Truncation and spectral variation in Banach algebras (Q323780)

In [J. Math. Anal. Appl. 393, No. 1, 144--150 (2012; Zbl 1251.46023)], \textit{M. Brešar} and \textit{Š. Špenko} proposed two interesting questions arising around Kaplansky's problem on spectrum preserving maps; namely, if \(\sigma\) and \(\rho\) denote the spectrum and the spectral radius, {\parindent=0.6cm\begin{itemize}\item[--] if \(\sigma(ax)=\sigma(bx)\) holds for all \(x\) in a Banach algebra \(A\), does this imply that \(a=b\)? \item[--] if \(\rho (ax) \leq \rho (bx)\) for all \(x\), determine the relation between \(a\) and \(b\). NEWLINENEWLINE\end{itemize}} The first question was solved in [\textit{G. Braatvedt} and \textit{R. M. Brits}, Quaest. Math. 36, No. 2, 155--165 (2013; Zbl 1274.46093)] in the positive for semisimple unital Banach algebras. Concerning the second question, there are some partial results, mainly when the algebra is prime.NEWLINENEWLINEIn [loc.\,cit.], some results were presented for elements on a unital \(C^{*}\)-algebra satisfying the condition \(\sigma(ax) \subset \sigma(bx)\cup \{0\}\) for all \(x\). It was proved that \(a=zb\), where \(z\) is a central projection of the bidual.NEWLINENEWLINEIn this paper, the authors prove that, when the condition \(\sigma (ax) \subset \sigma(bx)\) holds for all \(x\), then \(ax\) belongs to the bicommutant of \(bx\), for all \(x\). This allows them to obtain a precise description of the relation between \(a\) and \(b\) in many cases. In particular, they obtain, using different techniques, the aforementioned result for \(C^{*}\)-algebras. As the authors say, it is an interesting paper that gives another perspective on the consequences of this kind of conditions.