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Given a complex Hilbert space H, we study the manifold
of algebraic elements in
. We represent
as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine...
It has been previously established that a Cremona transformation of bidegree (2,2) is linearly equivalent to the projectivization of the inverse map of a rank 3 Jordan algebra. We call this result the “-correspondence”. In this article, we apply it to the study of quadro-quadric Cremona transformations in low-dimensional projective spaces. In particular we describe new very simple families of such birational maps and obtain complete and explicit classifications in dimension 4 and 5.
We obtain conditions on a JB*-algebra X so that the canonical embedding of X into its associated quasi-invertible manifold has dense range. We prove that if a JB* has this density property then the quasi-invertible manifold is homogeneous for biholomorphic mappings. Explicit formulae for the biholomorphic mappings are also given.
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