<p>Let <i>G</i> be a locally compact group and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$(\Phi ,\Psi )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo>,</mo> <mi mathvariant="normal">Ψ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> a complementary pair of Young functions satisfying the <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-condition. Let <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$A_\Phi (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi mathvariant="normal">Φ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the Orlicz analogue of the Figà-Talamanca Herz algebra <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$A_p(G).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi>p</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The dual of the algebra <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$A_\Phi (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi mathvariant="normal">Φ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the space of <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\Psi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ψ</mi> </math></EquationSource> </InlineEquation>-pseudomeasures, denoted by <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="71" /> </InlineMediaObject> <EquationSource Format="TEX">$PM_\Psi (G).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <msub> <mi>M</mi> <mi mathvariant="normal">Ψ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> For certain topologically introverted subspaces <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$PM_\Psi (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <msub> <mi>M</mi> <mi mathvariant="normal">Ψ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the Banach algebras <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$W_\Phi (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi mathvariant="normal">Φ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq13.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$B_\Phi (G),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi mathvariant="normal">Φ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> denoted by <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq14.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {B},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we characterise the maximal regular left/right/two-sided ideals of the Banach algebras <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq15.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}^{'}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow> <mi mathvariant="script">A</mi> </mrow> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq16.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {B}^{''}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow> <mi mathvariant="script">B</mi> </mrow> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </mmultiscripts> </mmultiscripts> </math></EquationSource> </InlineEquation> considered with the Arens product. We further characterise the minimal left ideals of <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq15.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {A}^{'}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow> <mi mathvariant="script">A</mi> </mrow> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mo>′</mo> </mmultiscripts> </mmultiscripts> </math></EquationSource> </InlineEquation> and prove the necessary and sufficient conditions for the existence of minimal ideals in the algebras <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="49" /> </InlineMediaObject> <EquationSource Format="TEX">$A_\Phi (G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>A</mi> <mi mathvariant="normal">Φ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40840_2025_1850_Article_IEq19.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {B}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

Ideals in the Dual of Introverted Subspaces of $\Psi $-Pseudomeasures

期刊论文
发表日期：

2025年3月12日

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Arvish Dabra,
Rattan Lal,
N. Shravan Kumar

摘要

Let G be a locally compact group and $(\Phi ,\Psi )$ $(Φ, Ψ)$ a complementary pair of Young functions satisfying the $\Delta _2$ $Δ_{2}$ -condition. Let $A_\Phi (G)$ $A_{Φ} (G)$ be the Orlicz analogue of the Figà-Talamanca Herz algebra $A_p(G).$ $A_{p} (G) .$ The dual of the algebra $A_\Phi (G)$ $A_{Φ} (G)$ is the space of $\Psi $ $Ψ$ -pseudomeasures, denoted by $PM_\Psi (G).$ $P M_{Ψ} (G) .$ For certain topologically introverted subspaces $\mathcal {A}$ $A$ of $PM_\Psi (G)$ $P M_{Ψ} (G)$ and the Banach algebras $W_\Phi (G)$ $W_{Φ} (G)$ or $B_\Phi (G),$ $B_{Φ} (G),$ denoted by $\mathcal {B},$ $B,$ we characterise the maximal regular left/right/two-sided ideals of the Banach algebras $\mathcal {A}^{'}$ $A^{^{'}}$ and $\mathcal {B}^{''}$ $B^{^{''}}$ considered with the Arens product. We further characterise the minimal left ideals of $\mathcal {A}^{'}$ $A^{^{'}}$ and prove the necessary and sufficient conditions for the existence of minimal ideals in the algebras $A_\Phi (G)$ $A_{Φ} (G)$ and $\mathcal {B}.$ $B .$

Ideals in the Dual of Introverted Subspaces of \(\Psi \)-Pseudomeasures

摘要