Let \(\mathcal {A}\) be a unital \(C^\) -algebra. In this chapter (and in the next one) we study the Poincaré half-space \(\mathcal {H}\) of \(\mathcal {A}\) \( \mathcal {H}:=\{h\in \mathcal {A}: Im(h) \hbox { is positive and invertible}\}. \) In the case of the \(C^\) -algebra \(\mathcal {B}(\mathcal {L})\) of bounded linear operators on the Hilbert space \(\mathcal {L}\) , the space \(\mathcal {H}\) is isomorphic to the tangent bundle of the set of inner products on \(\mathcal {L}\) . Also consider the following situation. Let E be a complex vector bundle over a compact space M provided with a Hermitian structure. Consider the \(C^*\) -algebra \(\Gamma (\textrm{End}(E))\) of continuous sections of the endomorphism bundle of E. The Poincaré space of \(\Gamma (\textrm{End}(E))\) consists of the elements of the form \(X+i a\) , where a is a Hermitian structure on E and X is infinitesimal deformation of a. The space \(\mathcal {H}\) is naturally isomorphic to the following spaces: 1. the Poincaré disk \(\mathcal {D}\) (Poincaré disk of a unital C-algebra) of :  Poincaré disk (of a unital C-algebra) \( \mathcal {D}:=\{z\in \mathcal {A}: \Vert z\Vert <1\}; \) 2. the space \(\mathcal {Q}_\rho \)   of -module projections q acting in \(\mathcal {A}\times \mathcal {A}\) which decompose the form \(\theta =\langle \rho \ \cdot ,\cdot \rangle \) induced by a fixed symmetry \(\rho \) in \(\mathcal {A}\times \mathcal {A}\) , in the following sense: \( \theta \hbox { is positive in } R(q) \hbox { and negative in } N(q). \) These spaces are homogeneous reductive spaces of the groups \(\mathcal {U}_\rho \) and of invertible \(2\times 2\) matrices with entries in \(\mathcal {A}\) which leave an appropriate form invariant. The isomorphisms between these spaces are equivariant with respect to these actions, and the actions are isometric. We regard these spaces as alternative presentations of \(\mathcal {H}\) , each one having its particular features which allow us to underline different aspects of the geometry of \(\mathcal {H}\) . For instance, from \(\mathcal {Q}_\rho \) , \(\mathcal {H}\) inherits the hyperbolic structure (it behaves as a manifold of non positive curvature).

Homogeneous Spaces of Operators

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发表日期：

2025年

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Esteban Andruchow,
Gustavo Corach,
Lázaro Recht

摘要

Let \(\mathcal {A}\) be a unital \(C^*\) -algebra. In this chapter (and in the next one) we study the Poincaré half-space \(\mathcal {H}\) of \(\mathcal {A}\) \( \mathcal {H}:=\{h\in \mathcal {A}: Im(h) \hbox { is positive and invertible}\}. \) In the case of the \(C^*\) -algebra \(\mathcal {B}(\mathcal {L})\) of bounded linear operators on the Hilbert space \(\mathcal {L}\) , the space \(\mathcal {H}\) is isomorphic to the tangent bundle of the set of inner products on \(\mathcal {L}\) . Also consider the following situation. Let E be a complex vector bundle over a compact space M provided with a Hermitian structure. Consider the \(C^*\) -algebra \(\Gamma (\textrm{End}(E))\) of continuous sections of the endomorphism bundle of E. The Poincaré space of \(\Gamma (\textrm{End}(E))\) consists of the elements of the form \(X+i a\) , where a is a Hermitian structure on E and X is infinitesimal deformation of a. The space \(\mathcal {H}\) is naturally isomorphic to the following spaces: 1. the Poincaré disk \(\mathcal {D}\) (Poincaré disk of a unital C-algebra) of : Poincaré disk (of a unital C-algebra) \( \mathcal {D}:=\{z\in \mathcal {A}: \Vert z\Vert <1\}; \) 2. the space \(\mathcal {Q}_\rho \) of -module projections q acting in \(\mathcal {A}\times \mathcal {A}\) which decompose the form \(\theta =\langle \rho \ \cdot ,\cdot \rangle \) induced by a fixed symmetry \(\rho \) in \(\mathcal {A}\times \mathcal {A}\) , in the following sense: \( \theta \hbox { is positive in } R(q) \hbox { and negative in } N(q). \) These spaces are homogeneous reductive spaces of the groups \(\mathcal {U}_\rho \) and of invertible \(2\times 2\) matrices with entries in \(\mathcal {A}\) which leave an appropriate form invariant. The isomorphisms between these spaces are equivariant with respect to these actions, and the actions are isometric. We regard these spaces as alternative presentations of \(\mathcal {H}\) , each one having its particular features which allow us to underline different aspects of the geometry of \(\mathcal {H}\) . For instance, from \(\mathcal {Q}_\rho \) , \(\mathcal {H}\) inherits the hyperbolic structure (it behaves as a manifold of non positive curvature).