AI 中文总结
研究由稳定理想对幺正\(C^*\) - 代数的托普利兹型扩张，利用谱度量空间构造，通过引入幺正2 - 压缩逼近等工具，证明在一定条件下，稳定理想基础代数的完备子算子系统序列收敛时，扩张中的相应序列也收敛，反之亦然。
AI 中文摘要
我们从非交换度量几何的角度研究由稳定理想对幺正\(C^*\) - 代数的托普利兹型\(C^*\) - 代数扩张。利用霍金斯和扎卡里亚斯（《数学物理通讯》350 (2017), 475 - 506）的谱度量空间构造，分析这些扩张与量子格罗莫夫 - 豪斯多夫距离的相互作用。证明了商空间或稳定理想基础的幺正代数的完备子算子系统能典范地确定扩张的完备子算子系统。引入幺正2 - 压缩逼近及其托普利兹型细化概念作为关键逼近工具。主要结果表明，在幺正2 - 压缩逼近条件和商空间的一个相容性条件下，如果稳定理想基础的幺正代数的完备子算子系统序列在量子格罗莫夫 - 豪斯多夫距离下收敛，那么扩张中的相应序列也收敛。在2 - 压缩托普利兹型细化条件下，从商空间到扩张也有类似结论。
英文摘要
We investigate the lifting of quantum Gromov-Hausdorff convergence through Toeplitz type $C^*$-algebra extensions by stable ideals in the framework of noncommutative metric geometry. Working with the spectral metric space construction of Hawkins and Zacharias (Comm. Math. Phys. 350 (2017), 475-506), we consider a sequence of complete sub-operator systems of the quotient or the unital $C^*$-algebra underlying the stable ideal, converging in the quantum Gromov-Hausdorff distance. We study whether this induces a corresponding convergent sequence of complete sub-operator systems of the extension. To address this problem, we construct complete sub-operator systems of the extension associated with those of the quotient and the unital $C^*$-algebra underlying the stable ideal. We also introduce the notion of unital $2$-contractive approximation together with its Toeplitz type refinement to provide the compatibility required by the commutator structure of the Dirac operator on the extension. We prove that, under this approximation hypothesis on the convergent sequence in the quotient or the unital $C^*$-algebra underlying the stable ideal, quantum Gromov-Hausdorff convergence lifts to the extension.
CommentsMinor modification. Earlier subsections 4.1 and 4.2 are converted to full sections. Section 6.3 is a new addition
