Computation of the \(\lambda_u\)-function in \(JB^\ast\)-algebras
In [ibid. 17, 139--147 (2011; Zbl 1227.46036)], the author began the study of the \(\lambda_{u}\)-function in the context of unital \(JB^\ast\)-algebras. In the setting of normed spaces, the \(\lambda\)-function was introduced by \textit{R. M. Aron} and \textit{R. H. Lohman} [Pac. J. Math. 127, No. 2, 209--231 (1987; Zbl 0662.46020)]. Let \(\mathcal{J}\) be a unital \(JB^\ast\)-algebra with unit \(e\) whose set of unitary elements is denoted by \(\mathcal{U}(\mathcal{J})\). Let \(u\) be a unitary element in \(\mathcal{J }\). For each \(x\) in the unit ball of \(\mathcal{J}\), \(\lambda_{u} (x)\) is defined by NEWLINE\[NEWLINE\lambda_{u} (x) = \sup \{ \lambda \in [0,1] : x= \lambda u + (1-\lambda) b,\quad u\in \mathcal{U}(\mathcal{J}),\quad b\in \mathcal{J},\quad \|b\|\leq 1\}.NEWLINE\]NEWLINE This function is a Jordan generalization of the \(\lambda_{u}\)-function introduced by \textit{G. K. Pedersen} [J. Oper. Theory 26, No. 2, 345--381 (1991; Zbl 0784.46043)]. Clearly, \(\lambda_{u} (x)\leq \lambda (x)\) and both functions coincide when every extreme point of the closed unit ball of \(\mathcal{J}\) is a unitary element.NEWLINENEWLINEThe main goal of this paper is to obtain some formulae to compute the \(\lambda_{u}\)-function for noninvertible elements of the closed unit ball in a \(JB^\ast\)-algebra in terms of the distance to the set \(\mathcal{J}_{inv}\) of invertible elements in \(\mathcal{J}\). Some new partial answers are established. Given an element \(x\) in a unital \(JB^\ast\)-algebra \(\mathcal{J}\), the function \(\alpha (x)\) is defined as the distance from \(x\) to the set \(\mathcal{J}_{inv}\). Among the results, the author shows that for a noninvertible element \(x\) in \(\mathcal{J}\), \(\lambda_u(x) \leq 1/2 (1 - \alpha(x))\). Further, if \(\alpha (x) = 1\), then \(\lambda_u(x) = 0\) (see Theorem 2.9). In the setting of \(C^\ast\)-algebras, Pedersen [loc. cit.] proved that \(\lambda_u(x) \leq 1/2 (1 - \alpha(x))\), for every non-invertible element \(x\) in a unital \(C^\ast\)-algebra \(A\). It remains an open problem if the last identity remains true for unital \(JB^\ast\)-algebras. This problem is intrinsically related to the questions left open by \textit{H. M. Tahlawi}, \textit{A. A. Siddiqui} and \textit{F. B. Jamjoom} [J. Math. Anal. Appl. 414, 734--741 (2014; \url{doi:10.1016/j.jmaa.2014.01.025})].
scientific article; zbMATH DE number 6220384
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