Geometric unitaries in JB-algebras

Abstract

In this note, we study geometric unitaries of JB-algebras (in particular, self-adjoint parts of C * -algebras). By using order theoretical arguments, we show that the geometric unitaries of a JB-algebra are precisely the central algebraic unitaries (or central symmetries).

Banach space characterizations of unitaries: A survey

“…On the other hand, the implication (2) ⇒ (3) follows from Remark 2.2(a), and the one (3) ⇒ (4) is clear. Therefore, the core of the theorem is the implication (4) ⇒ (1), which is in fact the main result in the Leung-Ng-Wong recent paper [48].…”

“…The proof of the implication (4) ⇒ (1) in Theorem 4.8, given in [48], combines order theoretical arguments with the previously known fact that central symmetries of a unital JB-algebra A are precisely the isolated points of the set of all extreme points of B A ([38, Proposition 1.3] together with Theorem 4.1). As pointed out in the introduction of [48], an alternative (and quicker) proof of the implication (4) ⇒ (1) in Theorem 4.8 can be provided by invoking Proposition 4.7 and a result taken from [38].…”

“…Then, we invoke deep results in the Godefroy-Indumathi and Godefroy-Rao papers [28,29], about the norm-weak upper semi-continuity of the pre-duality mapping of a dual Banach space, and derive three previously unnoticed relevant facts (see . We note that Corollary 5.8 (respectively, Corollary 5.9) can be regarded as an abstract version of a recent dual Banach space characterization of unitaries (respectively, of central symmetries) in von Neumann algebras (respectively, in JBW-algebras) proved in [2] (respectively, in [48]). …”

“…Wright and M.A. Youngson [63], and review in Theorem 4.8 the recent Leung-Ng-Wong determination of vertices of closed unit balls of JB-algebras [48].…”

“…However, in most cases, both the codification and the organization of results seem to us to be new. Let us also say that the present survey is an updated and enlarged version of an early short note of the author [58] which, although never published, has been quoted in several papers (see for example [25] and [48]). …”