A non commutative Kähler structure on the Poincaré disk of a C*-algebra

Abstract

We study the Poincaré disk $\d=\{z\in\a: \|z\|<1\}$ of a C${}^{\ast }$-algebra $\a$ as a homogeneous space under the action of an appropriate Banach-Lie group $\breve{(}\theta )$ of $2\times 2$ matrices with entries in $\a$. We define on $\d$ a homogeneous Kähler structure in a non commutative sense. In particular, this Kähler structure defines on $\d$ a homogeneous symplectic structure under the action of $\breve{(}\theta )$. This action has a moment map that we explicitly compute. In the presence of a trace in $\a$, we show that the moment map has a convex image when restricted to appropriate subgroups of $\breve{(}\theta )$, resembling the classical result of Atiyah-Guillmien-Sternberg.

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References in corpus (2)

• Poncaré half-space of a C*-algebra
• Projective geometry in the Poincaré disk of a $C^{\ast }$-algebra