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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.

MSC:

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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