿^𝑝(𝐺,Ω)) is dense in (𝐿^𝑝(𝐺),‖⋅‖_𝑝). Therefore 𝑇^∗_𝜔 (the adjoint of 𝑇_𝜔) is weak^∗ continuous and one to one linear map from 𝐿^𝑞(𝐺) into Ω⋅𝐿^𝑞(𝐺), where 1/𝑝+1/𝑞=1. Now, since 𝐵^∘ is 𝜎(𝐿^𝑞(𝐺),𝐿^𝑝(𝐺))-compact by the Alaoglu theorem and so 𝑇^∗_𝜔(𝐵^∘) is 𝜎(Ω⋅𝐿^𝑞(𝐺),𝐿^𝑝(𝐺,Ω))-compact, while 𝑇^∗_𝜔(𝐵^∘) is obviously convex. So we find that 𝑉_𝑝,𝜔=𝑓∈𝐿^𝑝(𝐺,Ω)∶𝑇_𝜔𝑓∈𝐵=𝑇_𝜔⁻¹(𝐵)=𝑓∈𝐿^𝑝(𝐺,Ω)∶𝑇_𝜔𝑓∈𝐵^∘∘=𝑓∈𝐿^𝑝(𝐺,Ω)∶𝑇^∗_𝜔𝑔𝑓≤1,forevery𝑔∈𝐵^∘=𝑇^∗_𝜔𝐵^∘∘. (3.3) Form which it follows that 𝑉^∘_𝑝,𝜔=𝑇^∗_𝜔𝐵^∘∘∘=𝑇^∗_𝜔𝐵^∘. (3.4)

We have the following characterization of the equicontinuous subsets of Ω⋅𝐿^𝑞(𝐺).

Theorem 3.3. Let 1≤𝑝<∞ and 𝑀 be a subset of (𝐿^𝑝(𝐺,Ω),𝜏_Ω)^∗=Ω⋅𝐿^𝑞(𝐺). The following are equivalent.(a)𝑀 is 𝜏_Ω-equicontinuous.(b)There are 𝜔∈Ω and an equicontinuous subset 𝑀^′ of (𝐿^𝑝(𝐺),‖⋅‖_𝑝)^∗=𝐿^𝑞(𝐺) so that 𝑀⊆𝜔⋅𝑀^′.(c)There are 𝜔∈Ω and 𝛼>0 such that sup{‖𝑓/𝜔‖_𝑞∶𝑓∈𝑀}≤𝛼<∞ whenever 1/𝑝+1/q=1.

Proof. (𝑎⇒𝑏) By (a), there is 𝜔∈Ω so that 𝑀⊆𝑉^∘_𝑝,𝜔, where 𝑉_𝑝,𝜔={𝑓∈𝐿^𝑝(𝐺,Ω)∶‖𝑓‖_𝑝,𝜔≤1}. According to Lemma 3.2, we have 𝑉^∘_𝑝,𝜔=𝑇^∗_𝜔(𝐵^∘), where 𝐵 is the closed unit ball of 𝐿^𝑝(𝐺). Hence 𝑀⊆𝜔𝐵^∘.
(𝑏⇒𝑐) There is 𝛼>0 so that 𝑀^′⊆𝛼𝐵^∘ by (b). So 𝑀⊆𝛼𝜔𝐵^∘, and sup_𝑓∈𝑀‖𝑓/𝜔‖_𝑞≤𝛼.
(𝑐⇒𝑎) If 𝑝=1, it is clear that 𝑀⊆𝑓∈𝐿^𝑝(𝐺,Ω)∶_𝐺𝑓(𝑥𝜔(𝑥)(𝑥)≤1)𝑑𝜆𝛼^∘, (3.5) and if 1<𝑝<∞, by Hölder’s inequality, for ℎ∈𝑀 and 𝑓∈𝑊=𝑓∈𝐿^𝑝(𝐺,Ω)∶𝑓_𝑝,𝜔≤1𝛼, (3.6) we have ℎ𝑓𝑑𝜆≤_𝐺𝐺ℎ𝜔ℎ𝑓𝜔𝑑𝜆≤𝜔_𝑞𝑓𝜔_𝑝≤1. (3.7) Hence, 𝑀⊆𝑊^∘, and this guarantees that 𝑀 is 𝜏_Ω-equicontinuous in both cases.

Proposition 3.4. Let Ω be a system of weights on 𝐺. Then, the set of extreme points of 𝑉^∘_𝑝,𝜔 is the set {𝜔𝑓∶𝑓∈𝐿^𝑞(𝐺),‖𝑓‖_𝑞=1} for 1<𝑝<∞, and {𝑓∈𝐿^∞(𝐺)∶|𝑓|=1𝑙.𝑎.𝑒.} for 𝑝=∞.

