The Banach–Lie algebra of multiplication operators on a JBW∗-triple

Abstract

The Banach-Lie algebra L(A) of multiplication operators on the JB * -triple A is introduced and it is shown that the hermitian part L(A) h of L(A) is a unital GM-space the base of the dual cone in the dual GL-space (L(A) h ) * of which is affine isomorphic and weak * -homeomorphic to the state space of L(A). In the case in which A is a JBW * -triple, it is shown that tripotents u and v in A are orthogonal if and only if the corresponding multiplication operators in the unital GM-space L(A) h satisfy 0 D(u, u) + D(v, v) id A , and that u is a pre-associate of v if and only if D(u, u) D(v, v).

THE BANACH-LIE *-ALGEBRA OF MULTIPLICATION OPERATORS ON A W*-ALGEBRA

“…Since the identity operator and it follows from Lemma 3.1(i) that D(a, a) is an element of the unital GM-space B(A) h . Therefore, by [2], Theorem 3.1(ii),…”

“…Then, the weak * -compact convex set S(B) consisting of elements X in the unit ball B For details of this result and those quoted above the reader is referred to [4,22]. The connection between the sets of hermitian elements in complex unital Banach algebras and partially ordered real Banach spaces is described in the next lemma, for the proof of which the reader is referred to [2], Lemma 2.3. A complex Banach space B that is also a Lie algebra, the multiplication in which is continuous in the norm topology is said to be a Banach-Lie algebra.…”

“…In this case, 1 is said to be the order unit in E [1,26]. The elementary proof of the result below can be found in [2], Lemma 2.1.…”

“…In the next lemma, the proof of which is given in [2], Lemma 2.4, the relationship that exists between the two real Banach spaces (L h ) * and (L * ) h is clarified.…”

“…As a consequence, the complete latticeŨ (A) is anti-order isomorphic to the complete lattice of subsystems. One motivation for an investigation of L(A) is that, in a recent paper, [2], it was shown that the order structure ofŨ (A) is closely related to the linear order structure of the hermitian part L(A) h of the Banach-Lie * -algebra L(A) of multiplication operators on A. A second motivation is that, for a W * -algebra A, the Banach-Lie * -algebra L(A) coincides with the set of generalized derivations of A.…”