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Banach manifolds of algebraic elements in the algebra ℒ (H) of bounded linear operatorsof bounded linear operators

José Isidro (2005)

Open Mathematics

Given a complex Hilbert space H, we study the manifold 𝒜 of algebraic elements in Z = ℒ H . We represent 𝒜 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...

Quadro-quadric Cremona transformations in low dimensions via the  J C -correspondence

Luc Pirio, Francesco Russo (2014)

Annales de l’institut Fourier

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 “ J C -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.

The density property for JB*-triples

Seán Dineen, Michael Mackey, Pauline Mellon (1999)

Studia Mathematica

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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