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It is easy to check that the hermitian matrices over the reals, complexes, and quaternions form special Jordan algebras that are formally real. One obtains another special formally real Jordan algebra (which we now call a spin factor JSpinn ) on the space R1 ⊕ Rn for n ≥ 2, by making 1 act as unit and deﬁning the product of vectors v, w in Rn to be given by the dot or inner product v • w := v, w 1. In a special Jordan algebra the algebraic structure is derived from an ambient associative structure xy via quasi-multiplication.

The domain D := D(J) of the triple J becomes a hermitian symmet−1 ric space under the Bergmann metric x, y p := tr(LBp,p x,y ). The automorphisms of the domain D ﬁxing 0 are linear and are precisely the algebraic automorphisms of the triple J. At the origin the exponential map exp0 : T0 (D) = J → D is a real analytic diﬀeomorphism given by the odd function exp0 (v) = tanh(v), and the curvature tensor is given by R(u, v)0 = Lv,u −Lu,v . The Shilov boundary of D is the set of maximal tripotents of J, and coincides with the set of all extremal points of the convex set D; it can be described algebraically as the set of z ∈ J with Bz,z = 0.

These are abstract manifolds, but every Hermitian symmetric space of “noncompact type” [having negative holomorphic sectional curvature] is a bounded symmetric domain, a down-to-earth bounded domain in Cn each point of which is an isolated ﬁxed point of an involutive biholomorphic map of the domain. Initially there is no metric on such a domain, but there is a natural way to introduce one (for instance, the Bergmann metric derived from the Bergmann kernel on a corresponding Hilbert space of holomorphic functions).