Let <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq7.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="100" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {H}_d:=\mathbb {C}^d\times \mathbb {R},$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">H</mi> <mi>d</mi> </msub> <mo>:</mo> <mo>=</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>d</mi> </msup> <mo>×</mo> <mi mathvariant="double-struck">R</mi> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="62" /> </InlineMediaObject> <EquationSource Format="TEX">$(d\in \mathbb {N}^)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">N</mi> </mrow> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq9.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">$2d+1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>d</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>-dimensional Heisenberg group and we denote by U(d) (the unitary group) the maximal compact connected subgroup of <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq10.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$Aut(\mathbb {H}_d),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>u</mi> <mi>t</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">H</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> the group of automorphisms of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq11.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {H}_d.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">H</mi> <mi>d</mi> </msub> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> Let <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq12.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="161" /> </InlineMediaObject> <EquationSource Format="TEX">$G_d:=U(d) < imes \mathbb {H}_d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>G</mi> <mi>d</mi> </msub> <mo>:</mo> <mo>=</mo> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> </mrow> <mo>⋉</mo> <msub> <mi mathvariant="double-struck">H</mi> <mi>d</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> be the Heisenberg motion group. In this work, we describe the <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq13.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$C^$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq14.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$C^*(G_d),$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>C</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <msub> <mi>G</mi> <mi>d</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> of <InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq15.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$G_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>G</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation> in terms of an algebra of operator fields defined over its dual space <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="43036_2024_417_Article_IEq16.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$\widehat{G_d}.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <msub> <mi>G</mi> <mi>d</mi> </msub> <mo stretchy="true">^</mo> </mover> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> This result generalizes a previous result in Ludwig and Regeiba (Complex Anal Oper Theory 13(8):3943–3978, 2019).

The $C^*$-algebra of the Heisenberg motion groups $U(d) < imes \mathbb {H}_d.$

期刊论文
发表日期：

2025年1月20日

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Hedi Regeiba,
Aymen Rahali

摘要

Let $\mathbb {H}_d:=\mathbb {C}^d\times \mathbb {R},$ $H_{d} : = C^{d} \times R,$ $(d\in \mathbb {N}^*)$ $(d \in N^{*})$ be the $2d+1$ $2 d + 1$ -dimensional Heisenberg group and we denote by U(d) (the unitary group) the maximal compact connected subgroup of $Aut(\mathbb {H}_d),$ $A u t (H_{d}),$ the group of automorphisms of $\mathbb {H}_d.$ $H_{d} .$ Let $G_d:=U(d) < imes \mathbb {H}_d$ $G_{d} : = U (d) ⋉ H_{d}$ be the Heisenberg motion group. In this work, we describe the $C^*$ $C^{*}$ -algebra $C^*(G_d),$ $C^{*} (G_{d}),$ of $G_d$ $G_{d}$ in terms of an algebra of operator fields defined over its dual space $\widehat{G_d}.$ $\hat{G_{d}} .$ This result generalizes a previous result in Ludwig and Regeiba (Complex Anal Oper Theory 13(8):3943–3978, 2019).

The \(C^*\)-algebra of the Heisenberg motion groups \(U(d) < imes \mathbb {H}_d.\)

摘要