Irish Math. Soc. Bulletin 59 (2007), 37–47 37 The Centre of Unitary Isotopes of JB*-Algebras

Abstract. We identify the centre of unitary isotopes of a JB∗-algebra. We show that the centres of any two uni-tary isotopes of a JB∗-algebra are isometrically Jordan *-iso-morphic to each other. However, there need be no inclusion between centres of the two unitary isotopes. 1. Basics We begin by recalling (from [3], for instance) the following concepts of homotope and isotope of Jordan algebras. Let J be a Jordan algebra, cf. [3], and x ∈ J. The x-homotope of J, denoted by J[x] , is the Jordan algebra consisting of the same elements and linear algebra structure as J but a different product, denoted by “.x”, defined by a.xb = {axb} for all a, b in J[x]. By {pqr} we will always denote the Jordan triple product of p, q, r defined in the Jordan algebra J as below: {pqr} = (p ◦ q) ◦ r − (p ◦ r) ◦ q + (q ◦ r) ◦ p, where ◦ stands for the original Jordan product in J. An element x of a Jordan algebra J with unit e is said to be invertible if there exists x−1 ∈ J, called the inverse of x, such that x ◦ x−1 = e and x2 ◦ x−1 = x. The set of all invertible elements of J will be denoted by Jinv. In this case, x acts as the unit for the homotope J[x−1] of J. If J is a unital Jordan algebra and x ∈ Jinv then by x-isotope of J, denoted by J [x], we mean the x−1-homotope J[x−1] of J. We denote the multiplication “.x−1 ” of J [x] by “ ◦x”. The following lemma gives the invariance of the set of invertible elements in a unital Jordan algebra on passage to any of its isotopes