Let G be a locally compact group, and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$VN^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msup> <mi>N</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the dual of the multidimensional Fourier algebra <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$A^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In this article, we define invariant means on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$VN^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msup> <mi>N</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and prove that the set of all invariant means on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$VN^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msup> <mi>N</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is non-empty. Further, we investigated the invariant means on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$VN^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msup> <mi>N</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for discrete and non-discrete cases of G. Also, we show that if H is an open subgroup of G, then the number of invariant means on <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$VN^n(H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msup> <mi>N</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is the same as that of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$VN^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <msup> <mi>N</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Finally, we study invariant means on the dual of the algebra <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$A_0^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>A</mi> <mn>0</mn> <mi>n</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the closure of Fourier algebra <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$A^n(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>A</mi> <mi>n</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in the cb-multiplier norm.

Invariant means on $VN^n(G)$

期刊论文
发表日期：

2026年4月21日

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Kanupriya,
N. Shravan Kumar

摘要

Let G be a locally compact group, and $VN^n(G)$ $V N^{n} (G)$ is the dual of the multidimensional Fourier algebra $A^n(G)$ $A^{n} (G)$ . In this article, we define invariant means on $VN^n(G)$ $V N^{n} (G)$ and prove that the set of all invariant means on $VN^n(G)$ $V N^{n} (G)$ is non-empty. Further, we investigated the invariant means on $VN^n(G)$ $V N^{n} (G)$ for discrete and non-discrete cases of G. Also, we show that if H is an open subgroup of G, then the number of invariant means on $VN^n(H)$ $V N^{n} (H)$ is the same as that of $VN^n(G)$ $V N^{n} (G)$ . Finally, we study invariant means on the dual of the algebra $A_0^n(G)$ $A_{0}^{n} (G)$ , the closure of Fourier algebra $A^n(G)$ $A^{n} (G)$ in the cb-multiplier norm.

Invariant means on \(VN^n(G)\)

摘要