AI 中文总结
研究卡普兰斯基第二测试问题，即\(T\oplus T\)与\(S\oplus S\)相似时\(T\)与\(S\)是否相似。核心方法是针对不同类型代数元素分析。贡献为对特定算子及代数元素给出肯定回答，并指出结果可用于量子态局部酉等价实现。
AI 中文摘要
卡普兰斯基关于相似性的第二测试问题是：若\(T\)和\(S\)是幺正巴拿赫代数\(\mathcal{B}\)中的元素，且\(T\oplus T\)在\(\mathbb{M}_2(\mathcal{B})\)中与\(S\oplus S\)相似，那么\(T\)在\(\mathcal{B}\)中是否与\(S\)相似？本文给出肯定回答：当\(T\)是\(\mathrm{I}_n\)型冯·诺依曼代数\(\mathcal{M}\)中具有性质\((J)\)的算子时；且\(1\leqslant n\leqslant 3\)时可去掉性质\((J)\)条件。对于具有本质有限维换位子的幺正巴拿赫代数中的元素\(T\)也有类似结果。还指出主要结果可应用于量子态的局部酉（LU）等价实现。
英文摘要
Kaplansky's second test problem on similarity asks: if $T$ and $S$ are elements in a unital Banach algebra $\mathcal{B}$ and $T\oplus T$ is similar to $S\oplus S$ in $\mathbb{M}_2(\mathcal{B})$, is $T$ similar to $S$ in $\mathcal{B}$? We answer this problem affirmatively if $T$ is an operator with property $(J)$ in a type $\mathrm{I}_n$ von Neumann algebra $\mathcal{M}$, i.e., $\{T\}'\cap\mathcal{M}$ contains a bounded maximal abelian family of idempotents. Moreover, the condition of property $(J)$ can be removed for $1\leqslant n\leqslant 3$. A similar result is proved if $T$ is an element in a unital Banach algebra $\mathcal{B}$ with essentially finite-dimensional commutant, i.e., the relative commutant of $T$ in $\mathcal{B}$ is finite-dimensional modulo its Jacobson radical. Finally, we point out that one of our main results can be applied to the implementation of local unitary (LU) equivalence of quantum states.
Comments54 pages