Proof. Fix 𝜔∈Ω and let 𝑇_𝜔∶𝐿^𝑝(𝐺,Ω)→𝐿^𝑝(𝐺) be the map defined in Lemma 3.2. From Lemma 3.2, it follows that for any extreme point ℎ of 𝑉^∘_𝑝,𝜔, there is an extreme point 𝑓 of 𝐵^∘ so that ℎ=𝑇^∗_𝜔(𝑓)=𝑓𝜔.
Conversely, let 𝜔∈Ω be arbitrary and let ℎ=𝜔𝑓, where 𝑓 is an extreme point of 𝐵^∘. Clearly, ℎ∈𝑉^∘_𝑝,𝜔, and if ℎ=𝑐𝑔+(1−𝑐)𝑘 for some 𝑔,𝑘∈𝑉^∘_𝑝,𝜔 and 0<𝑐<1, then there are 𝑚,𝑛∈𝐵^∘ such that 𝑇^∗_𝜔(𝑚)=𝑔 and 𝑇^∗_𝜔(𝑛)=𝑘. Thus, 𝑇^∗_𝜔(𝑓)=ℎ=𝑇^∗_𝜔(𝑐𝑚+(1−𝑐)𝑛) and since 𝑇^∗_𝜔 is one to one, 𝑓=𝑐𝑚+(1−𝑐)𝑛. However 𝑓 is an extreme point of 𝐵^∘, which implies that 𝑓=𝑚=𝑛, and hence ℎ=𝑔=𝑘, that is, ℎ is an extreme point of 𝑉^∘_𝑝,𝜔. Now the rest of the proof is easy to complete; see for example Section 2.14 in [13].

Let us recall that a locally convex space (𝐸,𝜏) is called semireflexive if (𝐸,𝜏)^∗∗=𝐸.

Theorem 3.5. Let Ω be a system of weights on 𝐺. Then (𝐿^𝑝(𝐺,Ω),𝜏_Ω) is semireflexive.

Proof. If 𝐹∈(𝐿^𝑝(𝐺,Ω),𝜏_Ω)^∗∗, then the restriction of 𝐹 to 𝐿^𝑞(𝐺,1/𝜔), for every 𝜔∈Ω, belongs to 𝐿^𝑞(𝐺,1/𝜔)^∗, where 𝐿^𝑞(𝐺,1/𝜔) was considered with the induced strong topology on (𝐿^𝑝(𝐺,Ω),𝜏_Ω)^∗. Now if {ℎ_𝛼}_𝛼∈𝐼⊆𝐿^𝑞(𝐺,1/𝜔) and ℎ_𝛼→ℎ for some ℎ∈𝐿^𝑞(𝐺,1/𝜔) in the norm ‖⋅‖_𝑞,1/𝜔, then for every weakly bounded set 𝐴 in 𝐿^𝑝(𝐺,Ω), _𝐺𝑓ℎ_𝛼−ℎ𝑑𝜆⟶0uniformlyon𝐴. (3.8)
This means that ℎ_𝛼→ℎ in the strong topology of (𝐿^𝑝(𝐺,Ω),𝜏_Ω)^∗. Hence, for every 𝜔∈Ω, there is a unique 𝑓_𝜔∈𝐿^𝑝(𝐺,𝜔) so that 𝐹(ℎ)=_𝐺𝑓_𝜔ℎ𝑑𝜆on𝐿^𝑞1𝐺,𝜔. (3.9)
Now note that if 𝜔,𝜈∈Ω with 𝜔≤𝜈, then 𝐿^𝑝(𝐺,𝜈)⊆𝐿^𝑝(𝐺,𝜔) and 𝐿^𝑞(𝐺,1/𝜔)⊆𝐿^𝑞(𝐺,1/𝜈). Therefore for every ℎ∈𝐿^𝑞(𝐺,1/𝜔), _𝐺𝑓_𝜔ℎ𝑑𝜆=_𝐺𝑓_𝜈ℎ𝑑𝜆, (3.10) and hence 𝑓_𝜔=𝑓_𝜈 almost everywhere. This implies that 𝐹∈lim_𝜔𝐿^𝑝(𝐺,𝜔),𝐼_𝜔,𝜈=𝐿^𝑝(𝐺,Ω). (3.11)
Conversely, if 𝑓∈𝐿^𝑝(𝐺,Ω), then it is obvious that the linear form 𝐹(ℎ)=_𝐺𝑓ℎ𝑑𝜆ℎ∈𝐿^𝑝(𝐺,Ω),𝜏_Ω^∗ (3.12) is continuous with respect to the strong topology on (𝐿^𝑝(𝐺,Ω),𝜏_Ω)^∗. So the canonical imbedding 𝐽∶𝐿^𝑝(𝐺,Ω)→(𝐿^𝑝(𝐺,Ω),𝜏_Ω)^∗∗ is onto. Hence 𝐿^𝑝(𝐺,Ω) is semireflexive.

4. 𝐿^𝑃(𝐺,Ω) As a Topological Algebra

In this section, we study conditions on a system of weights Ω for that 𝐿^𝑝(𝐺,Ω) with the convolution multiplication to be a topological algebra. We commence with some definitions.

If 𝑓 is a function on 𝐺, the left translate of 𝑓 by 𝑥∈𝐺 is the function given by 𝐿_𝑥𝑓(𝑦)=𝑓(𝑥⁻¹𝑦). A subset ℱ of functions on 𝐺 is called left translation invariant if 𝐿_𝑥𝑓∈ℱ for all 𝑓∈ℱ and 𝑥∈𝐺.

