Twisting exponential spectra
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(S4), we construct S,T∈B(X⊕ X) such that 1/2∈εB(X⊕ X)(ST) and 1/2∉εB(X⊕ X)(TS). Moreover, ltsrB(X⊕ X)=rtsrB(X⊕ 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 ngeqslant1, ltsrB(Xn)=rtsrB(Xn)= ⌈ ⌊dim K/2⌋/n⌉+1,&dim K<∞, ∞,&dim K=∞. 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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BibTeX
@misc{horvath2026twisting,
title = {{Twisting exponential spectra}},
author = {Bence Horváth and Tomasz Kania},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05362},
eprint = {2609.05362},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05362},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Twisting exponential spectra
AU - Bence Horváth
AU - Tomasz Kania
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05362
UR - https://arxiv.org/abs/2609.05362
AB - 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∈B(X⊕ X) such that 1/2∈ε_B(X⊕ X)(ST) and 1/2∉ε_B(X⊕ X)(TS). Moreover, ltsrB(X⊕ X)=rtsrB(X⊕ 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 ngeqslant1, ltsrB(X^n)=rtsrB(X^n)= ⌈ ⌊dim K/2⌋/n⌉+1,&dim K<∞, ∞,&dim K=∞. 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.
ER -