Twisting exponential spectra

Bence Horváth; Tomasz Kania

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arXiv · 2609.05362

Twisting exponential spectra

Bence Horváth·Tomasz Kania

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Abstract

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.

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Bence Horváth, Tomasz Kania. 2026-09-04. Twisting exponential spectra. https://arxiv.org/abs/2609.05362

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