The Banach-Lie algebra of multiplication operators on a \(JBW^{*}\)-triple
The main object of study in this paper is the Banach-Lie algebra of multiplication operators on a JB*-triple. Given two elements \(a\), \(b\) in a JB*-triple \(E\), the multiplication operator \(D(a,b)\) is the element of \(B(E)\) defined by \(D(a,b) (x) = \{ a,b,x\}\). Originally introduced by \textit{W. Kaup} in [Math. Z. 183, 503--529 (1983; Zbl 0519.32024)], the Banach-Lie algebra \(\mathcal{L} (E)\) of multiplication operators on \(E\) is defined as the norm-closed subspace of \(B(E)\) generated by all multiplication operators and the identity operator. The real linear subspace \(\mathcal{H} (E)\subset B(E)\) consists of all Hermitian operators on \(E\), i.e., of all \(T\in B(E)\) with \(\| \exp(i t T)\| =1\) for all \(t\in\mathbb{R}\). \(\mathcal{J}(E):=\mathcal{H}(E)\oplus i\mathcal{H}(E)\) is a closed complex Lie subalgebra of \(B(E)\) in the commutator product and admits a unique continuous involution * with \(\mathcal{H}(E) = \{T\in \mathcal{J}(E): T^*=T\}\). Actually, \(\mathcal{L} (E)\) is a closed *-invariant Lie subalgebra of \(\mathcal{J} (E)\) and hence \(\mathcal{L} (E)\) possesses a natural involution. In the paper under review, the authors show that the Hermitian part \(\mathcal{L} (E)_h\) of \(\mathcal{L} (E)\) is a unital GM-space of which the base of the dual cone in the dual GL-space \((\mathcal{L} (E)_h)^*\) is affine isomorphic and weak*-homeomorphic to the state space of \(\mathcal{L} (E)\). Although JB*-triples lack a global order structure, a partial order can be defined in the set of all tripotents in a JB*-triple. The main results of the paper connect the natural order in the set of tripotents in a JBW*-triple with the order of multiplication operators in \(\mathcal{L} (E)_h\). It is established that two tripotents \(u\) and \(v\) in a JBW*-triple \(E\) are orthogonal if and only if the corresponding multiplication operators in the unital GM-space \(\mathcal{L} (E)_h\) satisfy \[ 0\leq D(u,u) +D(v,v) \leq \text{id}_{E}, \] and that \(u\) is pre-associate to \(v\) (i.e., the Peirce-two subspace associated to \(u\) is contained in the Peirce-two subspace corresponding to \(v\)) if and only if \(D(u,u) \leq D(v,v)\). One of the basic tools employed in the proof are the pre-Hilbertian seminorms associated to those (bounded) positive sesquilinear forms defined by \((a,b) \mapsto \phi(D(a,b))\), where \(\phi\) is a state of the Banach-Lie algebra \(\mathcal{L} (E)\). These pre-Hilbertian seminorms played an important role in a Grothendieck type inequality proved for JB*-triples by \textit{A. Rodríguez Palacios} and the reviewer in [North-Holland Math. Stud. 189, 409--423 (2001; Zbl 1063.46506)].
scientific article; zbMATH DE number 5285680
scientific article