This paper presents abstract harmonic analysis foundations for structure of covariant function algebras of invariant characters of normal subgroups. Suppose that G is a locally compact group and N is a closed normal subgroup of G. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\xi :N\rightarrow \mathbb {T}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>:</mo> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">T</mi> </mrow> </math></EquationSource> </InlineEquation> be a continuous G-invariant character, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$1\le p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L_\xi ^p(G,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>ξ</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L^p$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation>-space of all covariant functions of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> on G. We study structure of covariant convolution in <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^p_\xi (G,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>ξ</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. It is proved that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^1_\xi (G,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>ξ</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a Banach <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$*$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-algebra and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L^p_\xi (G,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>ξ</mi> <mi>p</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a Banach <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L^1_\xi (G,N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>L</mi> <mi>ξ</mi> <mn>1</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-module. We then investigate the theory of covariant convolutions for the case of characters of canonical normal subgroups in semi-direct product groups. The paper is concluded by realization of the theory in the case of different examples.

Covariant Function Algebras of Invariant Characters of Normal Subgroups

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发表日期：

2026年5月16日

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Arash Ghaani Farashahi

摘要

This paper presents abstract harmonic analysis foundations for structure of covariant function algebras of invariant characters of normal subgroups. Suppose that G is a locally compact group and N is a closed normal subgroup of G. Let $\xi :N\rightarrow \mathbb {T}$ $ξ : N \to T$ be a continuous G-invariant character, $1\le p<\infty $ $1 \leq p < \infty$ , and $L_\xi ^p(G,N)$ $L_{ξ}^{p} (G, N)$ be the $L^p$ $L^{p}$ -space of all covariant functions of $\xi $ $ξ$ on G. We study structure of covariant convolution in $L^p_\xi (G,N)$ $L_{ξ}^{p} (G, N)$ . It is proved that $L^1_\xi (G,N)$ $L_{ξ}^{1} (G, N)$ is a Banach $*$ $*$ -algebra and $L^p_\xi (G,N)$ $L_{ξ}^{p} (G, N)$ is a Banach $L^1_\xi (G,N)$ $L_{ξ}^{1} (G, N)$ -module. We then investigate the theory of covariant convolutions for the case of characters of canonical normal subgroups in semi-direct product groups. The paper is concluded by realization of the theory in the case of different examples.