A weight function 𝜔 on a locally compact group 𝐺 is called left moderate if ℓ(𝑠)∶=esssup_𝑡𝜔(𝑠𝑡)𝜔(𝑡)<∞, (4.1) for all 𝑠∈𝐺. It is easy to see that ℓ(𝑠)>0, ℓ(𝑠𝑡)≤ℓ(𝑠)ℓ(𝑡); see [4] or [9]. Let us remark that any submultiplicative and any locally integrable left moderate measurable function is bounded and bounded away from zero on any compact subset of 𝐺; see Theorem 2.7 in [10]. In particular, ℓ is bounded on compact sets. The condition that 𝜔 is left moderate is equivalent to that the space 𝐿^𝑝(𝐺,𝜔) (for 1≤𝑝≤∞) being translation invariant; see for more details [4]. Observe that for 𝑓∈𝐿^𝑝(𝐺,𝜔) and 𝑥∈𝐺,𝐿_𝑥𝑓_𝑝,𝜔=_𝐺𝑓𝑥⁻¹𝑡𝜔(𝑡)^𝑝𝑑𝜆(𝑡)^1/𝑝=_𝐺𝑓(𝑡)𝜔(𝑥𝑡)^𝑝𝑑𝜆(𝑡)^1/𝑝≤_𝐺𝑓(𝑡)ℓ(𝑥)𝜔(𝑡)^𝑝𝑑𝜆(𝑡)^1/𝑝=ℓ(𝑥)𝑓_𝑝,𝜔. (4.2)

Lemma 4.1. Let Ω be a system of weights on 𝐺. Then 𝐿^𝑝(𝐺,Ω) is left translation invariant if and only if every element of Ω is left moderate.

Proof. The “if” part is clear by the remarks above. For the converse, we need only to note that 𝐿^𝑝(𝐺,Ω) is dense in (𝐿^𝑝(𝐺,𝜔),‖⋅‖_𝑝,𝜔) for 𝜔∈Ω.

Theorem 4.2. Let Ω be a system of locally integrable left moderate weights on 𝐺 and 𝑓∈𝐿^𝑝(𝐺,Ω). Then, the map 𝑥↦𝐿_𝑥𝑓 from 𝐺 into (𝐿^𝑝(𝐺,Ω),𝜏_Ω) is continuous.

Proof. Assume first that 𝑓∈𝐵_𝑐(𝐺) with 𝐾=supp(𝑓). Let 𝑥∈𝐺, 𝜔∈Ω, and (𝑥_𝛼) be a net in 𝐺 convergent to 𝑥. Choose a compact neighbourhood 𝑈 of 𝑥, then supp(𝐿_𝑥𝑓)⊆𝑈𝐾 whenever 𝑥∈𝑈. Let 𝑘=sup{𝜔(𝑠)∶𝑠∈𝑈𝐾}<∞. (4.3) Choose 𝛼₀ such that 𝑥_𝛼∈𝑈 for all 𝛼≤𝛼₀ and ‖𝐿_𝑥𝛼𝑓−𝐿_𝑥𝑓‖_𝑝≤𝜖/𝑘. Then 𝐿_𝑥𝛼𝑓−𝐿_𝑥𝑓_𝑝,𝜔=_𝑈𝐹𝑓𝑥_𝛼⁻¹𝑡−𝑓𝑥⁻¹𝑡𝜔(𝑡)^𝑝𝑑𝜆(𝑡)^1/𝑝≤𝑘_𝑈𝐹𝑓𝑥_𝛼⁻¹𝑡−𝑓𝑥⁻¹𝑡^𝑝𝑑𝜆(𝑡)^1/𝑝=𝑘𝐿_𝑥𝛼𝑓−𝐿_𝑥𝑓_𝑝≤𝜖, (4.4) for all 𝛼≥𝛼₀.
Finally, let 𝑓 be an arbitrary element of 𝐿^𝑝(𝐺,Ω) and 𝜖>0. Let 𝑀 be an upper bound for the function ℓ on the compact neighbourhood 𝑈 of 𝑥; recall that ℓ is submultiplicative. For every 𝜔∈Ω, there exists 𝑔_𝜔∈𝐵_𝑐(𝐺) such that ‖𝑓−𝑔_𝜔‖_𝑝,𝜔≤𝜖/3𝑀. By the first part, we can choose 𝛼₀ such that 𝐿_𝑥𝛼𝑔_𝜔−𝐿_𝑥𝑔_{𝜔𝑝,𝜔}≤𝜖3,𝑥_𝛼∈𝑈, (4.5) for all 𝛼≥𝛼₀. One can conclude that 𝐿_𝑥𝛼𝑓−𝐿_𝑥𝑓_𝑝,𝜔≤𝐿_𝑥𝛼𝑓−𝐿_𝑥𝛼𝑔_{𝜔𝑝,𝜔}+𝐿_𝑥𝛼𝑔_𝜔−𝐿_𝑥𝑔_{𝜔𝑝,𝜔}+𝐿_𝑥𝑔_𝜔−𝐿_𝑥𝑓_𝑝,𝜔≤ℓ𝑥_𝛼(𝑥)𝜖≤𝜖+𝜖/3+ℓ𝑀𝜖+𝜖3𝑀3+𝑀𝜖3𝑀=𝜖, (4.6) for all 𝛼≥𝛼₀. This finishes the proof.

