Truncation and Spectral Variation in Banach Algebras

TL;DR

This paper explores how spectral containment relations in Banach algebras imply strong algebraic properties like bicommutant inclusion, using subharmonic methods to connect spectral and algebraic structures.

Contribution

It introduces a new perspective on spectral containment implications in Banach algebras, extending recent studies with subharmonic methods.

Findings

Spectral containment $\sigma(ax) ext{ } extless ext{ }\sigma(bx)$ implies $ax$ is in the bicommutant of $bx$.

Spectral containment leads to strong commutation properties between elements.

Provides a new approach to spectral implications using subharmonic techniques.

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.