Spectral characterization of the kh-socle in Banach Jordan algebras (Q1645146)

This paper presents a characterization of \(\text{kh}(\text{soc}(J))\) in terms of the full spectrum. Namely, the following conditions are equivalent on a complex semisimple Banach Jordan algebra \(J\) with unit: {\parindent=0.7cm\begin{itemize}\item[(1)] \(x \in \text{kh}(\text{soc}(J))\), \item[(2)] \(\widehat{\sigma}(x+a) \setminus \widehat{\sigma}(a)\) is discrete for every \(a \in J\), and \item[(3)] \( \left\{ \lambda \in \mathbb{C} : 0 \in \sigma(a+\lambda x) \right\}\) is a closed discrete set for every \(a \in J\) with \(0 \notin \widehat{\sigma}(a)\). \end{itemize}}