We now focus on some systems of weights for that 𝐿^𝑝(𝐺,Ω) to be an algebra under usual convolution 𝑓∗𝑔(𝑡)=_𝐺𝑓(𝑠)𝑔𝑠⁻¹𝑡𝑑𝜆(𝑠)𝑓,𝑔∈𝐿^𝑝(𝐺,Ω), (4.7) whenever this integral makes sense. For 𝑝=1, it is well known that 𝐿¹(𝐺,𝜔) is a convolution algebra if and only if 𝜔 is weakly submultiplicative; that is, for all 𝑥,𝑦∈𝐺, 𝜔(𝑠𝑡)≤𝑐𝜔(𝑠)𝜔(𝑡), (4.8) for some 𝑐>0.

For any two weight functions 𝜔 and 𝜈 on 𝐺, we set Φ_[𝜔,𝜈](𝑥)=_𝐺𝜔(𝑥)𝜈𝑦𝜈𝑦⁻¹𝑥^𝑞𝑑𝜆𝑦. (4.9) In the case where 𝜔=𝜈, we simply write Φ_𝜔=Φ_[𝜔,𝜔].

The following lemma is similar to Lemma 2.2 in [9].

Lemma 4.3. Let 1<𝑝<∞ and Ω be a system of weights on 𝐺. If 𝐿^𝑝(𝐺,Ω) is a convolution algebra, then 𝜔^𝑝 is locally integrable for each 𝜔∈Ω.

The next result gives a sufficient condition for that 𝐿^𝑝(𝐺,Ω) to be a convolution algebra.

Theorem 4.4. Let 1<𝑝<∞ and Ω be a system of weights on 𝐺. If for every 𝜔∈Ω, there is a 𝜈∈Ω such that Φ_[𝜔,𝜈]∈𝐿^∞(𝐺), where 𝑞 is the conjugate exponent to 𝑝, then the space (𝐿^𝑝(𝐺,Ω),𝜏_Ω) is a complete locally convex algebra with continuous multiplication.

Proof. We must show that 𝑓∗𝑔_𝑝,𝜔≤𝑓_𝑝,𝜈𝑔_𝑝,𝜈 (4.10) for all 𝑓,𝑔∈𝐿^𝑝(𝐺,Ω). By Lemma 4.3, 𝐵_𝑐(𝐺) is dense in (𝐿^𝑝(𝐺,Ω),‖⋅‖_𝑝,𝜔), thus for any 𝜔∈Ω, it suffices to show that 𝑓∗𝑔_𝑝,𝜔≤𝑓_𝑝,𝜈𝑔_𝑝,𝜈, (4.11) for all 𝑓,𝑔∈𝐵_𝑐(𝐺). For this, let 𝑓,𝑔∈𝐵_𝑐(𝐺). Writing 𝑓∗𝑔(𝑥)=_𝐺𝑓𝑦𝑔𝑦⁻¹𝑥𝜈𝑦𝜈𝑦⁻¹𝑥𝜈𝑦𝜈𝑦⁻¹𝑥𝑑𝜆𝑦, (4.12) and using Hölder's inequality, we obtain (𝑥≤𝑓∗𝑔)_𝐺(|𝑓(𝑦)|𝜈(𝑦))^𝑝(|𝑔(𝑦⁻¹𝑥)|𝜈(𝑦⁻¹𝑥))^𝑝𝑑𝜆(𝑦)^1/𝑝_𝐺1𝜈𝑦𝜈𝑦⁻¹𝑥^𝑞𝑑𝜆(𝑦)^1/𝑞. (4.13) This shows that 𝑓∗𝑔(_𝐺𝑥)𝜔(𝑥)^𝑝(𝑥≤𝑑𝜆)_𝐺𝐺𝑓𝑦𝜈𝑦^𝑝𝑔𝑦⁻¹𝑥𝜈𝑦⁻¹𝑥^𝑝𝑑𝜆𝑦Φ_[𝜔,𝜈](𝑥)^𝑝/𝑞(𝑥)≤𝑓𝑑𝜆_{𝑝𝑝,𝜈}𝑔^𝑝_𝑝,𝜈Φ_{[𝜔,𝜈]∞}^𝑝/𝑞. (4.14) Whence 𝑓∗𝑔_𝑝,𝜔≤Φ_{[𝜔,𝜈]∞}^1/𝑞𝑓_𝑝,𝜈𝑔_𝑝,𝜈. (4.15) This completes the proof.

The following corollary is a direct consequence of Theorem 4.4.

Corollary 4.5. Let 1<𝑝<∞ and Ω be a system of weights on 𝐺 such that for every 𝜔∈Ω, Φ_𝜔∈𝐿^∞(𝐺). Then 𝐿^𝑝(𝐺,Ω) is a complete locally multiplicative convex algebra.

The next result provides us with a class of weights 𝜔 on the additive group ℝ^𝑛 for which the usual weighted Lebesgue space 𝐿^𝑝(ℝ^𝑛,𝜔) becomes a Banach algebra.

