Poncaré half-space of a C*-algebra

Abstract

Let $A$ be a C$*^$-algebra. Given a representation $A\subset B(L)$ in a Hilbert space $L$, the set $G^{+}\subset A$ of positive invertible elements can be thought as the set of inner products in $L$, related to $A$, which are equivalent to the original inner product. The set $G^{+}$ has a rich geometry, it is a homogeneous space of the invertible group $G$ of $A$, with an invariant Finsler metric. In the present paper we study the tangent bundle $T{G}^{+}$ of $G^{+}$, as a homogenous Finsler space of a natural group of invertible matrices in $M_2(A)$, identifying $T{G}^{+}$ with the {\it Poincaré halfspace} $H$ of $A$, $$H=\{h\in A:Im(h)\ge 0,Im(h)invertible\}.$$ We show that $\h\simeq TG^+$ has properties similar to those of a space of non-positive constant curvature.

33 pages, no figures

Cited by in corpus (2)

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