The \(\lambda_u\)-function in \(JB^*\)-algebras. (English) Zbl 1227.46036

R. M. Aron and R. H. Lohman introduced and studied in [Pac. J. Math. 127, No. 2, 209–231 (1987; Zbl 0662.46020)] a geometric function, called the \(\lambda\)-function, in the setting of normed spaces. Let \(X\) be a normed space and \(B(X)\) be its unit ball. \(X\) is said to have the \(\lambda\)-property if for each \(x\in B(X)\) there exist \(e\in\text{ext\,}B(X)\), \(y\in B(X)\) and \(0<\lambda\leq1\) such that \(x=\lambda e+(1-\lambda)y\). Denote by \(\lambda(x)\) the supremum of such \(\lambda\) for \(x\in B(X)\). In [J. Oper. Theory 26, No. 2, 345–381 (1991; Zbl 0784.46043)], G. K. Pedersen devoted a paper to the study of the \(\lambda\)-function, the \(\lambda\)- property and related topics, when \(X\) is a \(C^*\)-algebra. In the same paper, Pedersen introduced the following variant of the \(\lambda\)-function, called the \(\lambda_{u}\)-function, defined on the unit ball of a unital C\(^*\)-algebra, where \(u\) is a unitary element. Let \(A\) be a unital C\(^*\)-algebra with unit \(e\) whose set of unitary elements is denoted by \(\mathcal{U}(A)\). For each \(x\) in the unit ball of \(A\), \(\lambda_{u} (x)\) is defined by
\[ \lambda_{u} (x) = \sup \{ \lambda \in [0,1] : x= \lambda u + (1-\lambda) b,\quad u\in \mathcal{U}(A),\quad b\in A,\quad \|b\|\leq 1\}. \]
Clearly, \(\lambda_{u} (x)\leq \lambda (x)\) and both functions coincide when every extreme point of the closed unit ball of \(A\) is a unitary element. This definition also makes sense for every element in the unit ball of a unital JB\(^*\)-algebra.
The aim of the paper under review is to introduce and study the notion of a \(\lambda_{u}\)-function in the context of unital JB\(^*\)-algebras. The author studies the connection between the \(\lambda_{u}(x)\) and the existence of a suitable convex combination of unitary elements which coincides with \(x\). An interesting formula to compute the \(\lambda_{u}\)-function for invertible elements in a JB\(^*\)-algebra is also obtained.
As noted by the author, the study of the \(\lambda_{u}\)-function cannot be further extended to the more general setting of JB\(^*\)-triple systems because a JB\(^*\)-triple need not contain unitary elements. However, the study of the \(\lambda\)-property and the \(\lambda\)-function in the setting of JB\(^*\)-triples remains unexplored.


46H70 Nonassociative topological algebras
46B04 Isometric theory of Banach spaces
46L70 Nonassociative selfadjoint operator algebras
17C65 Jordan structures on Banach spaces and algebras
46L05 General theory of \(C^*\)-algebras
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