By De Simone A., Mundici D.
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Additional info for A Cantor-Bernstein Theorem for Complete MV-Algebras
Let J = Hn (C) be the formally real Jordan algebra of dimension n2 over the reals consisting of all Z ∈ Mn (C) with Z ∗ = Z. Then the positive cone Cone(J) consists precisely of the positivedeﬁnite matrices (the hermitian matrices whose Jordan form has only positive real eigenvalues). The structure group is generated by the two involutory transformations Z → −Z and Z → Z = Z tr together with the connected subgroup G = Strg(J)0 of all Z → AZA∗ for A ∈ GLn (C). The connected component K = Aut (J)0 of the automorphism group consists of all Z → U ZU ∗ = U ZU −1 for unitary U ∈ Un (C).
An isomorphism of Hermitian manifolds is a biholomorphic map of analytic manifolds whose diﬀerential is isometric on each tangent space. A hermitian symmetric space is a (connected) hermitian manifold having at each point p a symmetry sp [an involutive global isomorphism of the manifold having p as isolated ﬁxed point]. We henceforth assume that all our Hermitian manifolds are connected. 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.
4 Links with Diﬀerential Geometry Though mathematical physics gave birth to Jordan algebras and superalgebras, and Lie algebras gave birth to Jordan triples and pairs, diﬀerential geometry has had a more pronounced inﬂuence on the algebraic development of Jordan theory than any other mathematical discipline. Investigations of the role played by Jordan systems in diﬀerential geometry have revealed new perspectives on purely algebraic features of the subject. We now indicate what Jordan algebras were doing in such a strange landscape.
A Cantor-Bernstein Theorem for Complete MV-Algebras by De Simone A., Mundici D.