Proposition 4.6. Let 1<𝑝<∞ and 𝑛 be a natural number. Let 𝜛∶ℝ^𝑛→(0,+∞) be a function such that
(i)If ‖𝑥‖≤‖𝑦‖, then 𝜛(𝑥)≤𝜛(𝑦).(ii)𝜛⁻¹∈𝐿¹(ℝ^𝑛).(iii)There exists a positive number 𝑀 such that 𝜛(2𝑥)≤𝑀𝜛(𝑥) for all 𝑥∈ℝ^𝑛.
Then 𝐿^𝑝(ℝ^𝑛,√^𝑞𝜛) is a Banach algebra, where 𝑞 is the conjugate exponent to 𝑝.

Proof. For any 𝑥∈ℝ^𝑛, let 𝐴_𝑥={𝑦∈ℝ^𝑛∶2‖𝑦‖≥‖𝑥‖} and observe that 𝜛𝑦𝑥≥𝜛2,if𝑦∈𝐴_𝑥,𝜛𝑥𝑥−𝑦≥𝜛2,if𝑦∈ℝ^𝑛⧵𝐴_𝑥. (4.16)
Hence, for 𝑥∈ℝ^𝑛, Φ^𝑞_√𝜛(𝑥)=_ℝ𝑛𝜛(𝑥)𝜛𝑦𝜛=𝑥−𝑦𝑑𝑦_{ℝ𝑛⧵𝐴𝑥}𝜛(𝑥)𝜛𝑦𝜛𝑥−𝑦𝑑𝑦+_𝐴𝑥𝜛(𝑥)𝜛𝑦𝜛≤𝜛(𝑥)𝑥−𝑦𝑑𝑦𝜛(𝑥/2)_ℝ^𝑛_⧵𝐴𝑥1𝜛𝑦𝑑𝑦+_𝐴𝑥1𝜛‖𝜛‖𝑥−𝑦𝑑𝑦≤2𝑀₁. (4.17)
Thus, Φ^𝑞_√𝜛∈𝐿^∞(ℝ^𝑛), and now the result follows from Corollary 4.5.

Example 4.7. Let 1≤p<∞, 𝑞 be the conjugate exponent to 𝑝, and 𝑛∈ℕ. Set 𝜔(𝑥)=𝑎+‖𝑥‖^{𝑟𝑠/𝑞}𝑏^{(1/𝑞)ln(𝑐+‖𝑥‖𝑡)}𝑥∈ℝ^𝑛, (4.18) where 𝑠𝑟>𝑛,𝑏>1 and 𝑎,𝑐,𝑡>0. Then 𝐿^𝑝(ℝ^𝑛,𝜔) is a Banach algebra.

We are going to prove the converse of Theorem 4.4. For this, we fix some notation. If 𝑓,𝑔 be two complex-valued functions on 𝐺, then 𝑓⊗𝑔 denotes the function on 𝐺×𝐺 given by 𝑓⊗𝑔(𝑥,𝑦)=𝑓(𝑥)𝑔(𝑦) for all 𝑥,𝑦∈𝐺. Also for any two sets ℱ and 𝒦 of functions on 𝐺 we set ℱ⊗𝒦={𝑓⊗𝑔∶𝑓∈ℱ,𝑔∈𝒦}. For a locally compact group 𝐺, note that the cartesian product 𝐺×𝐺 is a locally compact group by defining the product (𝑥,𝑦)(𝑠,𝑡)=(𝑥𝑠,𝑦𝑡) for all 𝑥,𝑦,𝑠,𝑡∈𝐺.

We need the following easy lemma in the sequel.

Lemma 4.8. Let 1<𝑝<∞ and Ω be a system of weights on 𝐺 such that 𝐵_𝑐(𝐺)⊆𝐿^𝑝(𝐺,Ω). Then 𝐵_𝑐(𝐺)⊗𝐵_𝑐(𝐺) is dense in (𝐿^𝑝(𝐺×𝐺,𝜔⊗𝜔),‖⋅‖_{𝑝,𝜔⊗𝜔}).

Proof. Since 𝐵_𝑐(𝐺) is norm dense in 𝐿^𝑝(𝐺,𝜔), then 𝐵_𝑐(𝐺)⊗𝐵_𝑐(𝐺) is projective tensor norm dense in 𝐿^𝑝(𝐺,𝜔)⊗_𝜋𝐿^𝑝(𝐺,𝜔), where ⊗ is the projective tensor product. Hence 𝐿^𝑝(𝐺,𝜔)⊗𝐿^𝑝(𝐺,𝜔) is 𝜋-dense in 𝐿^𝑝(𝐺,𝜔)⊗_𝜋𝐿^𝑝(𝐺,𝜔). On the other hand, it is known that 𝐿^𝑝(𝐺,𝜔)⊗_𝜋𝐿^𝑝(𝐺,𝜔) is isometric with (𝐿^𝑝(𝐺×𝐺,𝜔⊗𝜔),‖⋅‖_{𝑝,𝜔⊗𝜔}). In fact, the linear map 𝜚∶𝐿^𝑝(𝐺,𝜔)⊗_𝜋𝐿^𝑝(𝐺,𝜔)⟶𝐿^𝑝(𝐺×𝐺,𝜔⊗𝜔),𝜚𝑓⊗𝑔𝑥,𝑦=𝑓(𝑥)𝑔𝑦 (4.19) for all 𝑓,𝑔∈𝐿^𝑝(𝐺,𝜔) and 𝑥,𝑦∈𝐺, can be extended to a surjective isometry; for more details, see for example [14]. Now we conclude that 𝐵_𝑐(𝐺)⊗𝐵_𝑐(𝐺) is ‖⋅‖_{𝑝,𝜔⊗𝜔}-dense in 𝐿^𝑝(𝐺×𝐺,𝜔⊗𝜔).

