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I have been reading the paper Determining elements in $C^∗$-algebras through spectral properties. The paper discusses about what would be the relation be between two elements $a$ and $b$ of a Banach algebra $A$ which satisfy $\sigma(ax) \subset \sigma(bx)$ or $\sigma(a+x) \subset \sigma(b+x)$ for all $x \in A$. Can anyone tell what exactly is the motivation behind this problem, what are its applications? Why should this problem be settled for any Banach algebra? The most I could find is that the authors say the problem is motivated by the Kaplansky problem.

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    $\begingroup$ I can't say anything on the actual question, but generally speaking, the fact that a problem is discussed in a published paper is scant evidence for there being any motivation for it. $\endgroup$
    – Christian Remling
    Commented May 26, 2022 at 17:00
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    $\begingroup$ It might be interesting to reverse the question: what was your motivation for reading the paper? $\endgroup$
    – Jochen Glueck
    Commented May 26, 2022 at 17:20
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    $\begingroup$ @JochenGlueck I personally love this theory as its neat and elegant. My question arose as to how to explain to mathematicians of another area as to why this theory is significant. $\endgroup$
    – user332905
    Commented May 27, 2022 at 4:26

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