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**Extra resources for Encyclopaedie der mathematischen Wissenschaften und Anwendungen. Algebra und Zahlentheorie**

**Example text**

With some eﬀort it can be shown that any JB-algebra can be imbedded in a unital one J := R ⊕ J using the spectral norm on J , and we will henceforth assume that all our JB-algebras are unital. Notice that the C ∗ and positivity conditions have as immediate consequence the usual formal reality condition x2 + y 2 = 0 ⇒ x = y = 0. Conversely, all ﬁnite-dimensional formally real Jordan algebras (in particular, the real Albert algebra H3 (K)) carry a unique norm making them JB-algebras, so in ﬁnite dimensions JB is the same as formally real.

This orphan turned out to have many surprising and important connections with diverse branches of mathematics. Actually, the child should never have been conceived in the ﬁrst place: it does not obey all the algebraic properties of the Copenhagen model, and so was in fact unsuitable as a home for quantum mechanics, not superﬁcially due to its ﬁnite-dimensionality, but genetically because of its unsuitable algebraic structure. M. Glennie discovered two identities satisﬁed by hermitian matrices (indeed, by all special Jordan algebras) but not satisﬁed by the Albert algebra.

D(J) can also be characterized as the unit ball of JC under a certain spectral norm. Positive Hermitian Triple Systems Ottmar Loos showed that there is a natural 1-to-1 correspondence between bounded homogeneous circled domains D in Cn and the ﬁnite-dimensional positive hermitian Jordan triples. A hermitian Jordan triple is a complex vector space J with triple product {x, y, z} = Lx,y (z) which is symmetric and C-linear in the outer variables x, z and conjugate-linear in the middle variable y, satisfying the 5-linear axiom {x, y, {z, w, v}} = {{x, y, z}, w, v} − {z, {y, x, w}, v} + {z, w, {x, y, v}} for a Jordan triple system.