The next theorem is our main result in this section.

Theorem 4.9. Let 1<𝑝<∞, 𝐺 be 𝜎-compact, and Ω be a system of weights on 𝐺. If the space (𝐿^𝑝(𝐺,Ω),𝜏_Ω) is an algebra with continuous multiplication, then for every 𝜔∈Ω there exists a 𝜈∈Ω such that Φ_[𝜔,𝜈]∈𝐿^∞(𝐺).

Proof. Choose an arbitrary 𝜔∈Ω. Then, by assumption, there exists some 𝜈∈Ω such that for every 𝑓,𝑔∈𝐿^𝑝(𝐺,Ω), ‖𝑓∗𝑔‖_𝑝,𝜔≤‖𝑓‖_𝑝,𝜈‖𝑔‖_𝑝,𝜈. Now for every ℎ∈𝐿^𝑞(𝐺,1/𝜔), 𝐹𝑓=_𝐺𝑓(𝑥)ℎ(𝑥)𝑑𝜆(𝑥)𝑓∈𝐵_𝑐(𝐺) (4.20) defines a continuous linear functional on 𝐵_𝑐(𝐺) with the norm ‖𝐹‖=‖ℎ‖_𝑞,1/𝜔. Also for every 𝑓,𝑔∈𝐵_𝑐(𝐺), 𝑓∗𝑔∈𝐵_𝑐(𝐺), and we have 𝐹𝑓∗𝑔=_𝐺𝑓∗𝑔(𝑥)ℎ(𝑥)𝑑𝜆(𝑥)=_𝐺𝐺𝑓𝑦𝑔𝑦⁻¹𝑥𝑦ℎ(𝑥)(𝑥)=𝑑𝜆𝑑𝜆_𝐺𝐺𝑓𝑦𝑔(𝑥)ℎ𝑦𝑥𝑑𝜆(𝑥)𝑑𝜆𝑦<∞. (4.21)
Set 𝐹(𝑓⊗𝑔)=𝐹(𝑓∗𝑔)=∫_𝐺×𝐺𝑓(𝑦)𝑔(𝑥)ℎ(𝑦𝑥)𝑑𝜆×𝜆(𝑥,𝑦) for 𝑓,𝑔∈𝐵_𝑐(𝐺). By Lemma 4.8, 𝐹 can be extended to a ‖⋅‖_{𝑝.𝜈×𝜈}-continuous functional on 𝐿^𝑝(𝐺×𝐺,𝜈⊗𝜈). Since 𝐺 is 𝜎-compact, by Exercise 15.14 in [1], it follows that the function (𝑥,𝑦)↦ℎ(𝑦𝑥) belongs to 𝐿^𝑞(𝐺×𝐺,1/(𝜈⊗𝜈)). But _𝐺𝐺ℎ𝑦𝑥𝜈(𝑥)𝜈𝑦^𝑞𝑑𝜆(𝑥)𝑑𝜆𝑦=_𝐺𝐺ℎ(𝑥)𝜈𝑦𝜈𝑦⁻¹𝑥^𝑞(𝑥)𝑦=𝑑𝜆𝑑𝜆_𝐺ℎ(𝑥)𝜔(𝑥)^𝑞Φ_[𝜔,𝜈](𝑥)𝑑𝜆(𝑥)<∞. (4.22) Since (ℎ/𝜔)^𝑞∈𝐿¹(𝐺) is arbitrary, we conclude that Φ_[𝜔,𝜈]∈𝐿^∞(𝐺); see Section 14 in [15] or Theorem  20.15 in [1].

As an immediate consequence of Theorem 4.9, we obtain the following corollary that partially answers a question raised in [3].

Corollary 4.10. Let 𝜔 be a weight on 𝜎-compact group 𝐺 and 1<𝑝<∞. Then 𝐿^𝑝(𝐺,𝜔) is a convolution algebra if and only if Φ_𝜔∈𝐿^∞(𝐺).

5. Ideals and the Spectrum of the Algebra 𝐿^𝑃(𝐺,Ω)

We commence this section with the following proposition.

Proposition 5.1. Let 1≤𝑝<∞ and let 𝐿^𝑝(𝐺,Ω) be a translation invariant algebra. Then(i)(𝐿^𝑝(𝐺,Ω),𝜏_Ω) has an approximate identity.(ii)(𝐿^𝑝(𝐺,Ω),𝜏_Ω) has a bounded approximate identity or an identity if and only if 𝐺 is discrete.

