We construct a new class of subspace lattices <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\cal L}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">L</mi> </mrow> </math></EquationSource> </InlineEquation> on an infinite-dimensional Hilbert space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\cal K}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">K</mi> </mrow> </math></EquationSource> </InlineEquation>. We show that the bounded cohomology groups <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^{n}({\rm Alg} \, {\cal L},\,{\cal B}({\cal K}))$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>H</mi> <mrow> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">g</mi> </mrow> <mspace width="thinmathspace" /> <mrow> <mi mathvariant="script">L</mi> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow> <mi mathvariant="script">B</mi> </mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="script">K</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> of the corresponding lattice algebras Alg <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\cal L}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">L</mi> </mrow> </math></EquationSource> </InlineEquation> with coefficients in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\cal B}({\cal K})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">B</mi> </mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="script">K</mi> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> are trivial for all n ≥ 1, and every derivation ϕ from Alg <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\cal L}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">L</mi> </mrow> </math></EquationSource> </InlineEquation> into Alg <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\cal L}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">L</mi> </mrow> </math></EquationSource> </InlineEquation> is an inner derivation under some conditions. We also prove that every Lie derivation δ from Alg <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\cal L}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">L</mi> </mrow> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${\cal B}({\cal K})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">B</mi> </mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="script">K</mi> </mrow> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is standard.

Cohomology Groups and Lie Derivations of a Class of Lattice Algebras

期刊论文
发表日期：

2025年11月15日

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Zhujun Yang

摘要

We construct a new class of subspace lattices ${\cal L}$ $L$ on an infinite-dimensional Hilbert space ${\cal K}$ $K$ . We show that the bounded cohomology groups $H^{n}({\rm Alg} \, {\cal L},\,{\cal B}({\cal K}))$ $H^{n} (A l g L, B (K))$ of the corresponding lattice algebras Alg ${\cal L}$ $L$ with coefficients in ${\cal B}({\cal K})$ $B (K)$ are trivial for all n ≥ 1, and every derivation ϕ from Alg ${\cal L}$ $L$ into Alg ${\cal L}$ $L$ is an inner derivation under some conditions. We also prove that every Lie derivation δ from Alg ${\cal L}$ $L$ into ${\cal B}({\cal K})$ $B (K)$ is standard.