发表机构
Helvetia Insurance Ltd.; Mathematical Institute Czech Academy of Sciences; Institute of Mathematics and Computer Science Jagiellonian University（赫尔维蒂亚保险公司; 捷克科学院数学研究所; 雅盖隆大学数学与计算机科学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究针对Klaja与Ransford的问题，利用Motakis构造的Bourgain-Delbaen空间，在有界算子代数中构造了ab与ba指数谱不同的例子，还得到了稳定秩相关结论并关联了指数谱交换性失效与矩阵稳定化映射核的关系。
AI 中文摘要
Klaja与Ransford构造了一个复杂的单位巴拿赫代数，其中ab和ba的指数谱在非零点处存在差异，并询问有界算子代数中是否也能出现这种情况。我们肯定地回答了该问题。对于通过Motakis构造得到的Bourgain-Delbaen空间X，其Calkin代数作为巴拿赫代数同构于C(S⁴)，我们构造了S、T∈B(X⊕X)，使得1/2属于B(X⊕X)(ST)的指数谱，且1/2不属于B(X⊕X)(TS)的指数谱。此外，B(X⊕X)的左稳定秩(ltsr)与右稳定秩(rtsr)均为2，这是此类例子的最小可能稳定秩。更一般地，若K为紧可度量化空间，X为Motakis构造的对应C(K)的任意空间，则对任意n≥1，B(Xⁿ)的左稳定秩与右稳定秩满足：当dim K有限时，为⌈⌊dim K/2⌋/n⌉+1；当dim K无限时，为无穷大。这也给出了所有稳定秩至少为2的有限稳定秩反例以及无限稳定秩反例。最后，我们将指数谱交换性的每一种失效情况与指数群上第一矩阵稳定化映射的核联系起来。
英文摘要
Klaja and Ransford exhibited a complex unital Banach algebra in which the exponential spectra of \(ab\) and \(ba\) differ away from zero, and asked whether this can occur in an algebra of bounded operators. We answer their question affirmatively. For a Bourgain--Delbaen space \(X\) obtained from Motakis' construction with Calkin algebra isomorphic as a Banach algebra to \(C(S^4)\), we construct \(S,T\in\B(X\oplus X)\) such that \[ \frac12\in\varepsilon_{\B(X\oplus X)}(ST) \quad\text{and}\quad \frac12\notin\varepsilon_{\B(X\oplus X)}(TS). \] Moreover, \(\ltsr\B(X\oplus X)=\rtsr\B(X\oplus X)=2\), which is the least possible stable rank for such an example. More generally, if \(K\) is compact metrisable and \(X\) is any space arising from Motakis' construction for \(C(K)\), then, for every \(n\geqslant1\), \[ \ltsr\B(X^n)=\rtsr\B(X^n)= \begin{cases} \left\lceil \lfloor\dim K/2\rfloor/n\right\rceil+1,&\dim K<\infty, \infty,&\dim K=\infty. \end{cases} \] This also gives counterexamples of every finite stable rank at least two and of infinite stable rank. Finally, we relate every failure of exponential-spectral commutativity to the kernel of the first matrix stabilisation map on the index group.
Comments27 pp
