论文标题
在可测量函数的代数上翻译不变的衍生物不存在
Non-existence of translation-invariant derivations on algebras of measurable functions
论文作者
论文摘要
令$ s（0,1）$为$*$ - 在单位间隔$（0,1）$上的所有类别可测量函数的代数，然后让$（\ Mathcal {a}，\ weled \ | \ | \ | \ cdot \ cdot \ cdot \ cdot \ cdot \ right | _ \ | _ \ | _ \ mathcal {a} $是一个完整的$ - $ s（0,1）$，其中简单函数是密度的，例如$ l_ \ infty（0,1）$，$ l _ {\ log}（0,1）$，$ s（0,1）$和ARENS代数$ l^ω（0,1）$，配备了自然$δ$ -Norms。我们表明，没有非平凡的推导$δ：\ mathcal {a} \ to s（0,1）$通勤，单位间隔的所有二元翻译。令$ \ MATHCAL {M} $为$ ii $（或$ i_ \ infty $）von Neumann代数，$ \ Mathcal {a} $是其Abelian von Neumann subalgebra，Let $ s（\ Mathcal {M}）$是所有测量$ callcal $ callcal $ callcal $ callcal $ calcal。 我们表明，任何非平凡的推导$δ：\ Mathcal {a} \ to s（\ Mathcal {a}）$都不能扩展到$ s（\ Mathcal {M}）$的派生。特别是，我们在\ cite {bks1}中回答一个未经处理的问题。
Let $S(0,1)$ be the $*$-algebra of all classes of Lebesgue measurable functions on the unit interval $(0,1)$ and let $(\mathcal{A},\left\|\cdot \right\|_\mathcal{A})$ be a complete symmetric $Δ$-normed $*$-subalgebra of $S(0,1)$, in which simple functions are dense, e.g., $L_\infty (0,1)$, $L_{\log}(0,1)$, $S(0,1)$ and the Arens algebra $L^ω(0,1)$ equipped with their natural $Δ$-norms. We show that there exists no non-trivial derivation $ δ: \mathcal{A} \to S(0,1)$ commuting with all dyadic translations of the unit interval. Let $\mathcal{M}$ be a type $II$ (or $I_\infty$) von Neumann algebra, $\mathcal{A}$ be its abelian von Neumann subalgebra, let $S(\mathcal{M})$ be the algebra of all measurable operators affiliated with $\mathcal{M}$. We show that any non-trivial derivation $δ:\mathcal{A} \to S(\mathcal{A})$ can not be extended to a derivation on $S(\mathcal{M})$. In particular, we answer an untreated question in \cite{BKS1}.