Proof. (i) Let 𝑈 be a fixed relatively compact neighbourhood of the identity element 𝑒, and let 𝒰 be the family of all neighbourhoods of 𝑒 contained in 𝑈 directed by reverse inclusion. Set 𝑒_𝑉∶=𝜒_𝑉/𝜆(𝑉), and note that since elements of Ω are locally integrable, 𝑒_𝑉∈𝐿^𝑝(𝐺,Ω). Given 𝜖>0 and 𝜔∈Ω, then by Theorem 4.2, there exists a neighbourhood 𝑊 of the identity such that ‖𝑓−𝐿_𝑡𝑓‖_𝑝,𝜔<𝜖 for 𝑡∈𝑊. Now, for 𝑉∈𝒰 with 𝑉⊆𝑊, and 𝑔∈𝐿^𝑞(𝐺,1/𝜔), we have 𝑒_𝑉∗𝑓−𝑓,𝑔=_𝐺𝑒_𝑉∗𝑓−𝑓(𝑥)𝑔(𝑥)𝑑𝜆(𝑥)≤_𝐺𝑉𝑓𝑡⁻¹𝑥−𝑓(𝑥)𝜆(𝑉)(𝑡𝑔(𝑥(𝑥)≤1𝑑𝜆))𝑑𝜆𝜆(𝑉)_V𝐿_𝑡𝑓−𝑓,𝑔𝑑𝜆(𝑡)≤sup_𝑡∈𝑉𝐿_𝑡𝑓−𝑓_𝑝,𝜔𝑔_𝑞,1/𝜔<𝜖𝑔_𝑞,1/𝜔. (5.1) Hence, ‖𝑒_𝑉∗𝑓−𝑓‖_𝑝,𝜔≤𝜖 for all neighborhoods 𝑉⊆𝑊, from which it follows that 𝑒_𝑉∗𝑓→𝑓 in 𝜏_Ω-topology.
(ii) Let (𝑒_𝛼)_𝛼 be a bounded left approximate identity for 𝐿^𝑝(𝐺,Ω). Fix an 𝜔∈Ω, then ‖𝑒_𝛼‖_𝑝,𝜔≤𝑀 for some positive number 𝑀. Let 𝑓∈𝐿^𝑝(𝐺,𝜔). Since 𝐿^𝑝(𝐺,Ω) is dense in 𝐿^𝑝(𝐺,𝜔) with the norm ‖⋅‖_𝑝,𝜔, then given 𝜖>0, there exists 𝑔∈𝐿^𝑝(𝐺,Ω) such that ‖𝑓−𝑔‖_𝑝,𝜔≤𝜖/3(𝑀+1). Choose 𝛼₀ such that ‖e_𝛼∗𝑔−𝑔‖_𝑝,𝜔≤𝜖/3 for all 𝛼≥𝛼₀. Then it follows that 𝑒_𝛼∗𝑓−𝑓_𝑝,𝜔≤𝑒_𝛼∗𝑓−𝑒_𝛼∗𝑔_𝑝,𝜔+𝑒_𝛼∗𝑔−𝑔_𝑝,𝜔+𝑓−𝑔_𝑝,𝜔𝜖≤𝑀3()+𝜖𝑀+13+𝜖3(𝑀+1)<𝜖, (5.2) for all 𝛼≥𝛼₀. This means that (𝐿^𝑝(𝐺,𝜔),‖⋅‖_𝑝,𝜔 has a bounded left approximate identity. But according to Theorem 4.2 in [9], this is equivalent to that 𝐺 is discrete.

The next theorem shows that closed ideals of the algebra (𝐿^𝑝(𝐺,Ω),𝜏_Ω) are exactly translation invariant subspaces.

Theorem 5.2. Let 1≤𝑝<∞ and 𝐿^𝑝(𝐺,Ω) be a translation invariant algebra. Then a closed linear subspace of 𝐿^𝑝(𝐺,Ω) is an ideal in 𝐿^𝑝(𝐺,Ω) if and only if it is two-sided translation invariant.

Proof. Suppose that 𝐼 is a 𝜏_Ω-closed two-sided translation invariant subspace of 𝐿^𝑝(𝐺,Ω). We have to show that 𝑔∗𝑓∈𝐼 and 𝑓∗𝑔∈𝐼 for all 𝑓∈𝐼 and 𝑔∈𝐿^𝑝(𝐺,Ω). Let ℎ∈𝐿^𝑞(𝐺,1/𝜔), for some 𝜔∈𝐺, such that ∫_𝐺𝑓(𝑥)ℎ(𝑥)𝑑𝜆(𝑥)=0 for all 𝑓∈𝐼. Then, for 𝑓∈𝐼 and any 𝑔∈𝐿^𝑝(𝐺,Ω), 𝑔∗𝑓(_𝐺𝑥)ℎ(𝑥)𝑑𝜆(𝑥)=_𝐺ℎ(𝑥)_𝐺𝑔𝑦𝑓𝑦⁻¹𝑥𝑦(𝑥)=𝑑𝜆𝑑𝜆_𝐺𝑔𝑦_𝐺𝐿_𝑦𝑓(𝑥)ℎ(𝑥)𝑑𝜆(𝑥)𝑑𝜆𝑦=0. (5.3) Since (𝐿^𝑝(𝐺,Ω),𝜏_Ω)^∗=Ω⋅𝐿^𝑞(𝐺), the Hahn-Banach theorem implies that 𝑔∗𝑓∈𝐼 for all 𝑓∈𝐼 and 𝑔∈𝐿^𝑝(𝐺,Ω). Thus 𝐼 is a left ideal, and using the right translation invariance of 𝐼, it is readily seen, in the same way, that 𝐼 is also a right ideal.
Conversely, let 𝐼 be a closed ideal of (𝐿^𝑝(𝐺,Ω),𝜏_Ω), and 𝑥∈𝐺. Let (𝑒_𝛼) be an approximate identity for 𝐿^𝑝(𝐺,Ω). Then for each 𝑓∈𝐿^𝑝(𝐺,Ω), we have 𝐿_𝑥𝑒_𝛼∗𝑓−𝐿_𝑥𝑓_𝑝,𝜔≤ℓ(𝑥)𝑒_𝛼∗𝑓−𝑓_𝑝,𝜔⟶0. (5.4) Hence, 𝐿_𝑥(𝑒_𝛼)∗𝑓→𝐿_𝑥𝑓 in 𝜏_Ω-topology. As 𝐼 is a 𝜏_Ω−closed left ideal, it follows that 𝐿_𝑥𝑓∈𝐼; that is, 𝐼 is left translation invariant. Similarly, it is shown that 𝐼 is also right translation invariant.

