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arXiv · 1808.03108

Truncation and Spectral Variation in Banach Algebras

Rudi Brits·Francois Schulz·Cheick Toure
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Abstract

Let $a$ and $b$ be elements of a semisimple, complex and unital Banach algebra $A$. Using subharmonic methods, we show that if the spectral containment $\sigma(ax)\subseteq\sigma(bx)$ holds for all $x\in A$, then $ax$ belongs to the bicommutant of $bx$ for all $x\in A$. Given the aforementioned spectral containment, the strong commutation property then allows one to derive, for a variety of scenarios, a precise connection between $a$ and $b$. The current paper gives another perspective on the implications of the above spectral containment which was also studied, not long ago, by J. Alaminos, M. Bre\v{s}ar et. al.

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Rudi Brits, Francois Schulz, Cheick Toure. 2018-08-09. Truncation and Spectral Variation in Banach Algebras. https://arxiv.org/abs/1808.03108

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