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Convex combinations of unitaries in \(JB^*\)-algebras. (English) Zbl 1227.46035

The Russo-Dye theorem is a celebrated result in C\(^*\)-algebra theory which affirms that the closed unit ball of a unital \(C^*\)-algebra is the closed convex hull of its unitary group. Alternative proofs and extensions of this result have been published during the last forty years. Some of the variants explore which elements in a unital C\(^*\)-algebra lie in the algebraic convex hull of its unitary group. This kind of results led a group of mathematicians, including R. V. Kadison and G. K. Pedersen [Math. Scand. 57, 249–266 (1985; Zbl 0573.46034)], C. L. Olsen and G. K. Pedersen [J. Funct. Anal. 66, 365–380 (1986; Zbl 0597.46061)] and others to study convex combinations of unitaries in unital C\(^*\)-algebras.
The paper under review continues the study of convex combinations of unitaries in an arbitrary unital JB\(^*\)-algebra developed by the author in [Indian J. Math. 48, No. 1, 35–48 (2006; Zbl 1115.46058), Arch. Math. 87, No. 4, 350–358 (2006; Zbl 1142.46020)] and [Int. J. Math. Math. Sci. 2007, Article ID 37186 (2007; Zbl 1161.46041)]. In this paper, the author exploits the notion of a unitary isotope to extend to the setting of JB\(^*\)-algebras some previous results established in the setting of C\(^*\)-algebras.
Let \(\mathcal{J}\) be a JB\(^*\)-algebra with unit \(e\). An element \(u\in \mathcal{J}\) is called unitary if it is invertible and \(u^*= u^{-1}.\) The set of unitaries in \(\mathcal{J}\) is denoted by \(\mathcal{U}(\mathcal{J})\).
Given an element \(x\) in a JB\(^*\)-algebra \(J\), we can consider the following numbers:
\[ u_{c} (x) = \min \left\{ n\in \mathbb{N} : x \text{ is a combex combination of } n \text{ unitaries in } J\right\} \]
and
\[ u_m (x) = \min \left\{ n\in \mathbb{N} : x = \frac1n \sum_{j=1}^{n} u_j, u_j \in \mathcal{U}(\mathcal{J})\right\}, \]
with \(u_c (x) = \infty\) when \(x\) cannot be written as a convex combination of unitary elements in \(J\). Clearly, \(u_c (x) \leq u_m (x)\).
In the first main result, the author shows that for every \(x\) in a unital JB\(^*\)-algebra \(J\) with \(\|x\|\leq \varepsilon <1\) and each unitary \(u\) in \(J\), there exist unitaries \(u_1\) and \(u_2\) in \(J\) satisfying \(u+ x = u_1 +\varepsilon u_2\). It is also proved that the distance from any positive element \(x\) to the set of unitaries in \(J\) is attained at the unit element of \(J\). When \(x\) is invertible, the distance from \(x\) to \(\mathcal{U}(\mathcal{J})\) is attained at the unitary element appearing in a suitable polar decomposition of \(x\) in some \(B(H)\). These results generalise a previous contribution due to C. L. Olsen and G. K. Pedersen [loc. cit.].
The result obtained by R. V. Kadison and G. K. Pedersen in [loc. cit.] is also generalised to the setting of JB\(^*\)-algebras, showing that for every unital JB\(^*\)-algebra \(J\) and \(0\leq \alpha \leq \beta\leq \frac12\), we have \(\alpha \mathcal{U}(\mathcal{J}) + (1-\alpha) \mathcal{U}(\mathcal{J}) \subseteq \beta \mathcal{U}(\mathcal{J}) + (1-\beta) \mathcal{U}(\mathcal{J})\). The final consequence of those results presented in Section 4 shows that any convex combination of unitaries in a unital JB\(^*\)-algebra \(J\) is the mean of the same number of unitaries in \(J\), and hence \(u_c (x) \leq u_m (x)\).
The proofs make use of the unitary isotopes of a JB\(^*\)-algebra, the Shirshov-Cohn Theorem to represent the JB\(^*\)-subalgebra generated by an element and the unit as a JC\(^*\)-algebra, and the precedents in C\(^*\)-algebra theory.

MSC:

46H70 Nonassociative topological algebras
46K70 Nonassociative topological algebras with an involution
17C65 Jordan structures on Banach spaces and algebras
46L05 General theory of \(C^*\)-algebras
46L45 Decomposition theory for \(C^*\)-algebras
46L70 Nonassociative selfadjoint operator algebras
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