We denote by Δ(𝐿^𝑝(𝐺,Ω)) the spectrum of (𝐿^𝑝(𝐺,Ω),𝜏_Ω) consisting of all 𝜏_Ω-continuous nonzero linear functionals Φ on L^𝑝(𝐺,Ω) which are multiplicative; that is,Φ𝑓∗𝑔=Φ𝑓Φ𝑔𝑓,𝑔∈𝐿^𝑝(𝐺,Ω). (5.5)

We conclude this work with the following result which is a characterization of the spectrum of (𝐿^𝑝(𝐺,Ω),𝜏_Ω).

Proposition 5.3. Let Ω be a system of weights on 6-compact group 𝐺. Then Δ𝐿^𝑝(𝐺,Ω)=Φ_𝜌∶𝜌∈𝐿^𝑞1𝐺,𝜔,𝜔∈Ω,𝜌𝑥𝑦=𝜌(𝑥)𝜌𝑦, (5.6) where Φ_𝜌𝑓=_𝐺𝑓(𝑥)𝜌(𝑥)𝑑𝜆(𝑥)𝑓∈𝐿^𝑝(𝐺,Ω). (5.7)

Proof. Let 𝜌∈𝐿^𝑞(𝐺,1/𝜔) for some 𝜔∈Ω such that 𝜌(𝑥𝑦)=𝜌(𝑥)𝜌(𝑦) for almost all 𝑥,𝑦∈𝐺. Then, Φ_𝜌 is ‖⋅‖_𝑝,𝜔-continuous and so 𝜏_Ω-continuous. Moreover, for 𝑓,𝑔∈𝐿^𝑝(𝐺,Ω), Φ_𝜌𝑓∗𝑔=_𝐺𝐺𝑓(𝑥)𝑔𝑦𝜌(𝑥)𝑦=𝑥𝑦𝑑𝜆𝑑𝜆_𝐺𝐺𝑓(𝑥)𝑔𝑦𝜌(𝑥)𝜌𝑦𝑑𝜆(𝑥)𝑑𝜆𝑦=Φ_𝜌𝑓Φ_𝜌𝑔. (5.8) That is, Φ_𝜌∈Δ(𝐿^𝑝(𝐺,Ω)).
Conversely, let Φ∈Δ(𝐿^𝑝(𝐺,Ω)). Then Φ is bounded on a 𝜏_Ω-neighbourhood of zero. Thus Φ is bounded on the set {𝑓∈𝐿^𝑝(𝐺,Ω)∶‖𝑓‖_𝑝,𝜔<1}∩𝐿^𝑝(𝐺,𝜔) for some 𝜔∈Ω. Therefore Φ can be extended to an element Φ in (𝐿^𝑝(𝐺,𝜔),‖⋅‖_𝑝,𝜔)^∗. It follows that there exists a function 𝜌∈𝐿^𝑞(𝐺,1/𝜔) such that Φ𝑓=_𝐺𝑓𝜌𝑑𝜆, (5.9) for all 𝑓∈𝐿^𝑝(𝐺,𝜔). Since for 𝑓,𝑔∈𝐵_𝑐(𝐺), Φ(𝑓)Φ(𝑔)=Φ(𝑓∗𝑔), we infer that _𝐺×𝐺𝑓𝑦𝑔(𝑥)𝜌𝑦𝜌(𝑥)𝑑𝜆×𝜆𝑥,𝑦=_𝐺𝐺𝑓𝑦𝑔(𝑥)𝜌𝑦𝜌(𝑥)𝑦(𝑥)=𝑑𝜆𝑑𝜆_𝐺𝐺𝑓𝑦𝑔(𝑥)𝜌𝑦(𝑥)=𝑦𝑥𝑑𝜆𝑑𝜆_𝐺×𝐺𝑓𝑦𝑔(𝑥)𝜌𝑦𝑥𝑑𝜆×𝜆𝑥,𝑦. (5.10) By an argument similar to the proof of Theorem 4.9, we deduce that 𝜌(𝑥𝑦)=𝜌(𝑥)𝜌(𝑦) for almost all 𝑥,𝑦∈𝐺.