Non-unital Banach Jordan algebras and C*-triple systems

Abstract

The definition of a suitable Jordan analogue of C*-algebras (which we call JB*-algebras in this paper) was recently suggested by Kaplansky (see (26)). The theory of unital JB*-algebras is now comparatively well understood due to the work of Alfsen, Shultz and Størmer (1) from which a Gelfand-Neumark theorem for unital JB*-algebras can be obtained (26). Independently, from work on simply connected symmetric complex Banach manifolds with base point, Kaup introduced the definition of C*-triple systems in (14) and subsequently in (7) it was shown that every unital JB*-algebra is a C*-triple system. In this paper, we wish to extend this result to show that every JB*-algebra is a C*-triple system.

Relatively Weakly Open Sets in Closed Balls of $C^*$-Algebras

“…On the other hand, noncommutative JB * -algebras whose Banach spaces are reflexive are perfectly determined [18,Theorem 3.5]. Keeping in mind such a determination, and the fact already commented on that noncommutative JB * -algebras are JB * -triples in a natural way (see [4,23]), the following result follows from Proposition 2.4. finite ∞ -sum of closed simple ideals which are either finite-dimensional or quadratic. Consequently, the Banach space of A is hilbertizable.…”

“…The basic reference for the theory of JB-algebras is Hanche-Olsen and Stormer [13]. By the main results in the papers of Wright [24] and Youngson [23], JB-algebras are in a bijective categorical correspondence with (commutative) JB * -algebras. The correspondence is obtained by passing from each JB * -algebra A to its self-adjoint part A sa .…”

C∗- and JB∗-algebras generated by a nonself-adjoint idempotent

“…As in the particular case of C * -algebras, already commented, J B * -algebras are J B * -triples under the triple product {·,·,·} determined by {a, b, a} := U a (b * ) (see [3,13]). For later reference, we remark that, if a J B * -algebra J has a unit 1, then for a, b ∈ J we have a · b = {a, 1, b} and a * = {1, a, 1}.…”

“…Let J e denote the J B * -subalgebra of J generated by e. By [13] and [12], there exists a C * -algebra A containing J e as a J B * -subalgebra. Therefore, by Lemma 5.4 and Proposition 2.4, K := σ (e) is a compact subset of ]1, ∞[.…”

Nonassociative unitary Banach algebras

“…It is well known that, if A is a non-commutative JB * -algebra, then A becomes a JB * -triple under its own norm and the triple product {· · ·} defined by {xyz} := U x,z (y * ) (see [8] and [33]). Now recall that, according to [15], real JB * -triples are defined as closed real subtriples of (complex) JB * -triples.…”