<p>Suppose that <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> is an infinite-dimensional separable Hilbert space. We construct a new class of Kadison–Singer lattices (KS-lattices) on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>, and characterize the bounded derivations on the corresponding Kadison–Singer algebras (KS-algebras). Moreover, if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation> is a nontrivial nest on <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$\xi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation> is a separating vector for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq6.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="27" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {N}''$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">N</mi> </mrow> <mrow> <mo>′</mo> <mo>′</mo> </mrow> </msup> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">L</mi> </math></EquationSource> </InlineEquation> is a KS-lattice generated by <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">N</mi> </math></EquationSource> </InlineEquation> and the rank-one projection <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$P_{\xi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>P</mi> <mi>ξ</mi> </msub> </math></EquationSource> </InlineEquation>, we prove that every Lie triple derivation from the KS-algebra <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq10.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{Alg}{\mathcal {L}}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Alg</mtext> <mi mathvariant="script">L</mi> </mrow> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10998_2025_631_Article_IEq11.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$B(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is standard.</p>

Characterizations of Kadison–Singer lattices and Lie triple derivations on Kadison–Singer algebras

期刊论文
发表日期：

2025年2月24日

查看全文

Guangyu An,
Tian Fang,
Danni Zhao

摘要

Suppose that $\mathcal {H}$ $H$ is an infinite-dimensional separable Hilbert space. We construct a new class of Kadison–Singer lattices (KS-lattices) on $\mathcal {H}$ $H$ , and characterize the bounded derivations on the corresponding Kadison–Singer algebras (KS-algebras). Moreover, if $\mathcal {N}$ $N$ is a nontrivial nest on $\mathcal {H}$ $H$ , $\xi $ $ξ$ is a separating vector for $\mathcal {N}''$ $N^{''}$ , and $\mathcal {L}$ $L$ is a KS-lattice generated by $\mathcal {N}$ $N$ and the rank-one projection $P_{\xi }$ $P_{ξ}$ , we prove that every Lie triple derivation from the KS-algebra $\textrm{Alg}{\mathcal {L}}$ $Alg L$ into $B(\mathcal {H})$ $B (H)$ is standard.